ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
EDITED BY G.-C. ROTA
Editorial Board
R.S. Doran, J. Goldman, T.-Y. Lam, E. Lutwak
Volume 44
Convex bodies: the Brunn-Minkowski theory

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
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41 Shoichiro Sakai Operator Algebras in Dynamical Systems: The Theory of
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Convex Bodies: The Brunn-Minkowski
Theory
ROLF SCHNEIDER
University of Freiburg
Cambridge
UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Victoria 3166, Australia
© Cambridge University Press 1993
First published 1993
Printed in Great Britain at the University Press, Cambridge
A catalogue record for this book is available from the British Library
Library of Congress cataloguing in publication data
Schneider, Rolf, 1940-
Convex bodies: the Brunn-Minkowski theory / Rolf Schneider,
p. cm. - (Encyclopedia of mathematics and its applications; v. 44)
Includes bibliographical references and index.
ISBN 0-521-35220-7
1. Convex bodies. I. Title. II. Series.
QA649.S353 1993
516.3'74-dc20 92-11481 CIP
ISBN 0 521 35220 7 hardback
KT

CONTENTS
Preface vii
Conventions and notation x
1 Basic convexity 1
1.1 Convex sets and combinations 1
1.2 The metric projection 9
1.3 Support and separation 11
1.4 Extremal representations 17
1.5 Convex functions 21
1.6 Duality 33
1.7 The support function 37
1.8 The Hausdorff metric 47
2 Boundary structure 62
2.1 Facial structure 62
2.2 Singularities 70
2.3 Segments in the boundary 80
2.4 Polytopes 94
2.5 Higher regularity and curvature 103
2.6 Generic boundary structure 119
3 Minkowski addition 126
3.1 Minkowski addition and subtraction 126
3.2 Summands and decomf)osition 142
33 Approximation and addition 157
3.4 Additive maps 166
3.5 Zonoids and other classes of convex bodies 182
4 Curvature measures and quermassintegrals 197
4.1 Local parallel sets 197
4.2 Curvature measures and area measures 200
4.3 The area measure of order one 214
4.4 Additive extension 219

'i Contents
4.5 Integral-geometric formulae 226
4.6 Local behaviour of curvature measures 261
5 Mixed volumes and related concepts 270
5.1 Mixed volumes and mixed area measures 270
5.2 Extensions of mixed volumes 284
5.3 S()ecial formulae for mixed volumes and quermassintegrals 289
5.4 Moment vectors and curvature centroids 303
6 Inequalities for mixed volumes 309
6.1 The Brunn-Minkowski theorem 309
6.2 The Minkowski and isoperimetric inequalities 317
6.3 The Aleksandrov-Fenchel inequality 327
6.4 Consequences and improvements 333
6.5 Generalized parallel bodies 343
6.6 Equality cases and stability 351
6.7 Linear inequalities 376
6.8 Analogous notions and inequalities 382
7 Selected applications 389
7.1 Minkowski's existence theorem 389
7.2 Uniqueness theorems for area measures 397
7.3 The difference-body inequality 409
7.4 Affinely associated bodies 413
Appendix Spherical harmonics 428
References 433
List of symbols 474
Author index 478
Subject index 484

PREFACE
The Brunn-Minkowski theory is the classical core of the geometry of
convex bodies. It originated with the thesis of Hermann Brunn in 1887
and is in its essential parts the creation of Hermann Minkowski, around
the turn of the century. The well-known survey of Bonnesen and
Fenchel in 1934 collected what was already an impressive body of
results, though important developments were still to come, through the
work of A.D. Aleksandrov and others in the thirties. In recent decades,
the theory of convex bodies has expanded considerably; new topics have
been developed and originally neglected branches of the subject have
gained in interest. For instance, the combinatorial aspects, the theory of
convex polytopes and the local theory of Banach spaces attract
particular attention now. Nevertheless, the Brunn-Minkowski theory has
remained of constant interest owing to its various new applications, its
connections with other fields, and the challenge of some resistant open
problems.
Aiming at a brief characterization of Brunn-Minkowski theory, one
might say that it is the result of merging two elementary notions for
point sets in Euclidean space: vector addition and volume. The vector
addition of convex bodies, usually called Minkowski addition, has many
facets of independent geometric interest. Combined with volume, it
leads to the fundamental Brunn-Minkowski inequality and the notion of
mixed volumes. The latter satisfy a series of inequaUties which, due to
their flexibility, solve many extremal problems and yield several
uniqueness results. Looking at mixed volumes from a local point of view, one
is led to mixed area measures. Quermassintegrals, or Minkowski
functional, and their local versions, surface area measures and curvature
measures, are a special case of mixed volumes and mixed area

viii Preface
measures. They are related to the differential geometry of convex
hypersurfaces and to integral geometry.
Chapter 1 of the present book treats the basic properties of conve>
bodies and thus lays the foundations for subsequent developments. Thij
chapter.does not claim much originality; in large parts, it follows the
procedures in standard books such as McMuUen & Shephard [26]
Roberts & Varberg [28], and Rockafellar [29]. Together with Sections
2.1, 2.2, 2.4, and 2.5, it serves as a general introduction to the metric
geometry of convex bodies. Chapter 2 is devoted to the boundary
structure of convex bodies. Most of its material is needed later, except
for Section 2.6, on generic boundary structure, which just rounds off the
picture. Minkowski addition is the subject of Chapter 3. Severa
different aspects are considered here such as decomposability, approxi
mation problems with special regard to addition, additive maps anc
sums of segments. Quermassintegrals, which constitute a fundamental
class of functional on convex bodies, are studied in Chapter 4, where
they are viewed as specializations of curvature measures, their loca'
versions. For these, some integral-geometric formulae are established in
Section 4.5. Here I try to follow the tradition set by Blaschke and
Hadwiger of incorporating parts of integral geometry into the theory ot
convex bodies. Some of this, however, is also a necessary prerequisite
for Section 4.6. The remaining part of the book is devoted to mixed
volumes and their applications. Chapter 5 develops the basic properties
of mixed volumes and mixed area measures and treats special formulae,
extensions and analogues. Chapter 6, the heart of the book, is devoted
to the inequalities satisfied by mixed volumes, with special emphasis or
improvements, the equality cases (as far as they are known) and stability
questions. Chapter 7 presents a small selection of applications. The
classical theorems of Minkowski and the Aleksandrov-Fenchel-Jessen
theorem are treated here, the latter in refined versions. Section 7.4
serves as an overview of affine extremal problems for convex bodies. Ii
this promising field, Brunn-Minkowski theory is of some use, but it
appears that for the solution of some long-standing open problems new
methods still have to be invented.
Concerning the choice of topics treated in this book, I wish to point
out that it is guided by Minkowski's original work also in the following
sense. Some subjects that Minkowski touched only briefly have later
expanded considerably, and I pay special attention to these. Example?
are projection bodies (zonoids), tangential bodies, the use of spherical
harmonics in convexity and strengthenings of Minkowskian inequalities
in the form of stability estimates.
The necessary prerequisites for reading this book are modest: the

Preface
usual geometry of Euclidean space, elementary analysis, and basic
measure and integration theory for Chapter 4. Occasionally, use is made
of spherical harmonics; relevant information is collected in the
Appendix. My intended attitude towards the presentation of proofs cannot be
summarized better than by quoting from the preface to the book on
Hausdorff measures by C.A. Rogers: 'As the book is largely based on
lectures, and as I like my students to follow my lectures, proofs are
given in great detail; this may bore the mature mathematician, but it
will I believe, be a great help to anyone trying to learn the subject ab
initio.' On the other hand, some important results are stated as
theorems but not proved, since this would lead us too far from the main
theme, and no proofs are given in the survey sections 5.4, 6.8, and 7.4.
The notes at the end of nearly all sections are an essential part of the
book. As a rule, this is where I have given references to original
literature, considered questions of priority, made various comments and,
in particular, given hints about applications, generalizations and
ramifications. As an important purpose of the notes is to demonstrate the
connections of convex geometry with other fields, some notes do take us
further from the main theme of the book, mentioning, for example,
infinite-dimensional results or non-convex sets or giving more detailed
information on applications in, for instance, stochastic geometry.
The list of references does not have much overlap with the older
bibliographies in the books by Bonnesen & Fenchel and by Hadwiger.
Hence, a reader wishing to have a more complete picture should consult
these bibliographies also, as well as those in the survey articles listed in
part B of the References.
My thanks go to Sabine Linsenbold for her careful typing of the
manuscript and to Daniel Hug who read the typescript and made many
valuable comments and suggestions.

CONVENTIONS AND NOTATION
Here we shall fix our notation and collect some basic definitions. We
shall work in n-dimensional real Euclidean vector space, E", with origin
o, scalar product (•, •) and induced norm| • |. We shall not distinguish
formally between the vector space E" and its corresponding affine space,
although our alternating use of the words 'vector' and 'point' is
deliberate and should support the reader's intuition. As a rule, elements
of E" are denoted by lower-case letters, subsets by capitals and real
numbers by small Greek letters. However, in later chapters the reader
will notice an increasing number of exceptions to this rule.
The vector x e E" is a linear combination of the vectors JCj, . . . ,
XicsE" a X = AjJCi -I- ... -I- k/iXii with suitable Aj, . . . , A^^ e R. If such A,
exist with Aj -I- . . . -I- A^^ = 1, then x is an affine combination of Jc j, . . . ,
Xi(. For A CE", linA (aff y4) denotes the linear hull (affine hull) of A;
this is the set of all linear (affine) combinations of elements of A and at
the same time the smallest linear subspace (affine subspace) of E"
containing A. Points xi, . . . , jc^^ e E" are affinely independent if none of
them is an affine combination of the others, i.e., if
^kjXi = o with A, 6 R and ^X^ = Q
i=\ i=\
implies that Aj = . . . = A^^ = 0. This is equivalent to the linear
independence of the vectors xi — x^, . . ., x^ — x^. We may also define a map
r: E" ^ E" X R by t{x) ■= (x, 1); then x,, . . . , x^e^" are affinely
independent if and only if r(jci), .. . , t^x^) are linearly independent.
For jc, y 6 E" we write
[x, y] ■■= {A - k)x + ky\0 ^ A ^ 1}
for the closed segment and
[x, y) := {A - X)x + Xy\Q ^ A < 1}

Conventions and notation xi
for a half-open segment, both with endpoints x, y. For A, fi C E" and
A 6 R we define
A + B ■= [a ■\- b\a € A, b € B),
XA := {Xa\a e A),
and we write -A for {-V)A, A- B for y4 + (-fi) and y4 + jc for
/I + {jc}, where jc e E". The set /i + fi is written A% B and called the
direct sum of /4 and B ii A and fi are contained in complementary
affine subspaces of E".
By cM, inty4, bdy4 we denote, respectively, the closure, interior and
boundary of a subset ^4 of a topological space. For y4 CE", the sets
relint A, relbd A are the relative interior and relative boundary, that is,
the interior and boundary of A relative to its affine hull.
The scalar product in E" will often be used to describe hyperplanes
and halfspaces. A hyperplane of E" can be written in the form
//„., = {X 6 l"\{x, u) = a}
with M e E"\{o} and orelR; here Hu.a=tiv.p if and only if
{v, ji) = (Am, ka) with A =j= 0. We say that u is a normal vector of Hu.a-
The hyperplane Hu,a bounds the two closed halfspaces
//-<, := {x 6 EKx, u) ^ a),
H„%:= {xe^"\{x,u)^a).
Occasionally we also use (•, • ) to denote the scalar product on E" x IR
given by
<(^- ^), (y, ri)) = {x. y) + §//.
An affine subspace of E" is often called a flat, and the intersection of
a flat with a closed halfspace meeting the flat but not entirely containing
it will be called a half-flat. A one-dimensional flat is a line and a
one-dimensional half-flat a ray.
The following metric notions will be used. For jc, yeE" and
0 =1= y4 C E", |jc - yl is the distance between x and y and
d{A, x) ■= inf {|jc - a\\a e a]
is the distance of x from A. For a bounded set 0^ A C^",
diamy4 := sup {\x - y\\x, y e a)
is the diameter of A. We write
Biz,p) := {x 6 l"\\x - z\^ p)
and
B(,iz,p) ■■= {x 6 E'-llac - z\< p)
respectively for the closed and open balls with centre z e E" and radius

xii Conventions and notation
p> 0. B" ■■= B(o. 1) is the unit ball and
5"-' := {x 6 E"||a:| = l}
the unit sphere of E".
By ^j'' we denote the A:-dimensional Hausdorff (outer) measure on
E", where 0^ k^ n. If A is a Borel subset of a /k-dimensional flat E''
or a ^-dimensional sphere S'' in E", then ^j''(A) coincides respectively
with the A:-dimensional Lebesgue measure of A computed in E'' or with
the A:-dimensional spherical Lebesgue measure of A computed in S''.
Hence, all integrations with respect to these Lebesgue measures can be
expressed by means of the Hausdorff measure ^''. In integrals with
respect to ^'" we often abbreviate d^"(jc) by dx. The n-dimensional
measure of the unit ball in E" is denoted by k„, and its surface area by
(o„, thus
K„ = Tf^^B") = — -, (v„ = X''-HS"-') = nK„ = ^—.
'i-D 'It)
Linear maps, affine maps and isometries between Euclidean spaces
are defined as usual. In particular, a map (p:E" —»E" is a translation if
q>(x) = X + t for X e ^" with some fixed vector f e E", the translation
vector. The set y4 -I- f is called the translate of A by t. The map q> is a
homothety if cp(x) = ax + t ior x eE" with some A > 0 and some f e E".
The set AA + t with A > 0 is called a homothet of A. Sets A, B are
called positively homothetic if A = KB + t with t e E" and A > 0, and
homothetic if either they are positively homothetic or one of them is a
singleton (a one-pointed set). A rigid motion of E" is an isometry of E"
onto itself, and it is a rotation if it is an isometry fixing the origin. Each
rigid motion is the composition of a rotation and a translation. A rigid
motion is called proper if it preserves the orientation of E", otherwise it
is called improper. A rotation of E" is a linear map; it preserves the
scalar product and can be represented, with respect to an orthonormal
basis, by an orthogonal matrix; this matrix has determinant 1 if and only
if the rotation is proper. The composition of a rigid motion and a
dilatation, by which we mean a map x >-^ kx with A>0, is called a
similarity.
By SO{n) we denote the group of proper rotations of E". With the
topology induced by the usual matrix norm it is a compact topological
group. The group of proper rigid motions of E" is denoted by G„ and
topologized as usual. Also, the Grassmannian G(n, k) of ^-dimensional
linear subspaces of E" and the set A{n, k) of ^-dimensional affine
subspaces of E" are endowed with their standard topologies.
The Haar measures on SO{n), G„, A(n, k) are denoted respectively

Conventions and notation xiii
by V, (J., ni^. We normalize v by v{SO(n)) = 1. The normalizations of the
measures n and Hic will be fixed in Section 4.5 when they are needed.
For an affine subspace £ of E" we denote by proJ£ the orthogonal
projection from E" onto E. We often write proJ£y4 =: A\E for A C E"
(since y4 is a set, no confusion with the restriction of a function, for
example f\E, can arise).
Some final remarks are in order. Since any A:-dimensional affine
subspace E of E" is the image of E'' under some isometry, it is clear
(and common practice without mention) that all notions and results that
have been established for E'' and are invariant under isometries can be
applied in E; similarly for affine-invariant notions and results.
The following notational conventions will be useful in several places.
If / is a homogeneous function on E", then / denotes its restriction to
the unit sphere 5""'. Very often, mappings of the type /: ^X x E" -» A/
will occur where X is some class of subsets of E". In this case we
usually abbreviate, for fixed Ke X, the function f{K, ■):E"-»A/ by
Ik-
Finally we wish to point out that in definitions the word 'if is always
understood as 'if and only if.

Basic convexity
1.1. Convex sets and combinations
A set y4 C E is convex if together with any two points x, y it contains
the segment [x, y], thus if
A - X)x + Xy e A lor x, y e A and 0 ^ A g 1.
Examples of convex sets are obvious; but observe also that B(,(z,p) U A
is convex if A is an arbitrary subset of the boundary of the open ball
Bo(z,p)- As immediate consequences of the definition we note that
intersections of convex sets are convex, affine images and pre-images of
convex sets are convex and if A, B are convex, then A + B and XA
(A 6 R) are convex.
Remarkl.1.1. For /i C E" and A, ^>0 one trivially has XA +
lxA'D{X + ix)A. Equality (for all A, ^ > 0) holds precisely if A is
convex. In fact, if A is convex and x e XA + ixA, then x = Xa + ixb with
a, b € A and hence
x = {X + A-r^ a + -^ b]e{X + ti)A;
\X + IX X + fj. I
thus XA + ixA = {X + fi)A. The other direction of the assertion is trivial.
A set y4 C E" is called a convex cone if A is convex and nonempty
and \i X € A, A^O implies Xx e A. Thus a nonempty set yl C E" is a
convex cone if and only if A is closed under addition and under
multiplication by non-negative real numbers.
By restricting affine and linear combinations to non-negative
coefficients, one obtains the following two fundamental notions. The point
jc 6 E" is a convex combination of the points JCj, . . . , jc^ e E" if there
are numbers Ai, . . . , A^^ e R such that

2 Basic convexity
k k
^ = 2A,vt„A,^0 (/= 1,...,A:), 2A, = 1.
i=I 1 = 1
Similarly, the vector jc e E" is a positive combination of the vectors x^,
. . . , xt 6 E" if
k
X = ^XiXi with A, SO (/ = 1, . . . , /t).
i=i
For y4 C E" the set of all convex combinations (positive combinations)
of any finitely many elements of A is called the convex hull (positive
hull) of A and is denoted by convy4 (poSy4).
Theorem 1.1.2. If A Ct" is convex, then convy4 = A. For an arbitrary
set y4CE", convy4 is the intersection of all convex subsets of E"
containing A. If A, B C E", then conv(y4 + fi) = convy4 + conv B.
Proof Let A be convex. Trivially, A Cconvy4. By induction we show
that A contains all convex combinations of any A: points of A. For k = 2
this holds by the definition of convexity. Suppose that it holds for A: - 1
and that x = A,jc, + . . . + k^Xk with jc,, . . . , jc^ e y4, Aj + . . . + A^ = 1
and Ai, . . . , A^ > 0, without loss of generality. Then
*-' A,.
since
= A - Ai) 2 1—V ^' ■•■ '^^^^ ^ ^
,= 1 1 - A^
A,. . t.' A.
= 1
and hence
1 -
~ ^k 1= 1 1 ~ ^k
A.-1 ^.
by hypothesis. This proves A =convy4. For arbitrary ACt" let D(A)
be the intersection of all convex sets /C C E" containing A. Since
A Cconvy4 and convy4 is evidently convex, we have D(y4) C convy4.
Each convex K with A C K satisfies conv A C conv K = K, hence
convy4 C D(A), which proves the equality.
Let /i, fi C E". Let jc 6 com (A + B), thus
k k
X = 2^/@, + bi) with a, e /i, fe, e fi. A, ^ 0, 2^, = 1
1=1 i=l
and hence jc = XA,a, + S^,''/ e convy4 + conv B. Let jc e convy4 +
conv fi,thus

1.1 Convex sets and combinations
with a, 6 A, bj e B, A,, ^y ^0, 2^ = Sa'/ = 1- We may write
and deduce that x e conv(y4 + B). ■
An immediate consequence is that conv(convy4) = convy4.
Theorem 1.1.3. If ACE" is a convex cone, then poSy4 = y4. For a
nonempty set A C E", poSy4 is the intersection of all convex cones in E"
containing A. If A, B C E", then pos(y4 + B) = posA + pos B.
Proof. As above. ■
The following result on the generation of convex hulls is fundamental.
Theorem 1.1.4 (Caratheodory's theorem). // ACE" and xeconvA,
then X is a convex combination of affinely independent points of A. In
particular, x is a convex combination of n + \ or fewer points of A.
Froof. The point x e conv A has a representation
k k
X = "^^iXj with Xj e A, A, > 0, "^^j = I
1=1 1=1
with some k eN, and we may assume that A: is minimal. Suppose that
Xj, . . . , jc^ are affinely dependent. Then there are numbers or,, . . . ,
a^ 6 R, not all zero, with
k k
^o:,Xj = o and 2<^i = 0.
1=1 1=1
We can choose m such that A^/or^ is positive and, with this restriction,
as small as possible (observe that all A, are positive and at least one q-, is
positive). In the affine representation
k
X
all coefficients are non-negative (trivially, if o-, ^ 0, otherwise by the
choice of m) and at least one of them is zero. This contradicts the
minimality of k. Thus JCj, . . . , x^ are affinely independent, which
implies k^ n + I. ■
The convex hull of finitely many points is called a polytope. A
k-simplex is the convex hull of A: + 1 affinely independent points, and
these points are the vertices of the simplex. Thus Caratheodory's

4 Basic convexity
theorem states that conv A is the union of all simplices with vertices in
A.
Another equally simple and important result on convex hulls is the
following.
Theorem 1.1.5 (Radon's theorem). Each set of affinely dependent points
(in particular, each set of at least n + 2 points) in E" can be expressed as
the union of two disjoint sets whose convex hulls have a common point.
Proof. U Xi, .. ., x/t are affinely dependent, there are numbers o-i,
. . . , a/t eU, not all zero, with
k k
^a,Xi = o and ^a, = 0.
1=1 1=1
We may assume, after renumbering, that or/ > 0 precisely for / = 1, . . . ,
/; then 1 S / < A: (at least one o-, is =1= 0, say > 0, but not all q-, are > 0).
With
or := or, + . . . + OTy = -(ary+, + . . . + a^) > 0
we obtain
X ■= 2.— X: = y I \X:
and thus x e conv{jci, . . . , Xj) A conv{jcy+i, . . . , X/^}. The assertion
follows. ■
From Radon's theorem one easily deduces Helly's theorem, a
fundamental and typical result of the combinatorial geometry of convex sets.
Theorem 1.1.6 (Helly's theorem). Let A^, . . . , A^C^" be convex sets.
If any n + I of these sets have a common point, then all the sets have a
common point.
Proof. Suppose that A: > n + 1 (for A: < n + 1 there is nothing to prove,
and for k = n + 1 the assertion is trivial) and that the assertion is
proved for A: - 1 convex sets. Then for i e A, . . . , A:} there exists a
point
Xi e Ai n ... n Aj n ... n Ak
where Aj indicates that Aj has been deleted. The A: S n + 2 points jcj,
. . . , jCjt are affinely dependent; hence from Radon's theorem we can
infer that, after renumbering, there is a point
X 6 conv {xi, . . . , Xj} n conv (jcy+i, . . . , x^}

1.1 Convex sets and combinations 5
for some ;' e A, . . . , A: - 1}. Because Xi, . . ., Xj e Aj+i, . . . , Ai^ v/e
have
X 6 conv{jCi, . . . , Xj} C Aj+i D .. . D Aj^,
similarly x e conv (jcy+i, .. . , jc^} C AiCt . . . A Aj. ■
Here is a little example (another one is Theorem 1.3.11) to
demonstrate how Helly's theorem can be applied to obtain elegant results of a
similar nature:
Theorem 1.1.7. Let M. be a finite family of convex sets in E" and let
AT C E" be convex. If any n + 1 elements of Jfi are intersected by some
translate of K, then all elements of JfL are intersected by a translate of K.
Proof. Let J/l= {Ai, . . . , ^4^^}. To any n + 1 elements of A, . .. , A:},
say 1, ..., n + l, there are f e E" and jc, e y4,-n (AT + r), hence
-t e K - Aj, for i = 1, . . . , n + l. Thus any n + l elements of the
family {K - Ai, ..., K - A^} have nonempty intersection. By Helly's
theorem there is a vector -feE" with -tsK-A; and hence
/i,n(/i: +f)=|=0for/e A, ...,/t}. ■
Next we look at the interplay between convexity and topological
properties. We start with a simple observation.
Lemma 1.1.8. Let y4 C E" be convex. If xs'miA and y€c\A, then
[x, y) C inty4.
Proof. Lc\ z = {\- K)y + kx with 0 < A < 1. We have B{x, p) C A for
some p> 0; put B(o, p) =■ U. First we assume y e A. Let w e W + z,
hence w = ku + z with ueU. Then x + ueA, hence w =
{I-k)y + k(x + u)e A. This shows that W + zCA and thus
z eintA.
Now assume merely that yeclA. Put V := [A/(l - X)]U + y. There
is some a e A A V. We have a = [A/(l - X)]u + y with u e U and hence
2 = A - X)a + k(x - u) e A. This proves that [x, y)C A, which
together with the first part yields [x, y)CmtA. ■
Theorem 1.1.9. If ACE" is convex, then intA and clA are convex. If
A C^" is open, then conv A is open.
Proof. The convexity of intA follows from Lemma 1.1.8. The convexity
of cl A for convex A and the openness of conv A for open A are easy
exercises. ■

6 Basic convexity
The union of a line and a point not on it shows that the convex hull
of a closed set need not be closed. This is different for compact sets, as
a first application of Carath^odory's theorem shows.
Theorem 1.1.10. If A C E", then convcl/i Cclconv/1. If A is hounded,
then convcM = clconVy4. In particular, the convex hull of a compact set
is compact.
Proof. convcM C clconvy4 is easy to see. Let A be bounded; then
n + l
{(Ai, . . . , A„+,, xi, ... , x„^.i)\Xi S 0, jc, 6 clA, ^Xi = 1}
1=1
is a compact subset of R"*' x(E")"*', hence its image under the
continuous map
n+l
(Ai, . . . , A„+i, Xi, . . . , JC„+i) !-» 2j^iXi 6 E"
1 = 1
is compact. By Carath^odory's theorem, this image is equal to convcM.
Thus cl conv ^4 C cl conv cl y4 = conv cl y4. ■
The set clconvy4, which by Theorem 1.1.9 is convex, is called for
short the closed convex hull of A. This is also the intersection of all
closed convex subsets of E" containing A.
To obtain information on the relative interiors of convex hulls, we
first consider simplices.
Lemmal.1.11. Let X\, . . . , X/^e^" be affmely independent; let S '■=
conv {xi, . . . , jc^} and x e aff 5. Then x e relint5 if and only if in the
unique affine representation
k k
X = '^^iX, with ^^i = I
;=i 1=1
all coefficients A, are positive.
Proof. Clearly we may assume that k = n + 1. The condition is
necessary since otherwise, because the representation is unique, an arbitrary
neighbourhood of x would contain points not belonging to S. To prove
sufficiency, let x be represented as above with all A, > 0. Since xi, . .. ,
jc„+i are affinely independent, the vectors r(jci), . . . , r(jc„+i) (see
'Conventions and notation') form a linear basis of E" x R, and for
y 6 E" the coefficients ^i, . . . , ^„+i in the affine representation
n+l n+l
>' = S l^i^i with S A'l = 1
1=1 ;=i

1.1 Convex sets and combinations 7
(the 'barycentric coordinates' of y) are just the coordinates of r(_>') with
respect to this basis. Since coordinate functions in E"*' are continuous,
the coefficients ^i, . . . , ^„+i depend continuously on y. Therefore,
d>0 can be chosen such that ^, > 0 (/ = 1, . . . , n + 1) and thus y e S
for all y with ly - jc| < d. This proves x e intS. ■
Theorem 1.1.12. If A CE" is convex and nonempty, then relinty4 =1= 0.
Proof Let dimaffy4 = k, then there are k + I affinely independent
points in A. Their convex hull S satisfies relint5 =|= 0 by Lemma LLll,
furthermore S C A and aff 5 = aff y4. ■
In view of this theorem, it makes sense to define the dimension,
dim y4, of a convex set A as the dimension of its affine hull. The points
of relint A are also called internal points of A.
The description of relint conv/i for an affinely independent set A
given by Lemma LLll can be extended to arbitrary finite sets.
Theorem 1.1.13. Let jc,, . . . , Xj, e E"; let P := conv{jc,, . . . , jc^} and
jc 6 E". Then x e relint P if and only if x can be represented in the form
k k
X = '^kiXi with A, > 0 (/ = 1, . . . , k), '^ki = 1.
1=1 1=1
Proof. We may clearly assume that dim P = n. Suppose that x e int P.
Put
* 1
y := 2 -r xr,
1 = 1 K
then y e P. Since x e int P, we can choose z e P for which x e[y, z).
There are representations
k k
z = '^HiX; with Hi ^ 0, '^iXi = 1,
(=1 1=1
X = {\ - K)y + Kz with 0 ^ A < 1,
which gives
k J k
X = ^X.;Xi with A; = A - A) — + kfii > 0, X'^i = 1-
1=1 f^ 1=1
Vice versa, suppose that
k k
X = "^XiXi with A, > 0, ^^i = I.
1=1 1=1
We may assume that Xy, ..., jc„+i are affinely independent. Put
A := Ai + . . . + A„+i and

Basic convexity
y ■■= 2 T ^'■
Lemma LLll gives y e intconv{jc,, . . . , jc„+i} C int P. If fc = n + 1,
then jc = >' 6 int P. Otherwise, put
I A,
JC,.
, = n + l 1 - A
Then z e P and x e [y, z) C int P by Lemma LL8. ■
Theorem 1.1.14. Let A C E" be convex. Then
(a) relinty4 = reHntcM,
(b) cl y4 = cl relint A,
(c) relbd A = relbd cl ^4 = relbd relint A.
Proof. We may clearly assume that dimy4 = n. Part (a): trivially,
inty4 C intcM. Let JceintcM. Choose _>'6inty4. There is zed A
with xe[y,z), and Lemma LL8 shows that JC6inty4. Part (b):
trivially, cM Dclinty4. Let xeclA. Choose y6inty4. By Lemma
1.L8 we have [y, x) CintA, hence JC6clinty4. Part (c): bdcly4
= cl(cM)\int(cM) = cM\int/i =bd/i, using (a). Then bdint/i =
cl(int/i)\int(int/i) = clA\intA = bdA, using (b). ■
We end this section with a definition of the central notion of this
book. A nonempty, compact, convex subset of E" is called a convex
body (thus in our terminology, a convex body need not have interior
points). By X" we denote the set of all convex bodies in E" and by Xq
the subset of convex bodies with interior points. For 0^ A CE" we
write X(A) for the set of convex bodies contained in A and Xo(A) =
XiA) n Xo. Further, "J.)" denotes the set of nonempty polytopes in E",
andV")=7y' r\Xa.
Notes for Section 1.1
1. The early history of the theorems of Carath6odory, Radon and Helly, and
many generalizations, ramifications and analogues of these theorems forming
an essential part of combinatorial convexity can be studied in the survey
article of Danzer, Griinbaum & Klee [32], which is still strongly
recommended. Various results related to Carath6odory's theorem can be found
in Reay A965). An important extension of Radon's theorem is Tverberg's
theorem (Tverberg 1966, 1981): Each set of at least (m - l)(n + 1) + 1 points
in E" (where m S 2) can be partitioned into m subsets whose convex hulls
have a common point. A survey of later developments is given by Eckhoff
A979). There one also finds hints about more recent developments of the
theorems of Carath^odory, Radon and Hclly in the abstract setting of
so-called convexity spaces.

1.2 The metric projection 9
2. It is clear how a version of Carath6odory's theorem for convex cones is to be
formulated and how it can be proved. A common generalization, a version of
Carath6odory's theorem for 'convex hulls of points and directions', is given
by Rockafellar [29], Theorem 17.1.
1.2. The metric projection
In this section, A CE" is a fixed nonempty closed convex set. To each
X eE" there exists a unique point p(A, x) e A satisfying
\x - p(A, x)\ ^ \x - y\ for all y e A.
In fact, for suitable p> 0 the set B(x, p) D y4 is compact and nonempty,
hence the continuous function yi-^ljc -yl attains a minimum on this
set, say at y^; then \x - y^l ^\x - y\ for all y e A. If, also, yi e A
satisfies \x - yi\ ^\x - y\ for all y e A, then z ='■ (yo + >'i)/2 e A and
\x - z\<\x - yo\, except if yo = yi- Thus yo ='■ p(A, x) is unique.
In this way a map p(A, •):E" ^> A is defined; it is called the metric
projection or nearest-point map of A. It will play an essential role in
Chapter 4 when the volume of local parallel sets is investigated. It also
provides a simple approach to the basic support and separation
properties of convex sets (see the next section), as used by Botts A942) and
McMullen & Shephard [26].
We have \x - p(A, x)\ = d(A, x), and for x e E"\A we denote by
"<^. '> - ^^(^
the unit vector pointing from the nearest point p(A, x) to x and by
R(A, x) ■■= {p(A, X) + ku(A, x)\k ^ 0}
the ray through x with endpoint p(A, x).
Lemma 1.2.1. Let x e E"\A and y e R(A,x); then p(A, x) = p(A, y).
Proof. Suppose that p(A, y) =1= p(A, x). If y e [x, p(A, x)), then
\x - p(A, y)\ ^\x - y\ -i- \y - p(A, y)\
<\x - y\ + \y - p(A, x)\
= \x - p(A, x)\,
which is a contradiction. If x e[y, p(A, x)), let q e[p(A, x), p(A, y)]
be the point such that the segment [x, q] is parallel to [y, p(A, y)]. Then
\x - q\ \y - PJA, y)\
\x - p(A, x)\ \y - piA, x)\ ^ ^'
again a contradiction.

10 Basic convexity
Theorem 1.2.2. The metric projection is contracting, that is,
\piA, X) - p(A, y)\ ^\x - y\ for x, y e I".
Proof. We may assume that v ■= p(A, x) - p(A, y) ^ o. We assert that
(x - p(A,x),v}^0. (*)
If this is false, then x $ A and the ray R(A,x) meets the hyperplanes
through p(A,y) that is orthogonal to y in a point z, and we deduce
from Lemma 1.2.1 that
\z - piA. y)\ <\z - p{A, x)\ =\z - piA, z)\,
which is a contradiction. Hence (*) holds. Analogously we get
(y - p(A, y),v) ^0. Thus the segment [x, y] meets the two
hyperplanes that are orthogonal to v and that go through p(A, x) and
p(A, y) respectively. Now the assertion is obvious. ■
Lemma 1.2.3. Let S be a sphere containing A in its interior. Then
p{A,S) = bdA.
Proof. p(A, S)CbdA is clear. Let x ebdA. For / e N choose Xj in the
interior of S (that is, of the ball bounded by S) such that jc,- $ A and
\Xi - x\< l/i. From Theorem 1.2.2 we have
\x - p(A, Xi)\ = \p{A, x) - p(A, Xi)\ ^ |jc - Xi\ < l/i.
The ray R(A, Xj) meets 5 in a point y, and we have p(A, y,) = p(A, Xj),
hence \x - p(A, y,)! < l/i. A subsequence (yi)jehi converges to a point
y e S. From \im p{A, y,) = x and the continuity of the metric projection
we see that x = p(A, y). Thus MAC p(A, S). U
The existence of a unique nearest-point map is characteristic of
convex sets. We prove this result here to complete the picture, although
no use will be made of it.
Theorem 1.2.4. Let A C E" be a closed set with the property that to each
point of ^" there is a unique nearest point in A. Then A is convex.
Proof. Suppose A satisfies the assumption but is not convex. Then
there are points x, y with [x, y]n A = {x, y), and one can choose p >0
such that the ball B = B((x + y)/2, p) satisfies BO A =0. By an
elementary compactness argument, the family C8 of all closed balls B'
containing B and satisfying (int B') D A = 0 contains a ball C with
maximal radius. By this maximality, there is a point p e C D A, and by
the assumed uniqueness of nearest points in A it is unique. If bd B and

1.3 Support and separation 11
bd C have a common point, let this (unique) point be q, otherwise let q
be the centre of B. For sufficiently small f > 0, the ball C + £(q - p)
includes B and does not meet A. Hence, the family Ti contains an
element with greater radius than that of C, a contradiction. ■
Note for Section 1,2
Theorem 1.2.4 was found independently (in a more general form) by Bunt
A934) and Motzkin A935); it is usually associated with the name of Motzkin. In
general, a subset /I of a metric space is called a Chebyshev set if for each point
of the space there is a unique nearest point in A. There are several results and
interesting open problems concerning the convexity of Chebyshev sets in normed
linear spaces. For more information, see Valentine [30], Chapter VII, Marti
[25], Chapter IX, Vlasov A973) and §6 of the survey article by Burago &
Zalgaller A978).
1.3. Support and separation
The simplest support and separation properties of convex sets seem
intuitively obvious, and they are easy to prove. Nevertheless, their many
applications make them a basic tool in convexity.
Let /I C E" be a subset and // C E" a hyperplane and let H*, H~
denote the two closed halfspaces bounded by H. We say that H
supports A at X \i X € A C\ H and either A <Z H* or A C H~. //is a
support plane of A or supports A if H supports A at some point x,
which is necessarily a boundary point oi A. If H =Hu,a supports A and
A C H~a- {y 6 ^"Ky, m) = Q'}, then H~^ is called a supporting half-
space of A and u is called an exterior or outer normal vector of both
//„.„ and //„.». If, moveover, //u^, supports A at x, then u is an
exterior normal vector of A at x. A flat E supports AatxiixeAOE
and E lies in some support plane of A.
Lemma 1.3.1. Let ACE" be nonempty, convex and closed and let
X 6 E"\y4. The hyperplane H through p(A, x) orthogonal to u(A, x)
supports A.
Proof Clearly // D y4 =|= 0. Let // ~ be the closed halfspace bounded
by H that does not contain x. Suppose there exists some y e A with
y i H~. Let z be the point in [p{A,x),y] nearest to x. Then
\x - z\<\x - p(A, x)\, which contradicts the definition of piA, x) since
z e A. This shows that AC H~. ■
Theorem 1.3.2. Let ACE" be convex and closed. Then through each
boundary point of A there is a support plane of A. If A ^ 0 is bounded,

12 Basic convexity
then to each vector u e E"\{o} there is a support plane to A with exterio.
normal vector u.
Proof. Let x ebdA. First let A be bounded. By Lemma 1.2.3 there is ;
point y 6 E"\A such that x = p(A, y). By Lemma 1.3.1 the hyperplant
through p(A, y) = x orthogonal to y - x supports K at x.
If A is unbounded, there exists through x a support plane H a
A n B(x, 1); let H~ be the corresponding supporting halfspace o)
A O B(x,l). If there is a point z€A\H~, then [z,x]CA, bui
[z, x) n B(x, 1) (^ H~, a contradiction. Hence H supports A.
Let A be bounded and u e E"\{o}. Since A is compact, there is i
point X e K satisfying (x, u) = sup {{y, u)\y e K}. Evidently (y e E"|
(y, u) = (x, u}} is a support plane to A with exterior normal vector u.
The existence of support planes through arbitrary boundary points is
characteristic for convex sets, in the following precise sense:
Theorem 1.3.3. Let A CE" be a closed set such that int A ^ 0 and such
that through each boundary point of A there is a support plane to A.
Then A is convex.
Proof Suppose that A satisfies the assumptions but is not convex. Then
there are points x, y e A and z e [x, y] with z $ A. Since ii\tA =)= 0 (and
n ^ 2, as we may clearly assume), we can choose a e int y4 such that x,
y, a are affinely independent. There is a point b ebdA n[a, z). By
assumption, through b there exists a support plane H to A, and a $ H
because a e inty4. Hence H intersects the plane aff (jc, y, a} in a line.
The points x, y, a must lie on the same side of this line, which is
obviously a contradiction. ■
We turn to separation. Let A, B CE" be sets and H^^a C E" a
hyperplane. The hyperplane //„ <^ separates A and B i( A C H~^ and
B C H* c,, or vice versa. This separation is said to be proper if A and B
do not both lie in //„,». The sets A and B are strictly separated by //„ <^
if y4 C int HZ,a and B C int H^^^ or vice versa, and they are strongly
separated by //„,» if there is an f > 0 such that Hu,a-e and H^^a+e both
separate A and B. Separation of A and a point x means separation of
A and {x}. We first consider this special case:
Theorem 1.3.4. Let A Cl" be convex and let x e E"\A. Then A and x
can be separated. If A is closed, then A and x can he strongly separated.

1.3 Support and separation 13
Proof. If A is closed, the hyperplane through p{A, x) orthogonal to
u{A, x) supports A and hence separates A and x. The parallel
hyperplane through {p{A, x) + x)/2 strongly separates A and x. If A is
not closed and jc ^ cl ^4, then a hyperplane separating cl A and x a
fortiori separates A and x. \i xecXA, then jcebdcM by Theorem
1.1.14, and by Theorem 1.3.2 there is a support plane to cM through x\
it separates A and x. ■
Corollary 1.3.5. Each nonempty closed convex set in E" is the intersection
of its supporting halfspaces.
Separation of pairs of sets can be reduced to separation of a set and a
point:
Lemma 1.3.6. Let y4, fi C E" be nonempty subsets. A and B can be
.separated (strongly separated) if and only if A - B and o can be
separated (strongly separated).
Proof. We consider only strong separation; the other case is analogous
(or put f = 0). Suppose that Hu^a strongly separates A and B, say
A C H~a-e and B C Hu.a+c for some £>0. Let x e A - B; thus
X = a - b with a e A, b e B. From (a, u) S a - e and {b, u) ^ a + e
we get (x, u) ^ -2e, so that A - B and o are strongly separated
by //„.-..
Suppose that A - B and o can be strongly separated. Then there are
u 6 E"\{o} and £• > 0 such that {x, u) S -Is for a\\x€ A- B. Let
a '.= sup {{a, u}\a e A},
^ := infiib, u)\b e B}.
For a e A, b e B we have {a, u) - {b, u) g -le, hence /3- Q'S2f.
Thus Hu,(a+p)l2 strongly separates A and B. ■
li A, B C E" are convex, then A - B is convex. If A is compact and
B is closed, then A - B is easily seen to be closed. The condition
o^ A - B IS equivalent to A C\ B = 0. Hence from Lemma 1.3.6 and
Theorem 1.3.4 we deduce:
Theorem 1.3.7. Let A, BC^" be nonempty convex sets with A D B = 0.
Then A and B can be separated. If A is compact and B is closed, then A
and B can be strongly separated.
The following examples should be kept in mind. Let
A := {(^,//NE2|^>0, //Sl/^},

14 Basic convexity
B := {(§,//) 6 E2|§>0,//g -1/^},
These are pairwise disjoint, closed, convex subsets of E^. A and B can
be strictly separated (by G), but not strongly. A - B and o cannot be
strictly separated. A and G can be separated, but not strictly.
On the other hand, convex sets may be separable even if they are not
disjoint. The exact condition is given by the following theorem.
Theorem 1.3.8. Let A, B C^" be nonempty convex sets. Then A and B
can be properly separated if and only if
relint/i n relint B = 0. (*)
Proof Suppose that (*) holds. Put C •= relint A - relint B. Then o $ C,
and C is convex. By Theorem 1.3.4 there exists a hyperplane //„ n with
CC //„.,)■ Let
p:=mi{{b,u)\be B},
then B C Hyp. Suppose there exists a point a e A with (a, u) > p. By
Theorem 1.1.12 there exists a point z e relint ^4, and Lemma 1.1.8 states
that [z, a) C relint A. Hence, there is a point a e relint ^4 with
(a, u) > p. There is a point b e B with (b, u) < (a, u) and then, by a
similar argument as before, a point b e relint B with (b, u) < (a, u).
Thus a - b e C and {a - b,u) > 0, a contradiction. This shows that
A C H~ jj. Thus A and B are separated by Hu,/). U A L) B lies in
some hyperplane, then this argument yields a hyperplane relative to
aff(y4U B) separating A and B, and this can be extended to a
hyperplane in E" that properly separates A and B.
Vice versa, let // be a hyperplane properly separating A and B, say
with A C H~ and B C H*. Suppose there exists x e rehnt/i D relint B.
Then x e H. Since A C H~ and jc e relint ^4, we must have AC H,
similarly B C H, a contradiction. Thus (*) holds. ■
Occasionally we shall have to use support and separation of convex
cones. For these we have:
Theorem 1.3.9. Let C C E" be a closed convex cone. Each support plane
of C contains o. If x e E"\C, then there exists a vector u e E" such that
(c, u) 5 0 for allc e C and (x, u) < 0.
Proof. Let // be a support plane to C. There is a point ye HOC.
Then Xy e C for all A> 0, which is impossible if o ^ H. Hence o e H.
The rest is clear from Lemma 1.3.1. ■

1.3 Support and separation 15
We shall now prove two more results in the spirit of the theorems of
Carath^odory and Helly. They are treated in this section since the first
of them needs support planes in its proof and the second one deals with
separation.
Theorem 1.3.10 (Steinitz's theorem). Let ACE" and jc e intconvy4.
Then x e intconv y4' for some subset A' C A with at most 2n points.
Proof. The point x lies in the interior of a simplex with vertices in
convy4; hence, by Carath^odory's theorem applied to each vertex,
X 6 intconv B for some subset B oi A with at most (n + 1)^ points. We
can choose a line G through x that does not meet the affine hull of any
M - 1 points of B. Let Xi, Xi be the endpoints of the segment
G n convfi. By Theorem 1.3.2, through Xj there is a support plane Hj
to convfi (/=1,2). Clearly JCy e conv(fi D//y); hence by Carath^o-
dory'.s theorem there is a representation
n n
Xj = Yi^jiy-fi with yii 6 B, kji s 0, X'^;/ = 1
i=i 1=1
(/ = 1,2), and here necessarily kj^ > 0 by the choice of G. With suitable
A 6 @,1) we have
n
oc = A - A)ac, + Ajtj = S[(l - ^)hiyM + ^hiyn]
1=1
6 relintconvly,,, . . . , ym, yai. • •• . yin)
by Theorem 1.1.13. Here relint can be replaced by int, since by the
choice of G the points y\i,...,y\n are affinely independent and for at
least one k also yn, . . . , yx„, yik are affinely independent. ■
Theorem 1.3.11 (Kirchberger's theorem). Let A, B C^" be compact sets.
If for any subset M G A^ B with at most n + 2 points the sets M H A
and MOB can be strongly separated, then A and B can be strongly
■separated.
Proof. First we assume that A and B are finite sets. For jc e E" define
(with r(A:) ••= (a:, 1))
Hj := {v 6 E" X R| ± {v, r(A:)) > 0}.
Let M C A U B and card M ^ n + 2. By the assumption there exist
w 6 E" and aeU such that {u, a) > a for a e M O A and (u, b) < a
for he MOB. Writing v.= (u,-a), we see that (y, r(a)) =
{u,a)-a>0; thus v e H^ for a€MC\A. Similarly v e HI for
beMf\B. Thus the family {H*\a e A) KJ {Hl\h e B) of finitely

16 Basic convexity
many convex sets in E" x R has the property that any n + 2 or fewer of
the sets have nonempty intersection. By Helly's theorem, the
intersection of all sets in the family is not empty. Since this intersection is open,
it contains an element of the form v = (u, -a) with u =)= o. For a e A
we have v e H^, hence (u, a) > a, and for b e B similarly {u,b)<a.
Hence A and B, being finite sets, are strongly separated by //„.».
Now let A, B be compact sets satisfying the assumption. Suppose that
A and B cannot be strongly separated. Then conVy4 and conv B cannot
be strongly separated. By Theorem 1.1.10 these sets are compact and
hence by Theorem 1.3.7 they cannot be disjoint. Let jc6convy4n
convfi. Then jc e convy4'f) conv fi' with finite subsets A'C A and
B' C B, which hence cannot be strongly separated. This contradicts the
result shown above. ■
We conclude this section with another application of a separation
theorem, which will be useful in the study of Minkowski addition.
Lemma 1.3.12. Let A, B CE" be nonempty convex sets. If
X 6 Telint(A + B),
then X can be represented in the form x = a + b with a e relint A and
b 6 relint B.
Proof. There is a representation x = y + z with y e A and z € B. We
may assume that x = y = z = o and also that dim(y4 + B)= n. Sirtce
o6 int(y4 + B), A + B and o cannot be separated. Hence, by Lemma
1.3.6 and Theorem 1.3.8, there is a point
a e relint/i n relint (-fi).
Then -a e relint B and o = a ~ a. ■
Notes for Section 1.3
1. Separation and support properties of convex sets in finite and infinite
dimensions are of fundamental imf)ortance in various fields such as functional
analysis, optimization, control theory, mathematical economy, and others.
For infinite-dimensional separation and support theorems we refer only to
Bourbaki [9], Marti [25], Holmes A975) and Bair & Fourneau [3]; see also
the survey article by Klee A969b).
A thorough study of several types of separation in (E" was made by Klee
A968).
2. A stronger version of Theorem 1.3.3 (existence of local support planes) is
associated with the name of Tietze. A survey of results of this type is given in
the article by Burago & Zalgaller A978).
3. Historical information on the theorems of Steinitz and Kirchberger can be
found in the survey article by Danzer, Griinbaum & Klee [32].

1.4 Extremal representations 17
Positive bases. Let BC^". Using Theorem 1.1.13 one easily sees that
pos i5 = (E" holds if and only if o 6 intconv B. The set B is called a positive
basis of (E" if posB = (E'' but posB'=f (E" for each proper subset B'CB.
Thus Steinitz's theorem implies that a positive basis of (E" contains at most 2n
vectors. If i5 is a linear basis of E", then B U (-B) is a positive basis, and up
to multiplication by positive numbers it is only in this way that the maximal
number 2m can be achieved. Positive bases have been investigated by Davis
A954), McKinney A962), Bonnice & Klee A963), Reay A965) and Shephard
A971).
1.4. Extremal representations
The purpose of this section is to represent a closed convex set as the
convex hull of a smaller set, and here the smallest possible sets will be
of particular interest. A first candidate for a smaller set with the same
convex hull is the relative boundary. Only the obvious trivial cases must
be excluded:
Lemma 1.4.1. If ACE" is a closed convex set with A =)= convrelbd/i,
then A is either a flat or a half-flat.
Proof. Clearly we may assume that dimy4 = n. There is a point
Ar6inty4 with Ar^convbdy4 (since otherwise y4 = inty4 U bdy4 =
convbdy4). By the separation theorem, 1.3.4, there is a closed halfspace
H~ such that x e H~ and con\bdACH*. Each point yeintH'
satisfies [at, y) D bd/i =0 and hence y e'mtA; thus H~ C A. By the
convexity and closedness of A, each translate of H" with a point of A
in its boundary is contained in A. Thus A is either equal to E" or is a
halfspace. ■
We will exclude the exceptional cases that are the subject of Lemma
1.4.1 by demanding that A be line-free, meaning that A does not
contain a line. Owing to Lemma 1.4.2 below, this is not a severe
restriction. First let ACE." be a closed convex set. Suppose that A
contains a ray Gj,_u '■= {x + Am|A^O} with a: e E" and u e E''\{o}. Let
y e A. Let z e Gyu and w e [x, z). The ray through w with endpoint y
meets G^u^ hence w e A. Thus [x, z) C A and hence z e A. This shows
that also Gy^u '^ A- For this reason, it makes sense to define
recA := (u e E"\{o}|G;<,„ C A} U {o}
where x e A; this set does then not depend upon the choice of a:. It is
evidently a closed convex cone, called the recession cone of A. One
may also write
recA = {u 6 l"\A -\- u C A}.

18 Basic convexity
Lemma 1.4.2. Each closed convex set A CE" can be represented in the
form y4 = y4 ® V, where V is a linear subspace of E" and A is a line-free
closed convex set in a subspace complementary to V.
Proof. Assume that A is not line-free. Then
V := recy4 f) (-recA)
(the lineality space of y4) is the linear subspace consisting of all vectors
that are parallel to some line contained in A. Let (/ be a linear
subspace complementary to V and put A ■= A D U; then A + V C A.
Let X e A. Through x there exists a line G C A; since it is parallel to
V, it meets (/ in a point y. Then x = y + (x - y) with y e A and
X - y e V; hence x e A + V. This proves A = A ® V. Clearly A is
closed, convex and line-free. ■
The representation by convex hulls of minimal sets requires some
definitions. Let y4 C E" be a convex set. A face of y4 is a convex subset
FCA such that each segment [at, y] C y4 with F D relint[Ar, y] =)= 0 is
contained in F or, equivalently, such that x,y e A and (x + y)/2 e F
implies x, y e F. If {z} is a face of A, then z is called an extreme point
oi A. In other words, z is an extreme point of A if and only if it cannot
be written in the form z = (I - k)x + ky with x, y e A and A e @, 1).
The set of all extreme points of A is denoted by ext A. An extreme ray
of y4 is a ray that is a face of y4. By extr A we denote the union of the
extreme rays of y4.
Theorem 1.4.3. Each line-free closed convex set A CE" is the convex hull
of its extreme points and extreme rays;
A = conv(exty4 U extry4).
Proof. The assertion is clear for n ^ 1. Suppose that n ^2, dimy4 = n
(w.l.o.g.) and the assertion has been proved for convex sets of smaller
dimension. By Lemma 1.4.1, A =convbd/4. By the support theorem,
1.3.2, each point x ebdA lies in some support plane H of A. By the
induction hypothesis, x lies in the convex hull of the extreme points and
extreme rays of H A A, and it is an immediate consequence of the
definition of a face that these are respectively extreme points and
extreme rays of A itself. The assertion follows. ■
Corollary 1.4.4. If A CE" is a line-free closed convex set then
A = convexty4 -I- recy4.

1.4 Extremal representations 19
Proof. By Theorem 1.4.3, a point x € A can be written as
km m
X = ^kjXj + 2 ^i^i^ K =0, 2^i = 1
,= 1 i=* + i 1 = 1
with Xj 6 exty4 and v; e extry4. Then y,, lying in some extreme ray, can
be written as f, = y,- + u, where y,, the endpoint of the ray, is an
extreme point of A, and u, e rec ^4. The assertion follows. ■
Corollary 1.4.5 (Minkowski's theorem). Each convex body K e X" is the
convex hull of its extreme points.
Here the set of extreme points cannot be replaced by a smaller set,
since a point at e AT is an extreme point of K if and only if i*C\{Ar} is
convex.
Extreme points can also be characterized in a slightly different way
that is sometimes useful. Let ACE" be closed and convex and let
X e A. A cap of A around at is a set of the form A A H*, where H * is
a closed halfspace with x eintH*.
Lemma 1.4.6. Let A CE" be closed and convex and let x e A. Then x is
an extreme point of A if and only if each neighbourhood of x contains a
cap of A around x.
Proof. Let at e ext y4. A given neighbourhood of x contains an open ball
^0 with centre x. Let A := conv(y4\Bo)- Then x $ A, hence there is a
hyperplane H strongly separating A and x. If H* is the closed
halfspace bounded by H and containing x, then y4 D //* is the required
cap.
If AT ^ ext y4, then x e relint [y, z] for suitable y, z e A. Each cap of A
around x contains y or z, hence sufficiently small neighbourhoods of x
cannot contain a cap around x. ■
The notion of an extreme point of a convex set A involves convex
combinations of points in A and is thus related to an 'intrinsic'
description of A. Looking at a convex set from outside, one is led to a
related but different class of special boundary points. The point at e y4 is
called an exposed point of A if there is a support plane H to A with
H A A = {x}. The set of exposed points of A is denoted by expy4.
Clearly exp A C ext y4, but even for A e jO- this inclusion is in general
strict. This is shown by the example of a rectangle with a semicircle
attached to one of its edges. However, each extreme point of a closed
convex set is a limit of exposed points. We formulate the corresponding
result for convex bodies only.

20 Basic convexity
Theorem 1.4.7 (Straszewicz's theorem). For K e T"
ext K C cl exp K,
hence
K = clconvexp K.
Froof. Let X eextK and let U be a neighbourhood of at. By Lemma
1.4.6. U contains a cap K D H* of K around x. Let G be the ray
orthogonal to H = bdH* with endpoint x and meeting H. To any
point :. e G there is a point y-sK with maximal distance from z-
Obviously, y.eexpK. If |z-Ar| is sufficiently large, then y^eH*
bv elementary geometry. Thus y.€Kr\H*(ZU and hence ext K C
clexp K.
Using Minkowski's theorem, we get
K = conv ext K C conv cl exp K
C cl conv exp K G cl conv ext K = K,
hence /C = cl conv exp /C. ■
Notes for Section 1.4
1. Minkowski's theorem and its aftermath. Minkowski's theorem was first proved
by Minkowski A911. §12). Of particular importance is its extension to
infinite-dimensional spaces: Each compact convex subset of a locally convex
Hausdorff linear space is the closed convex hull of its extreme points. This
was proved, in a more special case, by Krein & Milman A940); see, e.g.,
Bourbaki [9]. A certain converse is due to Milman A948); this states that in
the Krein-Milman theorem the set of extreme points cannot be replaced by a
set whose clo.sure does not contain the extreme points. We refer the reader to
Jacobs A971) for an elementary introduction to extreme points, including
applications, and to N. M. Roy A987) for a survey article on extreme points
of convex sets in infinite-dimensional spaces. An extension of the Krein-
Milman theorem to locally compact sets, of which Theorem 1.4.3 is the
finite-dimensional version, is due to Klee A957b): see Jongmans A968) for
another extension.
The Choquet theory of integral representations can be considered as a
further extension of the Krein-Milman theorem; see, e.g., Bauer A964),
Phelps A966, 1980), Choquet (I969a,b), Alfsen A971), and the survey article
by Sa§kin A973) which contains many references up to 1972.
2. Straszcwicz's theorem was first proved by Straszewicz A935); see also Wets
A974) and, for the infinite-dimensional case, Klee A958) and Bair A976a).
3. A good source for the theorems of Minkowski and Straszewicz and their
extensions to unbounded closed convex sets is Rockafellar [29].
Klee A959a) made an extensive study of the closed sets that are the
convex hull of a finite system of points and rays (polyhedral sets).
If ACE" is an unbounded line-free closed convex set, then Corollary
1.4.4 states that >1 = conv ext i4-n rec i4. Starting from this representation,
Batson A988) found characterizations of the sets A for which ext A is
bounded.

1.5 Convex function.<: 21
4. The following result is due to Dubins A962): see also Pranger A973). If
K G X" and M C E" is a flat of codimension k, then each extreme point of
K r\ M is & convex combination of k + I (or fewer) extreme points of A!^. In
fact, Dubins proved an infinite-dimensional version. A simplified proof and
an extension are due to Klee A963a).
f-. Continuous barycentre functions. By Minkowski's theorem, each point x € K,
where K e X", is a convex combination of extreme points of K. For a
polytope P, such a convex combination may involve all the extreme points of
P. Let P have the extreme points jr,, . . . , x^. A /c-tuple {q>i, . . . , q)^) of
real-valued continuous functions (p,, . . . , (jp* on f such that
* *
X = ^fMx,, <p,{x) so {i=l,...,k), 2<P.ix) = 1,
1=1 1=1
is called a continuous barycentre function. It is called extreme if for each
X € P, the set {x,\q),{x) >0) is affincly independent. The existence of
continuous barycentre functions was proved by Kalman A961) and the
existence of extreme continuous barycentre functions by Fuglede A986a).
Br0ndsted A986) showed that the extreme continuous barycentre functions
coincide with the extreme points of the convex set of all continuous
barycentre functions on P. Related investigations for general compact convex
sets were done by Fuglede {1986a). Answering a question of Kalman A961),
Wiesler A964) showed that not all the functions q>, can be convex.
1.5. Convex functions
The investigation of convex sets is closely tied up with convex functions.
We treat convex functions in a slightly more general fashion than would
be necessary for our later applications.
For convex functions it is convenient to admit as the range the
extended system IR = IRU {-<»,»} of real numbers with the usual rules.
These are the following conventions. For AeIR, -oo<A<oo,
x-l-oo = A-l-o<:> = o<:>-l-A=30,—30 — 30 = —=o-l-( —oo) = A— 00 = —oo-l-A
= -00, and finally Aoc = oo, 0 or -oo according to whether A > 0, A = 0
or A < 0. For a given function /: E" —»IR and for o- e IR we use the
abbreviation
{f=a} ■= {x el"\f(x) = a},
and {f^ a}, {/< a) etc. are defined similarly.
A function /: E" -»IR is called convex if / is proper, which means
that {/=-oo} = 0and{/=»}=)=E",andif
fid - k)x + ky) ^ i\ - k)f(x) + kf(y)
for all AT, y 6 E" and for 0 ^ A ^ 1. A function f: D^U with D C E" is
called'conveAT if its extension / defined by
fix) for X e D
00 for X 6 E''\0
fix) ■■--
is convex. A function / is concave if -/ is convex.

22 Basic convexity
Trivial examples of convex functions are the affine functions; these
are the functions /: E" ^ IR of the form f(x) = {u, x) + a with u e E"
and Q- 6 R. A real-valued function on E" is affine if and only if it is
convex and concave.
The following assertions are immediate consequences of the
definition. The supremum of (arbitrarily many) convex functions is convex if
it is proper. If /, g are convex functions, then f + g and af for o-^O
are convex if they are proper.
Remark. If / is convex, then
/(A,Ar, + . . . + ki,Xk) ^ Ai/(Ar,) + . . . + k,J(xi,)
for all Xi, ..., xi^eE" and all Aj, ..., A^ e [0,1] with A,-I-. . .
+ Xi; = 1. This is called Jensen's inequality; it follows by induction.
Convex functions have the important property (important for
optimization etc.) that each local minimum is a global minimum. In fact, let
/: E" —»IR be convex and suppose that at,) e E" and p > 0 are such that
f(xa) < =° and /(xq) ^ f(x) for \x - xq\ ^ p. For at e E" with \x - xo\
>plet
y-=^^xJi--^
\X - ATol \ \X - ATc
then \y - x^)\ = p and hence
fixo) ^ fiy) ^ T-^ fix) + (l - T~—r)f(xoh
\X — Xo\ \ \X - ATqI/
which gives /(xq) ^ f(x).
A convex function determines in a natural way several convex sets.
Let /: E" ^ R be convex. Then the sets
dom/ := {/< oc},
the effective domain of /, and for aeU the sublevel sets {f<a),
{/ ^ or}, are convex. The epigraph of /,
epi/ := {(X, 0 6 E" X R|/(x) ^ C}
is a convex subset of E" x IR. The asserted convexity is in each case easy
to see. Vice versa, a nonempty convex set y4 C E" determines a convex
function by
fO for AT 6 y4
'^(^^ ■ [oo forA:6E"M,
the indicator function of A.
We investigate the general analytic properties of convex functions,
starting with continuity.

1.5 Convex functions 23
theorem 1.5.1. Each convex function /:£"—»IR is continuous on
ntdom/anrf Lipschitzian on any compact subset of intdom/.
^roof. First we show the continuity. Let atq e intdom/. We can choose
1 simplex S with atq e int 5 C 5 C intdom/ and a number p>0 with
6(^0' P) ^ ^- For X € S there is a representation
n+l n+1
X = Yj Kxi with A; ^ 0, 2 ^. = 1'
1=1 (=1
where atj, . . . , Ar„+i are the vertices of S, and we deduce that
n + l
/(AT) ^ SA/(x,) ^ c := max{/(x,), • . . , /(:c„4i)}.
(=1
Now let y = xo + au with a e [0, 1] and |m| = p. From y = A - a)xn
+ Q'(Ari) + u) we get
/(y) ^ A - ar)/(Aro) + afi^x^ + u),
hence
f{y) - f(xo) ^ a(c - /(xo))
since ato + w e 5. On the other hand.
and hence
which gives
Thus
"'' = TT^>'^Tf^(""-"^
^(^") = TT^^(^)^Tf^^(^"-">'
/(xo) - f(y) ^ a(c - fix,)).
\f(y) - f(xo)\ ^ ^[c - f(x,)]\y - x,\
for y 6 B(xo, p), which shows the continuity of / at xq in a sharpened
version. Thus / is continuous on int dom/.
Now let CC intdom/ be a compact subset. By compactness, there
exists a number p>0 such that Cp •= C + B(o,p) C intdom/. On the
compact set Cp, the continuous function |/| attains a maximum a. Let at,
>' 6 C be given. Then
. ■ ^ •= >' + 1^ ^ ^1 (y - ^) e O
and
> = A - A)Ar + kz with A
p+ !>

24 Basic convexity
hence f(y) ^ A - k)f(x) + kf(z) yields
f(y) - fix) ^ k[fiz) - fix)] ^^\y-x\.
Interchanging x and y we get !/(_>') - fix)\ ^ b\y - x\ with b
independent of X and y. Thus / is Lipschitzian on C. ■
The differentiability of convex functions will be considered first for
the case n = I.
Theorem 1.5.2. Let f:U—*U be convex. Then on int dom f the following
holds. The right derivative f\ and the left derivative /( exist and are
monotonically increasing functions. The inequality f'\ = f, is valid, and
with the exception of at most countably many points, f{ = f, holds and
hence f is differentiable. Further, f, is continuous from the right and f[ is
continuous from the left (in particular, if f is differentiable on int dom/,
then it is continuously differentiable).
Proof. In the following, all arguments of / are taken from int dom/.
Let 0 < A < //. Then
fix + A) = fl^^—^ x + -ix + ti)]^ ^^—^ fix) + -fix + ti),
\ A* A* /A* A*
hence
fix + A) - fix) ^ fix + M) - fix)
A /z
Analogously,
/(. - A) = /(^, + A(, - J ^^fix) + |/(. - M),
\ A* A* /A* A*
hence
fix) - fix - M) ^ fix) - fix - A)
fi A
For arbitrary A, /z > 0,
/(^) = ^(lT7 (^ - z^) + T-frr (^ + ^)
JL
X + n
hence
jhf^^-'^^TT^f^^^'^^
fix) - fix - ji)^ fix + A) - fix)
H A

1.5 Convex functions 25
From the monotoneity and boundedness properties thus established one
deduces the existence of
fr(x) = hm ,
AiO ^
J\{x) - lim ,
AiO -A
as well as the inequality f{(x) ^ f'i(x).
Thus, for AT < y,
f{(x) g f'M ^ ^^y^ " f "^^ ^ f{(y) g n(y).
Hence /( and /r are increasing and thus have only countably many
discontinuities. At each continuity point of /( the above inequalities
yield /( = /r and hence the existence of the derivative /'.
Let X < y. We have
f(y) - fix) _ ,. f(y) - f(z) > .. .,,.
y - X zix y - z zix
hence
lim f'M ^ nix) ^ lim niz)
z Ix z Ix
by the monotoneity of f'f. Thus /r is continuous from the right.
Analogously one obtains that /[ is continuous from the left. ■
Remark 1.5.3. Let /:IR-»IR be convex; let atq e R and m be a number
with /{(xq) ^ m^ f'rixo)- As noted in the above proof, one has
f(x) - f(Xo) fg f-(xo) ^ m,ifx> ATo,
X - xa [^ fiixo) ^ m,\f X < xq,
thus f(x) S /(xq) + m(x - xq) for all x eU. This shows that the line
{(x, y) eUxU\y = f(x(,) + m(x - x,,)}
supports the epigraph of / at the point (atq, /(xq)).
Now let /: E" —»IR be a convex function. Theorem 1.5.2 implies, in
particular, for each point xeintdom/, the existence of all the
onesided directional derivatives
nx; u) := Mm f^^ ^ ""^^ - f^^\
;.io A
where m6E''\{o}. We first collect some remarks on these directional
derivatives in connection with sublinearity.

26 Basic convexity
A function /: E" —»IR is called positively homogeneous if
f{Xx) = kf(x) for all A ^ 0 and all x e E"
and / is called subadditive if
fix + y)^ fix) + fiy) for all x, y e E".
A function that is positively homogeneous and subadditive is called
sublinear. Every sublinear function is convex.
Lemma 1.5.4. Let /: E" —» R be convex and xeintdom/. Then the
directional derivative function
fix; -y.l" ^U
is sublinear.
Proof. Let u e E''\{o}. For A, r > 0 we may write
fix + Tku) - fix) _ ^ fix + Tku) - fix)
and r i 0 gives /'(at; Am) = ?-f'ix: u). For u, v eE" the convexity of /
yields
fix + r(M + I))) = /(i(.v + 2tu) + \ix + 2tv))
^ \fix + 2tu) + \fix + 2ry).
hence
fix + xiu + V)) - fix) ^ fix + 2ru) - fjx) ^ fix + Ixv) - fjx)
T 2r 2r
for T > 0, and r i 0 gives fix; u + v)^ fix; u) + fix; v). ■
Let /: E" —»IR be a sublinear function. The condition /(-m) = -/(")
is necessary and sufficient in order that / be linear on the subspace
lin {u}, and in this case we say that u is a linearity direction of /.
Lemma 7.5.5. Let /: E" -»IR be a sublinear function. The set L[f] of all
linearity directions off is a linear subspace. Let x e E". Then fix; •)~f
and
lin(L[/]U{x})C L[fix; ■)].
Proof Letu, v e L[/].Then/(M + v)^ /(u) + /(y) = -/(-u) - fi-v) ^
-/(-M-y)g fiu + v), the latter because 0 =/(o) = fiw-w)^
/(w) + /(-w). It follows that u + V e L[f] and hence that L[f] is a
linear subspace.
For M 6 E" and r > 0, the sublinearity of / yields

1.5 Convex functions 27
f^' ^ '^^ - ^(-) ^ fiu).
T
and r I 0 gives f'(x; u) = f(u).
Let u e L[f]. From
2f(x) ^ f(x + ru) + fix - Tu)
^ 2f(x) + T[f(u) + f(-u)]
= 2 fix)
it follows that
fix + TU) - fix) fix - TU) - fix)
+ = 0, (*)
and rjO gives /'(at; u) +/'(at;-u) = 0, hence u e L[f'ix; •)]. By
positive homogeneity, equality (*) holds also for u = at if 0<r<l,
hence r j 0 gives fix; x) + fix; -x) = 0 and thus x e L[f'ix; •)]. ■
We return to general convex functions and consider stronger
differentiability properties. For a convex function, the existence of all two-sided
directional derivatives (that is, Gateaux differentiability) implies (total
or Fr^chet) differentiability. In fact, this is already implied by the
existence of the partial derivatives. The ith partial derivative of / at at,
denoted by 3,/(Ar), is the two-sided directional derivative /'(Ar;e,),
where ej, . . . , e„ is a fixed orthonormal basis of E".
Theorem 1.5.6. Let /:E"—»IR be convex and Areintdom/. /// has
partial derivatives (of first order) at x, then f is differentiable at x.
Proof If / has partial derivatives at x, we can define the gradient of /
at X by
II
grad/(Ar) := S3,/(Ar)e,
and we then have to show that
lim ^ = 0, (*)
where
gih) := fix -f h) - fix) - {grddfix), h) for /j e E".
The function g is convex and finite in a neighbourhood of o. For
h = Xr=i'?'^' *^ have
/ 1 " \ 1 "
S(h) = g{— ^nme,] ^ — I^ginrjiC'i).

28
Basic coiivexitv
Using the inequality
{h, v) § \h\\v\ = \h\ Skil for " = S^j^m
1=1 1=1
we obtain
gC^) = Lli ——— = \h\ Z
i 1=1
1=1
gi^lte,)
nrji
(where the summands with r;, = 0 have to be replaced by 0) and a
similar inequality with h replaced by -h. Since g(h) + g(-h) =
g(o) = 0 by the convexity of g, we obtain for /j ^ o
1=1
gi-n^i^i)
nri
-8(-h) < g(h)
^1
i=l
ginn.ei)
nr]i
Since the partial derivatives of g at o exist and are zero, the left and
right sides of this chain of inequalities tend to 0 for /j—»o, hence (*)
follows. ■
f'@,) =
For differentiable functions one has some useful criteria for
convexity, which we now collect.
Theorem 1.5.7. Let D CU be an open interval and /: D —»IR a
differentiable function. Then f is convex if and only iff is increasing.
Proof. If / is convex, then /' is increasing, by Theorem 1.5.2. Let /' be
increasing, and let x, y e D, x < y. and A e @, 1). By the mean value
theorem, there exist numbers O^ e[x,(l - k)x + ky] and O2 e
[A - X)x + A.y,y] with
/((I - k)x + ky) - fix)
k(y - X)
f,,(, . f(y) - /((I - A)Ar + ky)
A - AKy - x)
From /'((^,) ^ /'(«2) we obtain
/((I - k)x + ky) ^ A - k)f(x) + kf(y). m
Corollary 1.5.8. Let DCU be an open interval and f.D^U a twice
differentiable function. Then f is convex if and only iff" ^ 0.
The extension of these criteria to differentiable functions on E" relies
on the simple fact, following from the definition of convexity, that a
function on E" is convex if and only if its restriction to an arbitrary line
is convex.

1.5 Convex functions 29
Theorem 1.5.9. Let D C E" be convex and open and let /: D —»IR be a
differentiable function. Then the following assertions are equivalent:
(a) / is convex;
(b) f(y) - fix) S <grad/(.v), y - x) ior x, y e D;
(c) <grad/(>')-grad/(Ar), _>'-Ar)SO for x, y e D.
Proof. Suppose (a) holds. Let x, y e D and 0 < A < 1. From
f(x + k(y - X)) = /((I - k)x + ky) ^ A - k)f(x) + kf(y)
we get
fix + k(y - X)) - fix)
(grad/(Ar), y - x)
^fiy)-fix)- {gTadfix),y-x}.
For A—»0 the left side converges to 0, hence (b) follows.
Suppose (b) holds. Interchanging the roles of x and y and adding
both inequalities, we get (c).
Suppose (c) holds. Let x, y e D. Let
g(A) := fix + kiy -x)) for 0 ^ A ^ L
For 0 = Aq < Ai = 1 the chain rule gives
g'(A,) - g'(Afl)
= <grad/(Ar + Ai(>' - x)), y - x) - <grad/(Ar + Ao(>' - x)), y - x)
Thus g' is an increasing function and, therefore, g is a convex function.
Since x, y e D were arbitrary, this implies that / is convex. ■
Theorem 1.5.10. Let D C E" be convex and open and let /: D —»IR be
twice differentiable. Then f is convex if and only if its Hessian matrix
Hess/(at) := C,3^/(Ar))^=,
is positive semidefinite for all x e D.
Proof. For x, y e D define
g(A) := fix + kiy - a)) for x + kiy - x) e D.
Then / is^convex if and only if all such functions g are convex, which is
equivalent to g" = 0. We need only check whether g"@) & 0 holds
generally. In fact, let A, 6@,1). Writing Xi ■= x + kyiy - x) and
f^i'^) ■= fixi +'^iy - xi)), we have g(Ai + A) =/2(A/A - A,)), hence
g"(Ai) S 0 if and only if h"iO) ^ 0.

30 Basic convexity
Now for X = X"=i^.^i. y = ^'i=\Viei we have
,-@) = hm ^•'*' - ^•""
A—O A
= lim - <grad/(Ar + A(_>' - x)) - grad/(Ar), y - x)
A—n A
= lim 2 T [3/(Ar + A(y - at)) - 3/(x)](^, - §,)
;.—0 i=\ A
n «
,= 1 y=i
from which the assertion follows. ■
For convex functions that are not necessarily differentiable, the
notions of gradient and differential have natural extensions that are
particularly important in optimization. In our context they are of
interest due to their simple geometric interpretation. If f:E"—*U is
convex and differentiable at x, then Theorem 1.5.9 states that
f(y) ^ fix) + <grad/(x), y - x) for all y e E".
This gives ri.se to the following definition. Let /: E" —»IR be convex and
let X 6 dom /. The set
dfix) := {v 6 E"!/(>') ^ f(x) + {v,y - x) for all y e E"}
is called the sub differential of / at at, and each element of 8/(Ar) is
called a subgradient of / at x. Clearly the .subdifferential of / at at is a
closed convex subset of E". It parametrizes the set of support planes to
the epigraph of / above at, in a sense made precise in the following
theorem.
Theorem 1.5.11. Let /; E" —> IR be convex and let x e dom/. The vector v
is a subgradient of f at x if and only if the vector (y, -1) e E" x IR k an
exterior normal vector to epif at (x, f(x)).
Proof, (v, -1) is an exterior normal vector to epi/ at (x,f(x)) if and
only if
{iy,n)-ix,fix)),i^^-l))^0
for all (y, rj) e epi/, and this is equivalent to
{(y,f(y))-(x,f(x))A^,-l))^0
for all y eE", hence to
f(y)^f(x)+ {v,y -x)
for ally 6 E". ■

Notes for Section 1.5 31
Differentiability of a convex function is equivalent to the uniqueness
of subgradients.
Theorem 1.5.12. Let f: E" —»IR be convex and let x e int dom/. Then f is
differentiable at x if and only if f has only one subgradient at x.
Proof. Suppose that / is differentiable at x. Let d be a subgradient of /
at X, then
A
for A > 0, hence f'(x; u)^ {v, u) for all ueE". Differentiability of / at
X implies f'(x; u) = -f'(x; -u) ^ -{v, -u) = {v, u), hence f'(x; u) =
{VyU}. Since this holds for all u e E", y is unique.
Vice versa, suppose that / has at most one subgradient at x. Let
M6E"\{o}. Put
g(k) := fix + Am) for a 6 R
and choose a number m with g,'@) ^ m ^ gr@)- By Remark L5.3, we
have g(A) S g@) + mk and thus f(x + Am) S f(x) + mk. Hence the line
G := {{x + Am, f{x) + mA)|A e R} C E" x R
satisfies G D intepi/ = 0. By Theorem 1.3.7 there exists in E" x IR a
hyperplane //(„.-!).„ separating G and intepi/ (the normal vector of
this hyperplane can in fact be assumed to be of the form {v, -1) since,
due to AT 6 int dom/, the hyperplane cannot be "vertical'). Since
//(„_,)„ supports epi/ at (at,/(at)), y is a subgradient of / at at and
hence unique. The inclusion G C //(„ _,) <, implies that
<(j), -1), (a: + Am, /(a:) + mk)) = a
for all A 6 IR and hence that m = {v,u). Since v is unique, m is also
unique and thus g is differentiable at 0. Since u e E"\{o} was arbitrary,
/ has two-sided directional derivatives at x and hence, by Theorem
1.5.6, is differentiable at AT. ■
Notes for Section 1.5
Standard references for convex functions are Rockafellar [29] and Roberts &
Varberg [28], which we have followed in many respects; see also Marti [25].
Differentiability almost everywhere of convex functions. Let / be a finite
convex function on an open convex subset D C E". By Theorem 1..5.1, / is
locally Lipschitzian and hence, by Radcmacher's theorem (e.g., Fedcrer
A969), p. 216), differentiable almost everywhere on D. This follows also
from Theorem 1.5.12 and Theorem 2.2.4, the latter applied to the epigraph

32 Basic convexity
of /, or from the one-dimensional case (Theorem 1,5.2) and Fubini's theorem
(Roberts & Varberg [28], p. 116). Rather precise information on the set of
non-differentiability is available from work of ZajiCek A979), who also has a
related result for continuous convex functions on a separable Banach space.
The convex function / is in fact twice differentiable almost everywhere.
Since the first differential need not exist in a full neighbourhood of the point
considered, this assertion must be interpreted carefully. One possible way of
defining the twice differentiability of / at x is by the existence of a
second-order Taylor expansion. This means that / is differentiable at x and
that there exists a symmetric linear map Af{x): E" -* E" such that
/(>) = fix) + (grad/(A:), y - x)
+ \{Af{x)iy -x),y-x)+ o(|>' - xl^)
for all y & D. Another possibility is to say that a map t7: D —»E" is a
subgradient choice for / if 9{x) e df{x) for each x & D. In this case the
function / is ealled twice differentiable at x if the family of subgradient
choices for / is uniformly differentiable at x, which means that there exist a
neighbourhood V of x, a linear map >1/(a:): E" —»E" and a function
^- K+ _> K+ with rpit) -♦ 0 for T -> 0 such that
\diy) - ^x) - Af(x)(y - x)\ g ^,(|y - x\)\y - x\
for >■ G V O D and each subgradient choice 0 for /. It follows from the work
of Aleksandrov A939c) (see also Bangert A979)) that for a convex function
twice differentiability in this sense and the existence of a second-order Taylor
expansion are equivalent.
With this definition, it is true that the convex function / is twice
differentiable almost everywhere. For n= 1, this is a consequence of the
differentiability almost everywhere of a monotone function, as was first
pointed out by Jessen A929). For n = 2, it was proved by Busemann & Feller
A936a), and by using their result and an induction argument, Aleksandrov
A939c) obtained the general case. Other proofs, more analytic in character,
are due to ReSetnjak A968b), who used techniques from the theory of
distributions, and to Bangert A977,1979). Essential tools of Bangert's proof
are the Lipschitz property of the metric projection (Theorem 1.2.2), Rade-
macher's theorem on the almost everywhere differentiability of Lipschitz
maps and a version of Sard's lemma for Lipschitz maps.
Asplund A973) applied Aleksandrov's theorem to show that a metric
projection on any closed (not necessarily convex) subset of R" is almost
everywhere differentiable.
3. Dudley A977) showed that a Schwartz distribution on E" is a convex function
if and only if its second derivative is a non-negative n x n matrix-valued
Radon measure. He also showed the absolute continuity of such a measure
with respect to (n - l)-dimensional Hausdorff measure, and other related
results.
4. The second-order suhdifferential. Given a finite convex function / on E",
Hiriart-Urruty A986) introduced for all x e E" and all v e df(,x) a closed
convex set containing o, which he (later) called the second-order subdifferen-
tial of / at (x, v), denoting it by d^f(x,v). Then
is called the second-order subdifferential of / at x. d^f(x) is a compact
convex set. If / is twice differentiable at x, then d^f(x) is the ellipsoid
associated with the ordinary second differential of / at x. The second-order
subdifferential was further investigated by Hiriart-Urruty & Seeger A989),

1.6 Duality 33
who also studied it in relation to a notion of generalized Dupin indicatrices
for non-snmooth convex surfaces.
1.6. Duality
Associated with convex sets, cones and functions, which satisfy some
weak conditions, are dual objects of the same kind. This duality permits
us, for example, to translate certain results on boundary points of
convex sets into results on support planes (or normal vectors), and vice
versa. But there are numerous other applications.
First, let K e X!a (the set of convex bodies with interior points) be a
body with o e int /C. We define
a:* := [x 6 l"\{x, y) ^ 1 for all y e K}.
(More properly, K* should be defined as the subset
{x* 6 (l")*\x*(y) ^ 1 for all y e K}
of the dual space (E")* of E". As, however, we are working with a fixed
scalar product in E", we shall always use the canonical isomorphism
between (E")* and E" induced by this scalar product to identify both
spaces.)
The next theorem shows that K*, which is called the polar body of
K, satisfies the same assumptions as K and that this correspondence is a
true duality.
Theorem 1.6.1. Let K e Jo and o e int/C. Then K* e Xq, oeint/C*.
and K**= K.
Proof. If ATi, ATjeA:* and A6[0, 1], then {(I - k)x^ + Xx2, y) ^l for
all y e K, hence A - A)Ari + Xxi e K*. Thus K* is convex, and it is
equally easy to see that K* is closed. For a ball B(o, e) one has
B(o, e)* = B(o, l/e) as an immediate consequence of the definition.
Further, K1CK2 implies KiD K*. We can choose £,p>0 with
B(o, £)C K C B(o, p); hence B(o, 1/p) C K* C B(o, l/e), showing that
o 6 int /C* and that K* is compact.
Let y e K. For arbitrary x e K* one has {x, y) ^ 1, hence ye K**.
Thus K C K** (observe that up to now neither convexity nor closedness
of K have been needed).
Let z 6 E'X/C. Since K is convex and closed then by Theorem L3.4
there is a hyperplane H^^a with K C H~„ and {z, u) > a; here a>0
since o e int /C. For all y e K we have {y, u/a) g 1, hence u/a e K*.
Now {z, u/a) > 1 shows that z ^ K**. This finishes the proof of
K**=K. m

34 Basic convexity
The following theorem describes the extent to which the forming of
polar bodies interchanges the operations of intersection and union.
Theorem 1.6.2. Ler A",, /Cj e "^o and o e int A", (/ = 1,2). Then
(Ai n a:2)* = conv(At U A|),
[conv(A, U K2)]* = Af n Kl
//Ai U A2 is convex, then A* U A| is convex, hence
(A, n A2)* = Af U Af,
(Ai U A2)* = At n Kl
Proof. From AiDAjCA, we get AfC(A,nA2)*, hence
conv(AtUA|)C (A, n A2)*. From A, C conv(A, U A2) we get
[conv(A,U A2)]*C Af, hence lconv(A, U Aj)]* C At n A|.
Applying both inclusions to A*instead of A, and using A** = A, we arrive at
the first two equalities of the theorem.
Suppose that A, U A2 is convex. Let at e E"\(At U Af). By Theorem
1.3.4 there are u, e E"\{o} and a, e U with {x, u,) > cv, and (y, u,) ^ cv;
for V 6 A* (hence cv, > 0), which implies m,/cv, e Af* = A, (i = 1, 2).
Since A, U A2 is convex, there is a point z e [ui/a'i, Ui/ai] n A, D A2.
Since z is a convex combination of "i/cvi and U2/a2, we get {x, z) > I
and hence a: $ (A, D A2)*. This proves (A, n A2)* C AtU A|. Since
the opposite inclusion is trivial, we get (Aj n A2)* = AtU A*, which
shows that A* U A* is convex. ■
A similar duality exists for convex cones. Let C C E" be a closed
convex cone. We define
C* := {x e E"Ka:, y) § 0 for all y e C}
and call this the dual cone of C. This is again a closed convex cone, and
one has C** = C. The proof is analogous to the corresponding one for
polar bodies, except that instead of Theorem 1.3.4 the separation
theorem, 1.3.9, has to be applied. Also, the proof of the following
theorem is so similar to that of Theorem 1.6.2 that it can be omitted.
Theorem 1.6.3. Let C], C2 C E" be closed convex cones. Then
(C, n c'2)* = ct + cf,
(Ci + C2)* = ct n ci
If C, U C2 is convex, then C* U C* is convex, hence
(Ci n c,)* = ct u c?,
(Ci u C2)* = a n d

1.6 Duality 35
To treat duality for convex functions, a few preliminaries are
necessary. To establish the dualities above, we had to use the fact that a
closed convex set is the intersection of its supporting halfspaces. In the
case of a convex function, this fact will be applied to the epigraph. One
considers, therefore, only functions whose epigraph is closed; such a
function is called, in brief, closed. Closedness of a function is evidently
equivalent to lower semicontinuity. The supporting halfspaces to the
epigraph of a convex function are of two kinds. Generally, a closed
halfspace in E" x IR containing the epigraph of a proper function is of
the form
{(X, ?) 6 E" X U\{ix, a (", n)) ^ ar}
with //SO. If T] = 0, the halfspace is called vertical. If it is not vertical,
we can assume that ?j = -1, so that the halfspace is given by
«("».-■)..= {(AT, OKaT, U) - CVS ?}
= epi h
with h(x) := {x, u) - a. Observing that epi/ C epi h for functions h, f
is equivalent to h^ f, we are led to the following lemma.
Lemma 1.6.4. Let f: E" —»IR be convex and closed. Then
/ = sup{/2|/2:E"-^Raffine, h^f).
Proof. The closed convex set epi/ is the intersection of its supporting
halfspaces. We have to show that it is already the intersection of its
nonvertical supporting halfspaces. Let
H\ := {(X, ?) 6 E" X R|/2i(x) := {x, u.) - a,^Q)
be a vertical supporting halfspace of epi/, and let (ato, ^o) e E" x IR
be a point such that (atq, ?o) ^ ^\- There is some non-vertical
supporting halfspace to epi/ (for instance, at a point (at,/(at)) with
X 6 relintdom/), say
Define h ■= A/jj -I- h2 with A > 0 chosen below. For x e dom/ we have
/2i(a:)S0 and h2{x)-^f{x). hence hi,x)^f{x). If .v6E"\dom/, then
/(at) = 3c and hence hi^x) S /(at) again. Thus
. • epi/C //- := {(a:, ?) 6 E" X R|/2(a:)S C}.
Since (ato, ?o) ^ H^, vie. have /2i(Aro) > 0 and hence A > 0 can be chosen
such that /2(Aro)> ?o and thus (ato, Co) ^ f^~ ■ We deduce the existence
of a non-vertical supporting halfspace (a translate of H~) to epi/ that
does not contain (atq, Co)- Thus epi/ is the intersection of all its

36 Basic convexity
non-vertical supporting halfspaces; equivalently / is the supremum of all
affine functions ^ /. ■
We are now in a position to treat the announced duality for convex
functions. For a closed convex function /: E" —»IR the conjugate function
is defined by
/*(m) := sup{<M, a:) - fix)\x el") for u e E".
The following reformulation of the definition sheds some light on the
intuitive meaning of the conjugate function. For given u e E"\{o}, each
real number a determines a non-vertical halfspace f/(~„ _i) ^ in E" x R,
and we have
(x, f(x)) 6 //(""„._,).a <=> {u, x) - f(x) ^ a,
where the latter inequality is satisfied for all at e E" if and only if
a = /*(m). Hence we may also write
f*iu) = inf {or 6 R|epi/ C //(".-i),,,},
which can be interpreted as saying that the conjugate function /*
provides a parametrization of the set of supporting halfspaces of the
epigraph of /.
Theorem 1.6.5. Let /: E" —»IR be a closed convex function. Then f* is a
closed convex function, and f** = f.
Proof From {f=x}j^l" it follows that {/* = -oo}=0. As the
epigraph epi/ has some non-vertical supporting halfspace
{(x.^)\{u,x)-a-^^}, we have {u,x)-f(x)^a for x el" and
therefore f*(u) ^ cv, hence {/* = «}=)= E". Thus /* is proper. Being the
supremum of a family of affine functions, which are convex and
continuous, /* is convex and lower semicontinuous, hence closed. The
definition of /* can be rewritten as
f*(u) = sup {{u,x} - a(;c,0 6epi/},
thus
f**(x) = sup {(at, u) ~ a\(u, a) e epi/*}.
Now (u, cv) eepi/* is equivalent to {• .u) - a^ f, hence Lemma 1.6.4
gives
f(x) = sup {h(x)\h: E" -^ U affine, h ^ f}
= sup {(at, u) - a\(u, a) e epi/*}
= f**ix). m

1.7 The support function 37
Note for Section 1.6
Infimal convolution. Under conjugation, addition of closed convex functions is
transformed into a process called infimal convolution (see Rockafcllar [29]). For
functions /i, ...,/„,: E" ^ R one defines
(/,D . . . Uf,„)(x) := inf {/.(x.) + . . . + Ux,„)\x, e E", xi + . . . + x„, = x).
If /,, ...,/„ are closed convex functions, then
(/i + ■■■ + /„,)* = /fD--.D/r„.
1.7. The support function
Since a closed convex set is the intersection of its supporting halfspaces,
such a set can conveniently be described by specifying the position of its
support planes, given their exterior normal vectors. This description is
provided by the support function, which is of fundamental importance in
the Brunn-Minkowski theory of convex bodies.
For a nonempty closed convex set /C C E" the support function
h(K, •) = Hk is defined by
h(K, u) ■■= sup {{x, u)\x e K} forueE".
For u 6 dom h(K, • )\{o} we put
H{K,u) := {x el"\{x,u) = h{K,u)},
H-(K, u) := {x 6 E"|(a:, u) ^ h(,K, u)),
F(K, u) := H(K, u) D K.
H(K,u), H~(K,u) and F(K, u) are respectively the support plane,
supporting halfspace and support set of K, each with exterior normal
vector u. Note that these definitions of support plane and supporting
halfspace extend the notions of Section 1.3. They obviously coincide
with the latter if F{K, u)^ 0, but for an unbounded set K it may
happen that F{K, u) = 0.
The intuitive meaning of the support function is simple. For a unit
vector M 6 5""' ndom/2(/C, •), h(K, u) is the signed distance of the
support plane to K with exterior normal vector u from the origin; the
distance is negative if and only if u points into the open halfspace
containing the origin.
From the definition of the support function we see immediately that
h(K, :)={z, •} if and only if K = {z). that hiK + t,u) =
hiK,u)+{t,u) for tel" and further that h(K, ku) = kh(K, u) for
A g 0 and hiK, u + v)^ h(K, u) + h(K, v). Hence. h(K, •)isa convex
function if /C ± E". It is also clear from the definition that h^ = hi if
and only if K C L and that, for any linear subspace £ of E", one has

38 Basic convexity
h(K\E, u) = h(K, u) for ueE. Further, we note that h(A.K.-) =
kh(K, Ofor A^Oand hi,-K,u)= hiK,-u).
For K j^E", the fact that a: e A: if and only if {x,u)^ h(K, u) for all
u 6 E" can also be expressed by saying that
that is, K is the subdifferential of its support function at o.
The support function of a compact convex set K CE" can also be
described as follows. Let
Ck ■■= {A(a:, -1) 6 E" X R|a: 6 a:, A ^ 0}.
Then Ck is a closed convex cone in E" x R, and its dual cone is the
epigraph of the support function of K,
C*K = epihK-
In fact, (_y, rj) e epi h^ if and only if r}^ {x, y) for all at e /C, and this is
equivalent to {(y, //), (Ajt, -A)) ^ 0 for all x e K and A ^ 0 and hence
to(>,^Na.
To simplify the exposition, the following considerations will be
restricted to compact convex sets, since in later chapters the support
function will be used only for these. For a convex body K, the
supremum in the definition of h(K, u) is attained and finite for each u,
and h(K, •) is a sublinear function.
Theorem 1.7.1. If f: E" —> IR is a sublinear function, then there is a unique
convex body K e X" with support function f.
The uniqueness is clear. For the fundamental existence result we give
three proofs, each of which has its merits. The first proof is very short
(provided the duality theorem for convex functions is taken for
granted), the second one is perhaps the simplest and the third is of
historical interest.
First proof By Theorem 1.5.1, the function / is continuous and hence
closed. For each real A > 0 its conjugate function /* satisfies
f*(x) = sup{{x,u) - fiu)\ueE")
= sup {(a:. Am) - /(Am)|m e E"}
= Ar(x),
which is only possible if f*(x) e {0, »} for at e E". By Theorem 1.6.5,
/* is closed and convex, hence it is the indicator function Ik of a closed
convex set K, which is not empty because {/* = <»}=)= E". Theorem
1.6.5 further implies

1.7 The support function 39
/(") = /**(") = sup {(a:, u) - f*(x)\x 6 E"}
= sup {{x, u}\x 6 K}
= h{K, u).
Clearly K is bounded. ■
Thus the support function of a convex body (or, more generally, of a
closed convex set) is the conjugate of its indicator function.
For the second and third proofs we define
K := {x e l"\{x, v) ^ /(y) for all v el").
Then K, being the intersection of closed halfspaces, one for each
normal direction, is convex and compact, and if K ^'Z>, we evidently
have h(K, u)'^ f(u) for ueE". Therefore, it remains to show that
a: =|=0and h(K,u)^fiu)ioTueE".
Second proof. Since / is sublinear and continuous, its epigraph is a
closed convex cone in E" x U. Let u e E"\{o} be given. Since
(u,/(u)) 6 bdepi/, Theorem 1.3.2 implies the existence of a support
plane //(y,,,)„ to epi/ through (u, f(u)) such that epi/C //(",,)(,. From
Theorem 1.3.9 it follows that a = 0. Clearly r; =)= 0 since dom/=E",
hence we may assume that r}= -I. Then we have
epi/ C //(-.,-:),„ = {(v, V) 6 E" X U\{iy, -1), (v, v)) g 0}
and thus {y, v) =/(y) for all v e E". From (u, f(u)) e //q. _,)o we get
{y,u) = f(u). This shows that y e A", hence A" 4= 0, and h(K,u)^
fin). m
Third proof. The assertion K ^ 0 will be proved by induction with
respect to the codimension of the linearity space L[f] (cf. Lemma
1.5.5). If dim L[f] = n, then / is linear, say f(u) = {z, u) with z e E",
and then K = {z}. Suppose that dimL[/] = k< n and that in the case
of greater dimension of the linearity space the assertion is true. Let
.V6E"\L[/]. By Lemma L5.5, dimL[/'(A:; ■)]^k + l, hence the
induction hypothesis applied to the function f'(x; •), which is sublinear
by Lemma 1.5.4, yields the existence of a point y e E" satisfying
{y, v) ^ f'(x; v) for ally e E".
By Lemma,1.5.5, f'{x; •) =/. hence yeK. The assertion A" + 0 is
proved.
Let u 6 E"\{o}. Since f'{u; •) is sublinear, the result already proved
yields the existence of a point y e E" with
{y, v) g f'iu; v) ^ /(y) for v e E".

40 Basic convexity
This shows that y e K, and from {y, -u) '^f'{u\ -u) = -f(u) we get
hiK,u)^{y,u)^fiu). m
The directional derivatives of support functions appearing implicitly in
the last proof have a simple intuitive meaning: they yield the support
functions of the corresponding support sets.
Theorem 1.7.2. For K e IC" and u e E"\{o},
h'K(u\ x) = h(F(K, u), x) forx e E".
Proof By Lemma 1.5.4 and Theorem 1.7.1. h'ldu; •) is the support
function of a convex body K' that satisfies K' C K because
h'f:(u; •)^hK (Lemma 1.5.5). Let yeK'. Then {y, u) ^ hiK, u);
from {y,-u} '^ h'K(u;-u) = -h(K, u) we get {y,u) = h(K,u) and
hence v e FiK, u). Thus K' C Fi,K, u).
Let yeF(K,u). Then {y, u) = hi,K, u) and {y, v)-^ h{K, v) for
V e E". Choosing v = u + ?jc (k>0,x eE") we get
, , ^ h(K, u + Xx)- h(K, u)
iy.^)^ J
and then {y, x) = h'f((u\ x). This proves F(K, u)C K' and thus the
theorem. ■
Corollary 1.7.3. Let K e X" and u e E"\{o}. The support function h^ is
differentiable at u if and only if the support set F(K, u) contains only one
point X. In this case,
X = grad/2^-(M).
Proof. By Theorem 1.5.6, differentiability of /j^ at u is equivalent to
partial differentiability at u, hence to /2'^(m; e,) =-/2'k(m;-e,) for the
vectors e-i, . . . , e„ of an orthonormal basis. By Theorem 1.7.2 this
is equivalent to the fact that F(K, u) is contained in the intersection
of n pairwise orthogonal hyperplanes, hence to F(K, u) = {x} for some
point X.
If FiK, u) = {x}, then {x, u) = hiF(K, u), v) = h'^iu; v) for o e E",
in particular {x, e,) = h'ldu; e,) = 8,/2k(m), thus x = grad hidu). ■
Corollary 1.7.3 also follows from Theorem 1.5.12, but we have
preferred to give a more direct proof. The connection to Theorem
1.5.12 is established by the following observation.

1.7 The support function 41
Theorem 1.7.4. If K e X" and u e E"\{o}, then the subdifferential of the
support function h^at u is precisely the support set F(K, u).
Proof. By the definition of the subdifferential, x e 8/2^(m) if and only if
h(K,v)^h(K,u)+{x,v-u) foryeE". (*)
If a: 6 F(K, u), then {x, u) = h(K, u) and {x, v) ^ h(K. v) for vel\
hence (*) holds. If (*) holds, then
h(K - x,v)^ hiK - X, u) for v e E".
Applying this to y = o and to v = 2u, we get h(K - x, u) = 0, thus
h(K,u)= {x,u); then h(K - x,i>)^0 for v e E" shows that oe
K - X, hence x e K. Thus x e F(K, u). ■
We now come to the property of the support function that, for
subsequent applications, is of prime importance, namely its additive
behaviour (in the first argument) under addition. If K and L are convex
bodies, then K + L is also a convex body. The convexity is clear, and
compactness follows from the compactness of K x L and the continuity
of addition as a map from E" x E" to E". The addition of convex bodies
is usually called Minkowski addition. A function /on X" with values in
some abelian semigroup is called Minkowski additive if
f(K + L) = f{K) + /(L) for K, L e X".
Important examples are obtained as follows.
Theorem 1.7.5. For K, L e X" one has
(a) hiK + L, ■) = hiK, ■) + hiL, •),
(b) HiK + L, ■)= HiK, ■)+ HiL, ■),
(c) FiK + L,-) = FiK, ■) + FiL, •),
Proof Let u e E"\{o}. There are points x e K with hiK, u) = (x, u)
and y e L with hiL, u) = (y, u). This implies hiK, u) + hiL, u) =
{x + y, u) ^ hiK + L, u). Each point z e K + L has a representation
z = x + y with X e K, y e L, and it follows that {z, u) =
{x, u) + {y, u) ^ hiK, u) + hit, u). Since z € K + L was arbitrary,
one has hiK + L, u)'^ hiK, u) + hiL, u). This proves (a), and (b) is
an immediate consequence. Equality (c) follows from Theorem 1.7.2
and (a) or by an easy direct argument. ■
A first important consequence of Theorem 1.7.5(a) is that the
equality K + M = L + M for convex bodies K, L, Me X" implies
K = L. Hence, iX",+) is a commutative semigroup with cancellation

42 Basic convexity
law. It also has a natural multiplication with non-negative real numbers,
{k,K)^KK, satisfying the rules A(A: + L) = A/C + AL, (A + /z)A: =
AA: + nK, KfxK) = ikn)K, IK = K. Thus X" together with Minkowski
addition and this multiplication is an abstract convex cone. By the map
(where /i^ = /2k|5""') this abstract cone is isomorphically embedded
into the real vector space C(S"~^) of continuous real functions on 5""'.
The additivity of the support function (in the first argument) carries
over to some important functionals derived from it. Let K ^IL". The
number
^(A:, m) := h{K, u) + h(K, -u), u e 5"'',
is the width of K in the direction u; this is the distance between the two
support planes of K orthogonal to u. The maximum of the width
function,
D(K) := max {^(A:, u)\u e 5""'}
is at the same time the diameter of K, thus D(K) = diam K. In fact, if
X, y e K are points with maximal distance, then the hyperplanes through
X and y orthogonal to x - y are support planes to K. On the other
hand, to each u e 5""' there are points x,y e K with w(K, u)^\x - y\.
The minimal width
A(K) := min{H'(A:, u)\u e 5""'}
is called, in brief, the width of K. (Of course, diameter and width are
not Minkowski additive; one merely has D(K + L)= D(K) + D(L)
and A(K + L) ^ AiK) + 4(L) for K, L e X".)
The mean value of the width function over 5""' is called the mean
width b, thus
b(K) := — |^„„/2(A:, u)dZ"-\u). A.7.2)
The vector-valued integral
siK) := ~ L,/2(/C, u)udX"-\u) A.7.3)
defines the Steiner point s(K) of K.
It is clear that mean width b and Steiner point s are Minkowski-
additive functions. They also have important invariance properties. The
mean width is rigid motion invariant, that is, it satisfies b(gK) = b(K)
for K e X" and each rigid motion g:E"—»E". This is obvious from
A.7.2), since for a rotation g we have

1.7 The support function 43
h(gK, u) = h(K, g-'u) A.7.4)
and the spherical Lebesgue measure is rotation invariant; further, for a
translation by a vector f e E" we have
h(K + t, u)= h(K, u) + {t, u).
The invariance of b then follows from
which holds since the integrand is an odd function on 5""'.
The Steiner point is equivariant under rigid motions, that is to say,
s{gK) = gs(K) for each rigid motion g. For a rotation this follows from
A.7.3) and A.7.4), and for translations one has to use
~ \,^_^{t, u}udX"-\u)= t forfeE". A.7.5)
For the proof, note that
L
with a real number a independent of t, since the integral is linear in t
and invariant under rotations and reflections fixing t. Choosing |f| = 1
we obtain, if (ej, . . . , e„) is an orthonormal basis of E",
= -i \u\'dX"-'iu) = K„.
We mention one further property of the Steiner point, namely
5(/i:) 6 relint/C for A: e 1^". A.7.6)
This will be clear when in Section 5.4 the Steiner point s{K) is
interpreted as the centroid of a certain measure concentrated on the
relative boundary of K (formula E.4.11) for r = n).
Besides the support function, other functions can be used to describe
a convex body analytically, but in our context these will be of less
importance.
Let K 6 .^0 be a convex body such that o e'xnXK. The function
g{K, •) defined by
g{K, x) ■■= min (A S 0|a: e kK} for x e I"
is called the gauge function of K, and
piK, x) ■■= max{k^O\Xx e K} for a: e E"\{o}
^ 1
" 8iK, X)

44 Basic coiivexily
defines the radial function p(K,•) of K. It i$ immediately clear from
the definitions that
K = {x e l"\giK,x)^ 1},
Ul
g(K,x) = ~- ifA:6E"\{o}, A>0, A;«- e bd A:
\kx\
and
piK, x)x ebdK for a: e E"\{o}. A.7.7)
The gauge function g{K, •) is sublinear; this follows easily from the
definition, but also from the following result.
Theorem 1.7.6. For K € IC" with os'xntK,
giK, •)= KK*, •)•
Proo/. Writing K ■= {x el"\hiK*,x)^ I}, we have giK,-) =
h(K*, •) by the definition of the gauge function. Let xeK. For
u e K* we have {u, x) ^ h(K*,x)^l, hence xeK** = K. Thus
K C K. Let X e K. There is some v e K* with hiK*, x)= {v,x)^l,
hence x e K. Thus K = K. ■
Remark 1.7.7. By Theorem 1.7.6 we have
which expresses a simple way to find the boundary points of the polar
body K* from the support planes of the body K. Further, let x ehd K,
let u be an outer unit normal vector of K at x, and let
)' = \y\uebdK*. Then
{y, x) = {u, x)/g(K*, u) = {u, x)/h(K, u) = 1
= g(K, x) = h(K*, x),
hence x is an outer normal vector to K* at y. (A general version of this
remark is contained in Lemma 2.2.3.)
Remark 1.7.8. If K e X" with oeint/C is centrally symmetric with
respect to o (i.e., K = -K), then
ll.vll^ := g(K,x) foTX e\E"
defines a norm \\-\\k on E" for which K is the unit ball. If we identify
(as we always do) the dual space of E" with E" itself via the scalar
product ( •, •), then the dual Banach space (E", || • \\*k) of the Banach
space (E", || • lU) has its norm given by

Notes for Section 1.7 45
1*^ =sup{{y,x}\x€\E",\\x\\K ^1}
= supl^y, x)\x e K}
= hiK, y) = giK*, y),
and its unit ball is K*.
We return to support functions and conclude this section with a useful
remark on differences of support functions.
Lemma 1.7.9. ///:5"~'—»IR is a twice continuously differentiate
function, there exists a convex body K e X" and a number r >0 such that
f=hK-h,B.. A.7.8)
Hence, the real vector space spanned by the differences of support
functions (restricted to S"~^) is dense in the space C(S"~^) (with the
maximum norm).
Proof We extend / to E"\{o} by putting f(x) := |Ar|/(Ar/|Ar|), and we
define gix) := |Ar| = h(B",x) for x e E". Then the function f + rg is
positively homogeneous of degree one, for any r eU. Let d^(/ + rg),
denote its second differential at x, considered as a bilinear form on E".
By homogeneity, d^(/+ rg)^(x, •) = 0, and for unit vectors x, y with
y J. AT we obtain d^(/ + rg).,(y, y) = d^f^iy, y) + r. By continuity and
compactness, d^fxiy, y) attains a minimum, hence we can choose r > 0
so that d'^if + rg)x(y,y)^0 for at 6 5""' and arbitrary y. By
homogeneity, d^(/+rg), is positive semidefinite for all Ar6E"\{o}.
Theorem 1.5.10 now shows that f + rg is convex and thus, by Theorem
1.7.1, that it is the support function of a convex body K that satisfies
A.7.8). The rest of the assertion is clear. ■
Notes for Section 1.7
1. Determination of a convex body by its support function. The second proof
given for Theorem 1.7.1 essentially goes back to Rademacher A921). It also
appears in Hille & Phillips A957), p. 253, where it is attributed to Fenchel.
Separation or support properties are used in several versions of the proof
appearing in the literature, at least implicitly, as in our first proof, which can
be found in Rockafellar [29]. These methods have the advantage of being
extendable to infinite dimensions. Our third proof is the one of Bonnesen &
Fenchel [8]. Yet other proofs are due to Aleksandrov A939a), who used the
topological theorem on the invariance of domain, and to McMuIlen A976a),
who found a particularly elementary approach involving Helly's theorem.
2. Continuous convex .sets. The nonempty closed convex sets K in E" for which
the support function
h(K, u) = sup{(x, u)\x G K} for u e 5""'

46 Basic convexity
(restricted to the unit sphere) is continuous (with the usual definition of
continuity for functions with values in IR U {<»}) have been called continuous
convex .sets by Gale & Klee A959). These authors have characterized those
sets in several ways and have shown that the system tS" of continuous convex
sets shares some properties with X". For instance. A, B e A" implies that
A + B &S". con\(A U B) e S" and that A and B can be strongly separated
if they are disjoint.
7>. The support function of an intersection. Let (K,),^i (where / is an arbitrary
index set) be a family of convex bodies in X" and suppose that their
intersection K is not empty. Then the support function of K can be
represented in the form
h(K,u) = ini\2h(K„u,Ju, = u],
Vic/ i€/ I
where the infimum is taken over all representations u = ^i<, with m, = o for
all but finitely many i e /. This was proved by Sandgren A9,'54), where the
result is attributed to F. Riesz, and was rediscovered by Kneser A970). it can
also be deduced from the more general Theorem 5.6 in Rockafcllar [29].
4. The semiaxis funciion. Let K e Xq be a convex body with oe'mtK. For
mg5""', the value h(K, u) of the support function is the distance of the
support plane to K with exterior normal vector u from o, while u/g{K, u) is
the boundary point of K on the ray with endpoint o and direction u. The
polarity K >-^ K* interchanges hyperplanes and boundary points, hence
support function and gauge function interchange their roles under this
polarity (Theorem 1.7.6). LeichtweiB A965) (see also LeichtweiB [23], §13)
suggested a way of describing convex bodies by a class of functions that serve
equally well for a body and its polar. Instead of support planes or boundary
points one has to consider support elements. A support element of J<^ is a pair
(a-, u) where x is a boundary point of K and u is an exterior unit normal
vector io K at X. Let Nor K denote the set of all support elements of K with
its natural topology. The map o: Nor K ^ S"'^ defined by
,, ,, Xo + U X
o((x, u)) := -—r, X(^ = —,
■Xn + u\ \x\
is a homeomorphism. To each ye 5""' there exists a unique two-sheeted
hyperboloid of revolution, with axis of revolution through o and of direction
i>. that touches K at a"'(()). Let a(K, o) denote the length of the semiaxis of
this hyperboloid. The semiaxis function a(K, ■) determines K uniquely, and
one has a(K*, •) =a(K, •)"'. The semiaxis function was further
investigated by Baum A973a, b), who also showed how to express the boundary of K
explicitly in terms of the semiaxis function and who obtained necessary and
sufficient conditions for a function to be a semiaxis function. Later Baum
A987) defined, for K e X" with o e int K, a function F^: E" -* U by
Fdu) ■■= sup{2(M, yy - ly\'\y e K, (u, y) g 0};
then the semiaxis function a(K, •) is the restriction of VfV to 5""'. The
supremum in the definition of F^ is attained at a unique point y g K. The
function F^ is convex and differentiate. Baum also considered infinite-
dimensional generalizations of these concepts. For boundary representations,
see Baum A989).
5. Support flats to convex bodies. The support function of a convex body
K G X" determines the position of its support planes, and the gauge function
of K (if o G int K) fixes its boundary points on rays through the origin. The
following common generalization was investigated by Firey A973). Let
<^ G {0, . . . , M - 1}. To each q-flat £, tangent to 5""', take a translate E',, in

1.8 The Hausdorff metric 47
the subspace spanned by Eg and o. What conditions are necessary and
sufficient for the family {E'^} to be the full set of supporting <7-flats to some
convex body K7 (Eg supports K ii Eg n K ^ 0 and Eg is contained in a
support plane of K.) Firey defined the ^-support function Hg of a convex
body by means of the distance of the flats Eg from o, and he answered the
question completely by characterizing the ^-support functions of convex
bodies containing o. He also treated some applications.
6. A convexity criterion. In the paper mentioned in the preceding note, Firey
A973) also extended Rademacher's plane polar coordinate test for convexity
(Bonnesen & Fenchel [8], p. 28) to higher dimensions: If a given function
/i:5"''^IR is extended to E" by positive homogeneity, that is, by
h(Xu) = Xh(u) for A SO, when is the extension subadditive and hence a
support function? Firey proved that this holds if and only if
(Mi,ei) ... {Ui, €„) /i(Mi)
det
{u„, e,)
{o,e,)
{u„, e„)
{", e„)
h(u„)
hi»)
(M,.e,) ... (M,,e„)
X det I ; i I g 0
^{u,„ e,) ... (m„, e„)/
for all choices of M], .. ., u„, v e 5""' for which o e pos {m,, . . ., m„}; here
(?i, . . . , e„ is an orthonormal basis of E".
In the plane, one has another simple criterion. A function /:IR-»IR of
period In is said to be a sub-sine function if the following holds: If f(a)
agrees with /I cos a-+ B sin a- at a=a\, o-i, where a\< ai< ax + n. then
/(a-) = Aco^a + B sin or for c(\< a< oci- Now there is a support function h
on E^ for which /(a-) = h(e\CO'i,a + easino-) if and only if / is a sub-sine
function (see Green 1950). Another formulation is found in Kallay A974).
7. Lemma 1.7.9 appears, more or less explicitly,- at several places in the
literature, e.g., Ewald A965), Valette A971), Schneider A971a). The last
part of the lemma was proved in a different way by Fenchel & Jessen A938).
Differences of support functions were investigated, by, e.g., Weil {1974a).
8. Power means of support function's. For K G X^ and x € K, let hi,(x) be the
(-fc)th power mean oi h(K - x. ■) over 5""'. Aleksandrov A967) obtained
several estimates for the maximum of h/,.
1.8. The HausdorfT metric
The set X" of convex bodies can be made into a metric space in several
geometrically reasonable ways. The Hausdorff metric is particularly
convenient and applicable. The natural domain for this metric is the set
<^-'" of nonempty compact subsets of E". We shall first treat the

48 Basic convexity
Hausdorff metric on this more general class of sets; this will involve no
extra effort and is useful in several respects.
For K, L e&" the Hausdorff distance is defined by
d{K, L) := max (sup inf \x - y\, sup inf \x - y\] A.8.1)
xeK y€L xeL y€K
or, equivalently, by
8(K, L) = min (A g 0|A: C L + kB", L C K + kB"}. A.8.2)
Then <5 is a metric on €*", the Hausdorff metric.
Since we are considering compact sets, sup and inf in A.8.1) may be
replaced by max and min. Of the two definitions, A.8.1) can be used in
arbitrary metric spaces if Iat - yl is replaced by the distance between x
and y in terms of the metric. The equivalence of A.8.1) and A.8.2) in
E" is seen as follows. Denote the right-hand side of A.8.2) by a. For
X e K one then has x e L + aB", hence x = y + ab with suitable y e L
and b e B"; hence \x — y\^ a and, therefore, inf^.^^, Iat - y| ^ a. Since
X e K was arbitrary, sup;,^^ infyg/, Iat - yl ^ a. Interchanging K and L,
we get 6(K,L)^a. Let 0<A<q' and, say, K(^L + kB". Then
X ^ L + kB" for some x e K, hence Iat - yl ^ A for all y e L. This
yields d(K, L) ^ A. Since A < o- was arbitrary, we get 6(K, L)^ a and
thus 6(K, L) = a, as asserted.
That d is in fact a metric is easy to see. To prove, for instance, the
triangle inequality, let K, L, M e (d" and put <5( A", L) = a, <5(L, M) = /3.
Then A: C L + aB" and L C A/ + fiB", hence A C A/ + fiB" + ocB" =
A/ + (a + fi)B"\ analogously, A/ C A + (a + /j)fi", hence b{K, M) ^
a+ p.
In the following, all metrical and topological notions occurring in
connection with 6" or X" are tacitly understood to refer to the
Hausdorff metric and the topology induced by it.
We want to show (in a stronger form) that the metric space Fi", 6) is
locally compact. A few preparations are needed.
Lemma 1.8.1. If (A,),g^ is a decreasing sequence in (^", that is, if
A,+i C Ki for I 6 N, then
X
lim Ki = C\^i-
Proof K '■= riJLiA; is compact and not empty. If the assertion were
false, then K„(^ K + eB" for all meN with some fixed £ > 0. Let
A„-= K„'^nt(K + cB"), then (/4„)„gM is a decreasing sequence of
nonempty, compact sets and hence has nonempty intersection A.
Clearly A r\ K =0, but A„ C Km implies A G K, a contradiction. ■

1.8 The Hausdorff metric 49
Theorem 1.8.2. The metric space (f/", d) is complete.
Proof. Let (/C,);^^ be a Cauchy sequence in ^^". Put A„ :=clUr=m^i'
Then (/4„),„gf^ is a decreasing sequence of nonempty compact sets (the
boundedness of U^, is an immediate consequence of the Cauchy
property), hence Lemma L81 yields A„^»A := n,g^>i, for m—»3c.
Hence, for given £ > 0, there exists «(, e N such that A„, C A + eS" for
m S «(), hence K,C A + cB" for / S no- Since (/C,),gfs>; is a Cauchy
sequence, there exists nj S «(> with KjC K, + eB" for i, j = ni. Thus
for /, m g n, one has [jJ=„Kj C K, + eB" and hence A„ C K; + eB",
which implies A C K, + e.B". We have proved that <5(/C„y4)^£ for
i^ rii, from which the assertion follows. ■
Theorem 1.8.3. In {f<", 6) each closed, bounded subset is compact.
In particular, the space F'",<5) is locally compact. By a known
compactness criterion in metric spaces. Theorem L8.3 is a con.sequence
of the following result.
Theorem 1.8.4. From each bounded sequence in (^" one can select a
convergent subsequence.
Proof. Let (/^"),gfsj be a sequence in 0" whose elements are contained
in some cube C of edge length y. For each m e N, the cube C can be
written as a union of 2""" cubes of edge length 2""'/. For K e 6" let
A„{K) denote the union of all such cubes that meet K. Since (for each
m) the number of subcubes is finite, the sequence (/^,),g^ has a
subsequence (/^;),g^ such that Ai{K])=.Ti is independent of /.
Similarly, there is a union T2 of subcubes of edge length 2~^y and a
subsequence (/^?),e?g of i^K]);^^ such that Ai^k]) = Ti. Continuing in
this way, we obtain a sequence {T^)„^^ of unions of subcubes (of edge
length 2~'"v for given m) and to each m a sequence (/^r);€K such that
A„AKr) = T„ (*)
and
(^DieN is a subsequence of (/^f),g^ for k < m. (**)
By (*) we have K'" <^ K"'+ KB" with A = 2-'" V^, hence
^KT, Kj).^2-'"y/7^ (i,j, meN) and thus by (**)
S(K^\ kI) ^ 2-'"V~^ for /, / 6 N and A: S m.
For A:„ := A:^ it follows that
6(K„, K^)^2-'"y/7iy for k ^ m.

50 Basic convexity
Thus iK„)„^f^ is a Cauchy sequence and hence convergent by Theorem
1.8.2. This is the subsequence that proves the assertion. ■
From now on we again restrict our considerations to the space X" of
convex bodies.
Theorem 1.8.5. X" is a closed subset of (/".
Proof. Let K ef;"\lC". Then there are points x, y e K and numbers
A e @,1), e>0 such that Biz,e)n K =0 for z = A - A)jc + ky. Let
K'ef^" satisfy 6{K,K')<e/2. There are points x',y'€K' with
\x' - x\< e/2 and \y' - v| < e/2, hence the point z' ■= A - A)jc' + A)-'
satisfies |^'-^|<e/2. If z'^ K', then there is a point weK with
\w - ^'1 < e/2, hence with Iw - ^| < e, a contradiction. Thus K' is not
convex. We have proved that f'."\X" is open. ■
Theorems 1.8.4 and 1.8.5 together yield the famous Blaschke
selection theorem:
Theorem 1.8.6. From each bounded sequence of convex bodies one can
select a subsequence converging to a convex body.
This theorem is a very useful tool in proving the existence of convex
bodies with various specific properties.
It is sometimes convenient to have a description of the convergence
of convex bodies in terms of convergent sequences of points.
Theorem 1.8.7. The convergence \\mi^o^K,= K in X" is equivalent to
the following conditions taken together:
(a) each point in K is the limit of a sequence (x,),^?^ with jc, e Kj for
/ e !^;
(b) the limit of any convergent sequence (xi^j^f^ with jc,^ e /C,^ for j eH
belongs to K.
Proof Assume that K,-^ K for i^^. Lei x e K. Put jc, := piK„x).
Then x, e K, and \x - x\ % diK, K) -^ 0 for / ^ ao. Thus (a) holds. Let
{ij)j^H be an increasing sequence in N and suppose that x,^ e K,^ (j e N)
and x,^ -^ jc for ;■ ^ oc, but jc $ /C. Then 5(jc, p)n{K + 'pB")'= 0 for
some p>0. But for / sufficiently large we have |jc,-jc|<p and
jc, e /C, C /C + pB". a contradiction. Thus (b) holds.

1.8 The Hausdorff metric 51
Now assume that (a) and (b) are satisfied. Let e >0 be given. We
have to show that
K C Ki + eB" for all large /, (*)
KiC K + eB" for all large /. (**)
If (*) is false, there exists a sequence (>',,)y6\j in K, converging to some
yeK, with ^(/C,, y,) ^ e (jeN). By (a), there exist jc, e Af, with
jc, —» y for / —» 00. Then |jc, - y, | —» 0 for / ^ sc, a contradiction. Hence
(*) holds. If (**) is false, there exists a sequence iyi)j^M with y, e Af,
and yj i K + eB" for ; e N. By (a) there exist points jc, e /C, with
Xi e K + eB" for sufficiently large ;. By the convexity of K^ (only the
connectedness is needed) there are points z, e K, nbd{K + eB"). The
sequence (z,)ye\j has a convergent subsequence whose limit must belong
to K by (b), a contradiction. Hence (**) holds. ■
Having endowed 6'" and .">Sf" with a topology, we should check the
continuity of the mappings that we have encountered in previous
sections. Some of them are even Lipschitz mappings, for trivial reasons.
This holds for the convex hull operator as a map from (k" to ."?". in fact
^convA:,convL) ^ 6iK, L).
and for addition and union as maps from &" x ^" to 6*", since
diK + K', L + L')^ diK, L) + 6{K', L'),
diK U K', LU L') ^ max {diK, L), 6{K', L')).
The intersection as a map from 6" x d" to 6" is not continuous. For
convex bodies, however, the following can be shown.
Theorem 1.8.8. Let K, L e DC" be convex bodies that cannot be separated
by a hyperplane. If Kj, L, e X" (i e N) are convex bodies with K,^ K
and Li^ L for i ^ x>, then Af, Ci Li^0 for almost all i and AT, n L,
-^ K n Lfor i^cc.
Proof We use the criterion 1.8.7. Let x e K n L (K n L^0 since
otherwise K and L could be separated). Define jc, := p(Af, D L„ x) for
those i for which K, n L, 4= 0- We assert that jc, exists for almost all /
and that x^—^x for i—*x. If this is false, there exists a ball B with
centre x (and positive radius) such that K,n L,n B = 0 for infinitely
many /..For sufficiently large / we have AT, D B ^ 0. since x e K and
Kj^ K; similarly LjCt B^0 for large /. Thus there is a sequence
iff,)i^K of hyperplanes H^^ separating KjdB and L^nB. A
subsequence converges to a hyperplane H, and H separates K n B and
L n B (as one easily sees using Theorem L8.7). Since x e K n L r\ B,

52 Basic convexity
necessarily x e H. From the convexity of K and L it now follows that
H separates K and L, a contradiction. Hence jc,^^ jc for i—* ^.
On the other hand, if iij)ii\ is an increasing sequence in N, if
A-, e Af,, n L, (;' e N) and x,^ -^ y for j^^, then clearly y e /C n L by
Theorem 1.8.7. Now it follows from this theorem that K,ri L,^ K d L
for / ^ ao. ■
The following example shows that the non-separability condition in
Theorem 1.8.8 cannot be omitted. Let K = L = L, = [x, y\, where
.V 4= y. and let Ki = [x, y,] be a segment with [x, yj n [jc, y] = {x} and
such that yi^y for /-^oo. Then it follows that Af,^ K and L,^ L,
but it does not follow that A:, n L,^ /C n L.
Next we .show that the metric projection is continuous in both
variables simultaneously, that is. the map
p;lC" X \E"-> E"
(K, x)^ piK, x)
is continuous. This is a consequence of the following more precise
result.
Lemma 1.8.9. Let K, L e X", x, y e E", and put D := diam(A: U L U
{x, y}). Then
\p{K, x) - piL, y)\ ^ |jc - y| + V 5DdiK, L) .
Proof. Taking Theorem 1.2.2 into account (and the triangle inequality),
we only have to show that
\p{K, x) - PiL, x)\ ^ V^DdiK, L).
If X e K r\ L, this is trivial. We assume therefore, say, x ^ K. Put
d{K, x) = d and d{K, L) = 6. The ball B{p{K, x), 6) contains a point of
L, hence ^(L, jc) ^ ^ + 6 and thus p{L, x) e B{x, d+ &). Let H~ be
the supporting halfspace of K with exterior normal vector u{K,x)-
Then L is contained in the translate W" + 6u{K, x), hence
p{L, x) e B{x, ^ + C) n [H- + du{K, x)\.
Elementary geometry now yields
\p{K, x) - PiL, x)\ S V^dd + 6" ^ V5D6,
since d ^ D and d^ D. ■
Also the support function h: X" x E" ^ K is continuous as a function
of two variables; it is in fact locally Lipschitz continuous, as a
consequence of the following result.

1.8 The Hausdorff metric 53
Lemma 1.8.10. For K, L e IC" with K,LCRB" (« > 0) and for u,
\h{K, u) - hit, i>)\ ^ R\u - v\ + max{|w|, \v\NiK, L).
Proof. Put d{K, L) = 6 and choose a point xeK with h{K,u) =
{x,u). From x e K d L + 68" we get {x,v) ^ h{L + 68".v) =
h{L,v) + 6\u\, hence h{K, u) - hit, v) ^ (x, u - v) + 6\v\ g |jc||w - i;|
+ d\v\ ^ R\u - v\ + d\i)\. The assertion follows after interchanging
(K, u) and iL,v). ■
In terms of the support function, the convergence of convex bodies
can be treated very conveniently, as a result of the following simple
observation. Here || • || denotes the maximum norm for real functions on
Theorem I.8.II. For K, L e IT",
diK, L) = sup \h{K, u) - h{L, u)\ =: \\ht: - hi\\.
Proof Let 6{K, L) g a. Then KC L + aB", hence hiK, u) g
hiL + a8", u) = h{L, w) + a for w e 5"~'. Interchanging K and L, we
get \h{K, u) - h{L, u)\ ^ a for w e S"'^ and thus H^^' " ^lII = «. The
argument can be reversed. ■
From Theorem 1.8.11 and the definitions A.7.2) and A.7.3) it follows
immediately that the mean width b and the Steiner point s are Lipschitz
maps. In both cases, the optimal Lipschitz constants are easily
determined. From A.7.2) one obtains
\h{K) - b{L)\ ^ 2diK, L)
for KyLeJC", with equality if and only if K = L + a8" or
L= K + a8" with a^O. For given K, L e X", take o e 5""' so that
siK) - s(L)| = (siK) - i(L),;;); then
\siK) - siL)\ = — i„JhiK, u) - hiL, u)]{u, v}dX"-\u)
^ ^ diK. L) I \{u, o}\d^,"-\u).
The last. integral is twice the (n - l)-dimensional measure of the
orthogonal projection of 5""' on to a hyperplane orthogonal to o (this is
easy to see and is a special case of E.3.32)). Thus
\siK) - siL)\ ^-^^^^ 6{K, L). A.8.3)

54 Basic convexity
It is clear that ior K j= L equality in A.8.3) cannot be attained, and it is
easy to see that the constant cannot be replaced by a smaller one.
Theorem 1.8.11 shows that the map cp defined by A.7.1) maps X"
not only isomorphically as a convex cone into the vector space CiS"~^),
but also isometrically with respect to the Hausdorff metric on X" and
the metric induced on C{S"~^) by the maximum norm. In particular,
convergence of convex bodies is equivalent to uniform convergence on
5"~' of the corresponding support functions. This uniform convergence
follows in any case from pointwise convergence.
Theorem 1.8.12. If a sequence of support functions converges pointwise
(on E" or, equivalently, on S"~^), then it converges uniformly on 5"~' to
a support function.
Proof. Let {h{Ki, • ))i6M be a pointwise converging sequence of
support functions. Clearly the limit function, say h, is subhnear and, hence,
a support function by Theorem 1.7.1. For each we 5""', a{u) ■ =
s\ip,h{Ki, u) is finite. We can choose a number /? > 0 with
I I t^u.aC) ^ RB",
uiS"-'
then Ki C RB" for / e N. Let hj be the restriction of hiKj, ■) to S"~^;
then \hi{u) - hi{v)\ ^ R\u - v\ by Lemma 1.8.10. Choose a finite set
5C5"~' such that to each u e S"~^ there is some veS with
\u - d| < e/3R. Since S is finite, there exists noeN such that
\hiiv)- hiiv)\< e/3 for ij^n^ and all veS. Now let u e S"~\
Choose V e S such that |w - d| < e/3R. Then
Mu) - hjiu)\ ^ \hiiu) - hM + \hiiv) - hjiv)\ + \hjiv) - hjiu)\
^ R\u - v\ + e/3 + R\u - v\< e
for /, ;■ ^ no and hence |/2,(w) - h{u)\ = e for / ^ no- Since u e 5""' was
arbitrary, the uniform convergence on S"~^ of the sequence (/z,),^:^ to h
is proved. ■
Having introduced a metric on the set JC" of convex bodies, we can
now consider approximation. The approximation of general convex
bodies by simpler ones such as polytopes or bodies with differentiable
boundaries is a useful tool for many investigations. Here we note only
the most basic facts on approximation by polytopes. Further
approximation results will be treated in Sections 2.4 and 3.3.
Theorem 1.8.13. Let K e X" and e > 0. Then there is a polytope P e X"
with PC KG P+ eB", hence with 6{K, P) ^ e.

1.8 The Hausdorff metric 55
Proof. Cover K by finitely many balls with radius e and centres in K,
and let P be the convex hull of their centres. Evidently P satisfies the
requirements. ■
Remark. In the preceding proof we may impose the condition that the
centres of the balls have rational coordinates. Hence, the space (,X", d)
is separable, that is, it has a countable dense subset.
Lemma 1.8.14. Let Af,,/w2 e .X" and /w2Cint/wi. Then there is a
number t)>0 such that each convex body K e X" with d{Ki, K) < t)
satisfies KiC K.
Proof. Since K2 C int K^, the function h{Ki, •) - h{K2, •) is positive
on E"\{o} and hence, being continuous, attains a positive minimum t]
on S"~\ Let K e X" be a convex body with diKi,K)<T). Then
\hiKi,u)- hiK,u)\<-n and hence hiK2, u)^ hiK^u) - ■n< hiK, u)
for ueS"-^;thus K2CK. ■
Theorem 1.8.15. Let K e X" and o e int K. For each A > 1 there exists a
polytope P e X" with PC K CkP.
Proof Choose a number p > 0 with B(o, p) C int K and a number e
with 0<eg(A- l)p. By Theorem 1.8.13 and Lemma 1.8.14 we can
choose e small enough that there is a polytope P with
B(o, p)C P C K C P + eB".
For w e 5""^ we deduce
hiXP, u) = h{P, u) + (A - \)h{P, u)
^ h{P, u) + e = hiP + eB", u)
^ h{K, u),
thus KdXP. ■
We prove a further continuity result. The volume functional V„ on
X" is defined as the restriction of the n-dimensional measure .^"
to_X".
Theorem 1.8.16. The volume functional V„ is continuous on X".
Proof_Lti KeX" be given. If V„(A:) = 0 and K e X" satisfies
6{K, K) = a^l (without loss of generahty), then K lies in a hyper-
plane and ^ C a: + aB". Hence, V„(^) ^V„iK + aB") ^ c{K)a with

56 Basic convexity
ciK) independent of a, by an easy estimation using Fubini's theorem.
Suppose, therefore, that V„{K)>0. Then we may assume that
oe int a:. Let e><) be given. Choose A> 1 with (A" - 1)A"V„(A:)< e
and p> 0 with pB" C int /C. By Lemma 1.8.14, there is a number a > 0
with a^(A-l)p and such that pB" C K for all R e X" satisfying
6(K, K)< a. Suppose that the latter holds. Then
a: C ^ + aB" C ^ + (A - l)pB" C ^ + (A - 1)^ = A^
and similarly K <ZXK. It follows that
V„{K) ^ V„{XK) + )."V„{K),
hence
V„{K) - V„{K) % (A" - l)V„{K) ^ (A" - 1)A"V„(/C),
V„{K) - V„{K) ^ (A" - l)V„{K) g (A" - \)X"V„{K),
thus
\VM) - yAK)\ g (A" - \)k"V„{K) <e. ■
Notes for Section 1.8
1. Hausdorff metric and topologies on spaces of subsets. The Hausdorff metric
is usually considered in a more general context. Let {X, d) be a metric
space. Then
(%C, D) := max {sun inf d{x, >•), sup inf d{x, y))
xcC yil) Xib yiC
defines the Hausdorff metric 6 on the set of nonempty closed and bounded
subsets of X. It was introduced by Hausdorff A914, pp. 293ff), who also
described the convergence with respect to this metric in terms of closed
limits (see also Hausdorff 1927, p. 149).
Let (^{X) denote the set of nonempty compact subsets of X. Different
common topologies introduced on spaces of subsets of a topological space
coincide, on Q{X), with the topology induced by the Hausdorff metric. This
is true for the Vietoris topology and for the topology of closed convergence.
The latter, which is usually considered on the set of closed subsets of a
locally compact, second countable Hausdorff space, has applications in
mathematical economics (Hildenbrand 1974) and stochastic geometry (Ma-
theron 1975). For properties of the metric space ((^(X), S) we refer the
reader to Kuratowski A968), §42, and for general investigations on
topologies for spaces of subsets to Michael A951) and the monograph of Klein &
Thompson A984).
2. Selection theorems. The Blaschke selection theorem for convex bodies was
first stated and used by Blaschke [5] (pp. 62ff). A general theorem, which
implies the compactness of ((-ti^X), h) for a compact metric space X. appears
in Hausdorff A927), p. 147; see also Kuratowski A968), §42 and Rogers
A970), p. 91. An elegant proof for compact subsets of E" was given by
Hadwiger A949a), [18], Section 4.3.2; we have followed his approach. A
ver>' general form appears in Dierolf A970).
By Lemma 1.8.10, one has \h{K, u)- h{K, v)\ g R\u - v\ for K e T"
with K C RB" (for given /? > 0) and for «, d 6 E". Therefore, the family
{hfclK 6 X", K C RB")

Notes for Section 1.8 57
is equicontinuous; further, it is uniformly bounded and closed in 0E""')
(with the maximum norm). By the Arzel^-Ascoli theorem, it is compact.
This is another proof of the Blaschke selection theorem. Two different
methods of deducing the Blaschke selection theorem from the Arzel^-Ascoli
theorem were described by Heil A967a); he also deduced the selection
theorem for compact sets in this way.
Selection theorems for star-shaped sets and for more general sets were
treated by Hirose A965), DreSevie A970), Beer A975) and Spiegel A975).
Convergence in sense of the Hausdorff metric. Theorem 1.8.7 can be
generalized and refined in the following way. We consider the space &" of
nonempty compact sets in E". Define
X{K, L) := max min |.r - >'| for K, L e 6",
SO that d(K, L) = max{X(K, L),?.(L, K)}. For K e &" and a sequence
('^/)/€N in (^", the following assertions may hold.
A) \iK, K,)-^0 for !^ 00.
B) Each point .r e A^ is the limit of a sequence (.r,),,:^ with x, e K,{i e N).
C) Each open set intersecting K intersects K, for almost all /.
(T) X{K„K)^0 for!-*=c.
B') The limit of any convergent sequence (^,,),e% with x,^ e K,, (j e N)
belongs to K, and the sequence (A^,),^^ is bounded.
C') Each closed set having empty intersection with K has empty
intersection with almost all K,.
Then the following can be shown: The conditions A), B), C) are
equivalent; the conditions A'), B'), C') are equivalent. Hence
lim /c, = /C « B) and B') •» C) and C')-
The proof is left to the reader.
Convergence of closed convex sets. On the set of (not necessarily bounded)
closed convex subsets of E" several different types of convergence have been
considered, which coincide on X". A comparison of four such types of
convergence is made in Salinetti & Wets A979). See also Wijsman
A964,1966) and Beer A989).
Topology of hyperspaces. For a nonempty subset X of E", the space X{X)
(of convex bodies contained in A"), metrized by the Hausdorff metric, is also
called the cc-hyperspace of X. The topology of X{X), in dependence on the
geometry of X, has received some attention. The following results are due
to Nadle'r, Quinn & Stavrakas A975,1977, 1979); see also Chapter XVIII of
Nadler A978). If ^ e i:" and dim^ §2, then X{X) is homeomorphic to
the Hilbert cube /. = nr=i[-l/2M/2']. For «g2, :^(int 5") and X{1")
are homeomorphic to /« with a point removed. If A* C E^ is compact,
connected and such that X{X) is homeomorphic to /», then X is topologic-
ally a 2-cell. Curtis, Quinn & Schori A977) found a complete
characterization of the polyhedral 2-cells whose cc-hyperspaces are homemorphic to /».
Embedding theorems for spaces of convex sets. The map
<p:X"^ 0E"-')
. • K^h^
given by A.7.1) maps the cone of convex bodies in E" isomorphically, and
by Theorem 1.8.11 also isometrically, onto a closed convex cone in a Banach
space. For certain spaces of convex subsets of infinite-dimensional vector
spaces, similar embedding theorems are known. R^dstrOm A952) proved the
following result.

58 Basic convexity
Let L be a real normed linear space and M a set of closed, bounded
convex sets in L with the following properties.
A) M is closed under addition and multiplication by non-negative scalars.
B) U A e M and S is the unit ball of L, then >1 + 5 is closed.
C) M is metrized by the Hausdorff metric.
Then M can be embedded as a convex cone in a real normed linear space N
in such a way that
(a) the embedding is isometric,
(b) addition in N induces addition in M,
(c) multiplication by non-negative scalars in N induces the corresponding
operation in M.
R^dstrom showed that the conditions on M are satisfied, for instance, by the
set of all finite-dimensional compact convex sets and by the set of all
compact convex sets.
RSdstrom proof was by means of an abstract construction, but it was
pointed out later that one can also use support functionals in a similar way
to their use in the finite-dimensional case. For locally convex spaces this was
done by Hormander A955), who proved also a generalization of Theorem
1.7.1, characterizing the support functionals of closed convex sets in such a
space (essentially by an argument that can be viewed as extending Rade-
macher's method, which is given as the second proof of Theorem 1.7.1).
More general embedding theorems of a similar kind were obtained by
Godet-Thobie and The Lai A970), Urbanski A976), Tolstonogov A977) and
Schmidt A986).
Essential for the existence of such embeddings is the cancellation law
demanding that A + C == B + C implies A = B. An algebraic study of the
validity of this cancellation law in arbitrary real vector spaces under suitable
convexity and other assumptions on the subsets A, B, C was carried out by
Jongmans A973).
7. Representation of semigroups as systems of compact convex sets. If the
multiplication (A, K)i->XK = {kx\x e K) {K e JC") is considered for all real
AeR, then the rules listed before A.7.1) remain the same except that
(A -I- fi)K = XK + fiK holds only for Xfi^O. The structure (X", +) with this
multiplication is then an example of an abelian R-semigroup with
cancellation law. Ratschek & Schrbder A977) have characterized the abstract
R-semigroups that can be represented in this way by systems of compact
convex sets, with E" replaced by a suitable locally convex space.
8. Different metrics for convex bodies. In the theory of convex bodies, the
Hausdorff metric was first used by Blaschke [5] (§§17,18). It is, therefore,
sometimes called the Blaschke-Hausdorff metric. In the geometry of convex
bodies, other metrics have been proposed also. The following are defined on
IK5 only. The symmetric difference metric d^ is given by
dHK, L) := X"(K A L) for K, L e IC^.
where K d^ L- (K\L) U (L\K) denotes the symmetric difference of K and
L. This metric is invariant under volume-preserving affinities of E". Two
affine-invariant metrics are defined on ICH as follows:
d"(K, L) := log(l -I- 2inf {A '^Q\K (Z L + ADL, L <Z K -\- AD/C}),
where DK ; = K - A^ is the difference body of K, and
d^(K, L) := ^'"(K A L)IX"(K U L),
A detailed investigation and comparison of the four metrics d, 6^, d^, 6^
on Xo was carried out by Shephard & Webster A965). In particular, they
showed that all these metrics induce the same topology on X„, but not the

Notes for Section 1 8 59
same uniform structures. Further, they studied the completeness and
uniform continuity of some functions with respect to these metrics.
Dinghas A957a) showed that the symmetric difference metric (for not
necessarily convex sets) is decreased under simultaneous symmetrization at
an affine subspace.
Another class of metrics is obtained if in Theorem 1.8.11 the maximum
norm is replaced by an Lp norm: For 1 g p < oo and K, L e X" let
dp{K,L) .= |J^ __);,^ - hi,PAX"-'Y-
then dp is a metric on T", called the Lp metric. These metrics were
investigated by Vitale A985a). One has 6p(K, L)^ w^^odiK, L) trivially,
and Vitale established estimates in the other direction. As a consequence of
a sharp, but more complicated, inequality he showed, for instance, that
Cp(K, L)[6{K, L))'"*"-"/" S dp(K, L)
for K, L 6 .T", where the factor Cp(K, L) depends (for fixed n and p) only
on diam(A^ U L). He deduced that all of the dp metrics A = p = », with
6^ = S), induce the same topology on X" and yield complete metric spaces
in which closed, bounded sets are compact. For ^i, see also Florian A989).
9. Isometries. Because the Hausdorff metric on X" has a simple geometric
meaning, the study of the metric space (.T", 6) deserves some interest of its
own. A natural first question asks for the isometries that this space permits.
Suppose that the map i: X" —> X" is an isometry with respect to the
Hausdorff metric. Under the assumption that i is surjective, it was proved by
Schneider A975a) that i is induced by an isometry of E", in the sense that
there is a rigid motion g:E"-*E" so that i(K) = gK for all K e X".
Without assuming surjectivity, Gruber & Lettl A980) have shown that one
can find a rigid motion g of E" and a convex body L such that
i(K) = gK + L for all K e X". Every isometry of XS with respect to the
symmetric difference metric is induced by an equi-affinity of E" (Gruber
1978). For the subspace XS(B") of ^-dimensional convex bodies contained
in the unit ball, every isometry with respect to the symmetric difference
metric is induced by a rotation of the ball (Gruber 1982). The corresponding
result for the Hausdorff metric on X"(B") follows from Bandt A986).
Similar investigations for non-convex sets and for subsets of other spaces
were carried out by Gruber A980a, b). Gruber & Lettl A979), Lettl A980),
Gruber & Tichy A982) and Bandt A986).
10. Metric convexity and metric segments. For a metric space it is, furthermore,
of interest to study its properties in terms of the geometric notions
introduced by Menger A928) (see also Blumenthal 1953). The space A^", d)
is a metric segment space, since it follows from Theorem 1.8.11 that
d(K, A - a)K + at) = a6(K, L)
for K, L e X" and a 6 [0,1). (Some generalizations to infinite-dimensional
spaces were investigated by Bantegnie 1975). Shephard & Webster A965)
showed that (.TJ, 6^) is metrically convex (in Menger's sense), but not a
metric segment space, while (X"uf>^) and (XS,d^) are not metrically
convex.
In (Jie space (.'/ST", d) there is in general a great variety of metric segments
joining two given bodies K and L. A study of their totality was begun by
Jongmans A979). Answering a question left open by him, Schneider A981c)
showed the following. Let K, L e X" be convex bodies joined by only one
metric segment (equivalents, A = \(K + L) is the only convex body
satisfying d{K, A) = d(A, L) = \6(K, L)). Then either K = L + pB" or

60 Basic convexity
L= K +pB" with .some p § 0, or else dim K < n and L = K + t where the
vector t is orthogonal to aff K.
11. Metric entropy. A characteristic of a totally bounded metric space (X, d)
that is designed to describe its "ma.ssivity", is the metric entropy. This is, by
definition, the function H given by
W(f) ;= log,V(£) for f > 0,
where N(e) is the minimal number of closed d-balls of radius f covering X.
For the spaces (.T"(A), (S) and (X^A),6^), where 0 =)= A C E" is a
bounded open set. the asymptotic behaviour of the metric entropy for £—>0
was investigated by Dudley A974). For (X"(B"),d), Bronsht'ein A976)
showed more preci.sely that
c-,(«)t"-"i'- g W(f) g Cj(«)t-"-">- for £ g cy(n),
where c,(n) is a po.sitive constant depending only on n. In particular, the
exponent of entropy, defined by
liminf(loglog;V(f)/|log£l),
f JO
is equal to (n - l)/2 for the space (X"(B"), d) (see also Bandt 1988).
12. Diameters of sets of convex bodies. It is an interesting and, in general,
difficult task to determine the diameter, with respect to the Hausdorff
metric, of a given bounded subset of X" that is defined by some geometric
conditions. As an example, McMullen A984) proved that the space of all
convex bodies in T" with mean width b equal to that of a line segment of
length 2 and with Steiner point s at the origin, has diameter 1. As a
corollary, McMullen obtained the sharp e.stimate
d(K, L) S ma\{b(K). b(L)} + U(K) - s(L)\
for K, Le X".
13. Affine equivalence cla.sses of convex bodies. Let i-/Xo denote the set of all
affine equivalence classes of ^-dimensional convex bodies in E". Thus L
belongs to the equivalence class [A^] 6 < ■IXn if and only i{ K = aL for some
non-degenerate affine transformation ar of E". If < -fXu is equipped with the
quotient topology (induced by the standard topology on Xu), then ^-ZXa is
compact. This was proved by Macbeath A951). In the course of the proof,
Macbeath introduced the function
p(K, L) := mf{X"(aL)/:^/'(K)\K C at),
a
where a ranges over the affinities of E", and showed that
A([Kl [L]) = logfKK, L) + logp(L, K), [K], [L] 6 , ^X,,
defines a metric A on < -/Xd that induces the quotient topology.
The exponent of entropy for the space (i -/Xc,, 4) was determined by
Bronshtein A978a); it is equal to (n - l)/2.
14. Affine equivalence classes of compact .sets. Let < ■i(^" denote the set of affine
equivalence classes of nonempty compact subsets of E". Webster A965)
showed how the Hausdorff metric can be used to define a metric on < ■((?"
that makes i ■fO" into a compact space. This metric, however, does not
induce the quotient topology on 11'('/" (obtained from {(^",6) and the
affine-equivalence relation), since the latter fails to be metrizable. Let , iP",
U ■fO"+) be the subset of all affine equivalence classes containing an element
C with dimconvC=« (respectively, ^>"(C)>0). Then Webster's metric
and the quotient topology on t V^" induce the same topology on < ■(&", which
is compact. On the other hand, there does not exist a metric on < k^\ that

Notes for Section 1.8 61
makes < iOZ compact and induces the quotient topology. Web.ster's results
essentially answered a que.stion posed by Griinbaum [37] (p. 263).
15. Applications of the Steiner point. The exact Lipschitz constant in A.8.3) was
determined by PrzesJawski A985), Saint Pierre A985) and Vitale A985a).
The properties of the Steiner point, in particular Minkowski additivity,
the inclasion property A.7.6), and the Lipschitz continuity A.8.3), make it a
useful tool for some applications. For instance, the Steiner point has been
used for the construction of continuous selections of multi-valued mappings
(Linke 1980, Przeslawski 1985) and in the theory of random sets (Vitale
1984, 1990, Gine & Hahn 1985a, b). Motivated by this, Vitale A985b)
showed the non-existence of a continuous extension of the Steiner point to
all convex bodies in infinite dimen.sional Hilbert spaces.
16. Minkowski additive selections. Generalizing the Steiner point map, Zivaljevic
A989) studied Minkowski additive functions /: T" -^ E" satisfying f(K)e K
for all K 6 JC". In particular, he characterized all such Minkowski additive
selections / for which f(K) 6 ext K for all K e IC".
17. Isometry invariant measures on !X". The following question was posed by
McMullen (see Gruber & Schneider [36], p. 268): Is there a natural and
useful (isometry) invariant measure on the space T" with the Hausdorff
metric? It appears (depending on the interpretation of useful') that the
answer is no. Bandt & Baraki A986) have shown: For n > 1 there is no
a-finite Borel measure on (X",d) that is invariant with respect to all
isometries from the whole space into itself and is not identically zero.

Boundary structure
2.1. Facial structure
The notions of face, extreme point and exposed point of a convex set
were defined in Section 1.4. In the present section we shall study the
boundary structure of closed convex sets in relation to these and similar
or more specialized notions. We shall assume in the following that
K C E" is a nonempty closed convex set.
An /-dimensional face of K is referred to as an i-face. 7(K) is the
set of all faces and Ifiif^) the set of all /-faces of K. A face of
dimension (dim K) - I is usually called a facet. The empty set 0 and K
itself are faces of K; the other faces are called proper. It follows from
the definition of a face and from Lemma 1.1.8 that the faces of K are
closed. If f 4= /C is a face of K, then f n relint A: = 0. (If z e f n
relint K, we choose y e K\F. There is some x e K with z e relint [x, y].
Then \x.y]CF, a contradiction.) In particular, FCrelbdAf and
dimf < n.
Theorem 2.1.1. If F, is a face of K for i e I (a nonempty index set), then
\\,eiF, is a face of K. If F is a face of K and G is a face of F, then G is a
face of K.
Proof. This follows immediately from the definition of a face. ■
Theorem2.1.2. If Fi, F2e'J(K) are distinct faces, then relint Fj n
relint Fi = 0. To each nonempty relatively open convex subset A of K
there is a unique face F e '7(/w) with A C relint F. The system
{relint F|/-" e 'J{K))
is a disjoint decomposition of K.

2.1 Facial structure 63
Proof. Assume that F^, Fie <7^{K), Fi 4= ^2 and z e relint Fj n
relint Fi- There is some x e FfXFi, say. We can choose y e f 1 such that
z e relint [x,y]. From [x, y] C F^ C K, z e F2 and F2 e <J{K) it follows
that [a:, y] C Fi, a contradiction. Thus relint f 1 n relint Fi = 0.
Let /I 4= 0 be a relatively open convex subset of K. Let F be the
intersection of all faces of K containing A\ then F is a face of K.
Suppose there is some point x e /IXrelint F. Then there is a support
plane H to F with x e H and H n F ^ F, Since a: e /I C f and A is
relatively open, it follows that AC. H. Now H D F '\s clearly a face of
F and hence a face of K. But then f C W n F by the definition of F, a
contradiction. Thus A Crelintf. It follows from the first part that F is
the only face with this property.
In particular, the set A may be one-pointed. Thus each point x e K
is contained in relint F for a unique face F e 'T^iK). ■
In the proof above we have used the obvious fact that each support
set H n K, where W is a support plane of Af, is a face of K. The
support sets are also called exposed faces of K, The singleton {x} is an
exposed face if and only if x is an exposed point. We denote by 'V(Af)
the set of all exposed faces and by r'ii(K) the set of all i-dimensional
exposed faces of K. Clearly not every face is an exposed face: not every
extreme point is an exposed point, and taking direct sums yields
examples with higher-dimensional faces. However, each proper face F
of K is contained in some exposed face. In fact, since F D relint K =0,
there exists a hyperplane H separating F and K. From F C K it follows
that F C H, hence F is contained in the exposed face H r\ K.
Theorem 2.1.3. If Fi is an exposed face of K for / e / 4= 0, then A,6/F, is
either empty or an exposed face of K.
Proof. Let F := H/g/F; and assume that F4=0- After a translation, we
may assume that o e F. For i e / there exists a vector w, e IE"\{o} such
that Fi = K n Hu^xi and K C HZ,^- We may assume that wi, . . , , w,
are linearly independent and that any other w, linearly depends on these
vectors. Then f = n'=iF,. Let w'=2i'=i"c Then KCH'q and
o € K n Hufl, hence Hu,o supports K. For a: e F we have {x,u,) =0
for iel and hence {x,u)=0; thus FcH^^nK. For y e K\F,
{y,Wy)<Ofor some ; e {1,. . , , r}, hence {y,u}<0 and thus
y $ HuX). This proves that H,,^ C\ K = F and therefore that F e ''^{K). ■
One should observe that the second part of Theorem 2.1.1 has no
counterpart for exposed faces; An exposed face of an exposed face of K

64 Boundary structure
need not itself be an exposed face of K, but it is, of course, a face
of K.
From now on in this section we assume that K e 'K".
Exposed faces behave well under polarity, as we shall now see.
Suppose that K e Xo and o e int AiT. In Section 1.6 the polar body of K
was defined as
K* = {xe l"\{x, y) ^ 1 for all y e K).
For a subset F C K we now define the conjugate face of F by
F ■= {x e K*\{x, y) = 1 for all y e f}.
One should keep in mind that that F depends also on K and not only
on f; but this slightly incomplete notation is quite common. If we write
F, this means (F) , where the right-hand circumflex refers to K ■
Instead of {y}" we write y.
Theorem 2.1.4. Let oe mt K and 0 ^ F C K. Then the following
assertions (a), (b), (c) are equivalent.
(a) F lies in an exposed face of K,
(b) f 4=0,
(c) F is an exposed face of K*.
If (a)-(c) hold, then F is the smallest exposed face of K containing F.
Proof. For y e bd Af, the hyperplane [x e l."\{x, y) = 1} supports K*
(e.g., by Theorem 1.7.6); hence y = F{K*, y) is an exposed face of K*.
For y e int /C, (a:, y) < 1 for all x e K*, hence y = 0. By definition,
F= flUe K*\{x,y) = \)= fly.
yeF >'€F
Thus F is either empty or (by 2.1.3) an exposed face of K*.
Suppose that F lies in the exposed face S ■= K r\ W„ i (without loss
of generality), where Kd H^i. Then ue F and hence F^0. Vice
versa, if w e f, then u e K* and {u, y) = 1 for all yeF, hence F lies
in an exposed face of K. Thus (a), (b) and (c) are equivalent.
Suppose again that F lies in the exposed face S = K n W„ i. Let
z e F. Then z e K** = K and (z, x) = 1 for all x e F, and in particular
(z, u) = I, thus z e S. This shows that F C 5 for each exposed face S of
K containing^ f. Since F is itself an exposed face of K containing F, it
follows that F is the smallest exposed face containing F. ■
Remark 2.1.5. For a proper face F oi K that is not an exposed face it
follows from Theorem 2.1.4 that F =t= F.

2.1 Facial structure 65
By Theorem 2.1.2, each point x e K belongs to the relative interior
of a uniquely determined face F^ of K. This leads to a classification of
the points of K according to the dimension of the containing face. The
point X is called an r-extreme point of K if dimF^^ ^ r. Equivalently, x
is r-extreme if and only if there is no (r + l)-dimensional convex set
A C K with X e relint A. Further, the point x is called an r-exposed
point of K ii X is contained in an exposed face F e S(K) with
dim F ^ r. The set of all the r-extreme (r-exposed) points of K is called
its r-skeleton {exposed r-skeleton) and denoted by ext, K (exp, K). We
have e\i(jK = e\iK, expo Af = exp Af; evidently exp, Af C ext, Af and,
trivially, ext, K C ext, K and exp, K C expj Af for r < s.
Theorem 1.4.7, which is due to Straszewicz and which states that each
extreme point is a limit of exposed points, can be extended to r-extreme
and r-exposed points. This extension was obtained by Asplund A963).
His proof, however, is difficult to understand, and we prefer to give a
proof due to McMullen A983b). It requires the following lemma.
Lemma2.1.6. Let G e '7,iK) (for some r e {0, . . . , n - 1}) and q e
relint G. // there is a (n - r - l)-f}at supporting K precisely at q, then
GeS.iK).
Proof. The assumption demands the existence of an (n - r - l)-flat L
and of a support plane H to K such that L C H and L n K = {q}. Let
F ■■= H n K. From g e relint G we have G C F. Suppose that F ^ G.
Let X e F\G. We may assume that ^ = o. By L n G = {o} and o e
relint G we have L n lin G = {o}; from dim G = r and dim L =
n - r - 1 we get L + lin G = //. Hence, there is a representation x =
y + z with >> e lin G and o ^ z e L. Since o e rehnt G, there is a
number A > 0 with -ky e G. Then
thus o^[X/(l + X)]z e L n K, which contradicts the assumption. We
conclude that F = G and thus G e cS,(K). ■
Theorem2.1.7. For r = 0, . . . , n - 1,
ext, K C clexp, K.
Proof. For r = 0 this is true by Theorem 1.4.7. Assume that r S 1 and
that the assertion is true for r - 1 instead of r. Let p e ext, K and a
neighbourhood V of p be given. By Theorem 2.1.2 there is a unique
face F e ''7iK) such that p e relint F, and p e ext, K implies dim F ^ r.

66 Boundary structure
If dim F < r, then p e ext,_i K and by the induction hypothesis we have
p e clexp,_, K C clexp, K. Assume, therefore, that dim F = r. Let M
be an (n - r)-flat through p that is complementary to aff F. Then
p e ext(M n Af), hence by Theorem 1.4.7 there is a point
^ e V n exp(M n K). Let G e ■7sif<) be the face with q e relint G.
Since qeexpiMHK) and ^e relint G, we must have s^r. If
s<r, then ^eext,_i/w, and the induction hypothesis yields
q e clexp,_i K C clexp, Af. If s = r, let L be an (n - r - l)-flat in M
that supports M n K precisely at q. Lemma 2.1.6 now shows that
G e ry,iK), hence q e exp, K. Since V was an arbitrary neighbourhood
of p, the assertion is proved. ■
We conclude this section with a simple remark on the topology of the
r-skeleton. If dim K = 2, it is easy to see that ext K is closed, but for
dim K = 3 this is not true in general. For example, let K be the convex
hull of a circular disc C and of an orthogonal segment S such that there
is a point x e relbd C D rehnt5. Then a: is a limit of extreme points of
K but is itself not an extreme point.
It is, however, easy to see that ext, K is a G^ set (an intersection of
countably many open sets): For /c e N let A/, be the set of the centres of
all (r + l)-dimensional closed balls of radius l/k contained in Af. It is
not difficult to show that A^ is closed, and obviously
ext, K = C]i[K + Bo(o, 1^)]^^).
Notes for Section 2.1
1. Characterization of sets of extreme points. If S = ext K for some K e X",
then clearly (a) cl S C conv 5, (b) cl S is compact, (c) S n conv A C A for all
AGS. Bjorck A958) proved that for a nonempty set .S C E" these three
conditions are also sufficient for the existence of a convex body K e X" such
that S = ext A^, and he considered an infinite-dimensional extension of this
result.
2. A semicontinuity property of skeletons. Let {K,),^:^ be a sequence in 7f"
converging io K e. ~K". Then, for r = 0 " - 1,
clext^ K C liminfext^ K,.
(By definition, .v 6 liminf/I, if every neighbourhood of x meets A, for almost
all !.) For a proof, see Schneider [43] (Theorem B.13) and §8).
3. The face-function. Let K 6 To- By Theorem 2.L2. each point x e K is
contained in the relative interior of a unique face of K: denote this face by
F,. The function F: x >-> F> is called the face-function of K: its restriction to
bd K is the face-function of bd /C. Klee & Martin A971) have investigated the
semicontinuity properties of the face-function. For X = K or hd K. \ci Xi
(Xu) be the set of all points x 6 A" at which F \% lower (upper) semicon-
tinuous. Klee and Martin studied these sets and showed the following:

Notes for Section 2.1 67
Theorem. The face-function of bd K is lower semtcontinuous almost
everywhere in the sense of category and upper semicontinuous almost everywhere
in the sense of measure. However, when n^3 it may be lower
semicontinuous almost nowhere in the sense of measure and upper semicontinuous almost
nowhere in the sense of category.
Actually, upper semicontinuity almost everywhere in the sense of measure
for n >3 was only conjectured, but the assertion was completed by Larman
A971) who proved the following:
Theorem. The union of the relative boundaries of the proper faces of K of
dimension at least 1 has zero (n - l)-dimensional Hausdorff measure.
Lower semicontinuity of face-functions is related to the continuity of
so-called envelopes of functions. For a continuous function f: K —*R one
defines the envelope f^ by
/e := sup {gig: E" ^Raffine, gS fon K}.
Let Kc denote the set of all points of K at which every envelope is
continuous. Kruskal A969) gave a three-dimensional example with ^^. =t= K.
Klee & Martin A971) proved Kc = K tor n = 2, and Eifler A977) showed
that K^ = K, for K e .i:j.
Stable convex bodie.i. The convex body K e IZ^ is called stable if the map
(x, )') i-» \(x + y) from K x K onto K is open. The following assertions are
equivalent:
(a) K is stable;
(b) the face-function of K is lower semicontinuous;
(c) all skeletons ext, K, r = 0, . . . , n, ot K are closed.
For the proof, see Papadopoulou A977) or Debs A978); the survey by
Papadopoulou A982) explains why stable convex sets are of interest and
contains further results. For n >2, the convex bodies K with properties (a),
(b), (c) are also characterized by the fact that the metric space consisting of
the proper faces of K with the Hausdorff metric is compact. Similarly, the
metric space of all exposed faces is compact if and only if all the exposed
r-skeletons of K are closed, 0 = r g « - 2. These results are due to Reiter &
StavrakasA977).
Let K 6 JC", and for x e relbd K let 5^, be the smallest exposed face of K
containing x. In his investigation of the convexity of Chebyshev sets. Brown
A980) conjectured the following. If F is an exposed face of K, then there
exists an X 6 relbd F such that either S, = {x} or S, = F. He proved this for
n^5.
Topology of skeletons. In general, the topological structure of ext K and
exp K for K e X^ is not easy to describe. (For the situation in the
infinite-dimensional case, see Klee A955), Bensimon A987).) The following is
quoted from Klee A958) (where n = 3): 'Find a useful and simple
characterization of the class '/■„ of all subsets X of the unit sphere 5"~' such that
there is a homeomorphism of 5"' onto the boundary of a convex body K
mapping X into ext K\ The following partial answer was given by Collier
A975): Let A' be a subset of a compact 0-dimensional metric space Z, and
let « g 3 be' an integer. There is a homeomorphism of Z into the boundary
of a convex body K e TJ mapping X onto ext K if and only if A" is a G,-, set
with at least n + 1 points. On the other hand, non-trivial examples of subsets
of 5^ that do not belong to Cf'j can be constructed by means of the following
result of Collier A976): U K € Xl, then each component of (clext K)\ext K
is a subset of a 1-dimensional face of K.

AS Boundary structure
Bronshtein A981) considered the collection -//?„ ,v of sets M in E" such
that there exist K e X^' and a homeomorphism of c\ M into K mapping M
onto ext K. He showed that .///„„+, contains all compacta of E", further that
each locally compact, bounded set is in -''//„.,,+2 but not necessarily in .///„ „+i,
and siinilar results.
The following is known about exp K. For dim K = 2, exp K is & G/, set;
but Corson A965) constructed a three-dimensional convex body K for which
exp K is not the union of a C/, set with an F,, set and contains no dense G^
set. He also constructed a body K e Xl for which exp K is the union of a
countable number of closed sets each of which has no interior with respect to
exp K. Choquet, Corson & Klee A966) proved for K e X^ that exp K is the
union of a G/, set, an F„ set and n ~ 2 sets each of which is the intersection
of a Gj set with an f „ set. For K e Xl they showed that exp K is the union
of a G/, set and a set that is the intersection of a Gf, and an F„ set.
Klee A958) proved for smooth K 6 Xq that exp K is a G^ set and
ext K\cxp K is a first category F„ set in ext K. For « g 2 he showed the
existence of a smooth body K e X'd such that ext K is closed and ext K
\exp K is dense in ext K, and proved the existence of a body K e Xl such
that hd K\extK, cxtK\expK and exp A^ are all dense in bd K, see also
Edelstein A965).
7. Paths in the l-skeleton. The 1-skeleton of a polvtope has well-known
connectivity properties, which are important, for instance, for the
edge-following procedures of the simplex algorithm. Larman & Rogers A970, 1971)
initiated a programme of extending these properties to general convex bodies.
In their 1970 paper they proved the following theorem: Let a and b be two
distinct exposed points of a convex body K e X(,. Then there are n simple
arcs Pi, ..., P„ in the 1-skeleton of K, each joining a to 6, such that
P,n Pi = {a, h} for 1 S / < ; S «. In their 1971 paper, Larman and Rogers
obtained, in two refined versions, the following result on the existence of
increasing paths: Let L be a non-constant linear function on E" and let
K 6 Xll. Then there is a continuous map s of the closed interval [0, 1) to the
exposed 1-skeleton of K with
L(i@))= inf L(x), L(i(l)) = supL(.r),
L(i(/i)) < L(i(;2)), when 0 g /, < /, § 1-
This line of research was continued by Gallivan A979). Gallivan & Larman
A981) and Gallivan & Gardner A981). Similar investigations for
infinite-dimensional compact convex sets were carried out bv Larman & Rogers A973),
Larman A977) and Dalla A986).
8. It was pointed out by Fedotov A979) that a convex body K with o 6 int X
can have a face F with F^ F (Remark 2.1.5); this corrects an erroneous
statement in Bourbaki [10] (Chapter IV, §1, Exercise 2a).
9. Hausdorff measures of skeletons. For a polvtope P C E" and for r 6 {0, 1.
.... n} it is natural to consider the total r-dimensional volume, r]r{P). of the
r-faces of P. Inequalities for the.se functional were inve.stigated by Eggles-
ton, Grunbaum & Klee A964) and by Larman & Mani A970). The definition
i1r{K) := ^''{exirK),
where ^■' is the rdimen.sional Hausdorff measure, extends the function t], to
arbitrary convex bodies K e X". A different way of extending r;^ from
polytopes to general convex bodies was proposed by Eggleston, Grunbaum &
Klee A964). They showed that t]r is lower semicontinuous on the set of
^-dimensional polytopes and defined

Notes for Section 2.1 69
C(/C) := liminfrj,(P)
for K 6 JC". where liminf is taken over all sequences of polytopes converging
to K. Then t, is lower semicontinuous on X". The comparison of rj, and l.^
is non-trivial for 1 g r g « - 2. Burton A979) showed that t], is lower
semicontinuous and that t,,^T},. It has been asked whether t,, = t}, In
general (Problem 76 posed by Schneider in [36]). For r = n -2 this was
proved by Larman A980), but the other cases remain open.
Eggleston, Grunbaum & Klee A964) also asked for a characterization of
those convex bodies K e X" for which !^,(K)<'^. It was observed by
Schneider [43] (p. 19) that a convex body K for which t,r(K)<x or
r],(K)<^ has the property that (in the sense of Haar measure) almost every
(n - r)-flat intersects K in a polytope. This property, however, is not so
restrictive as one might expect. Dalla & Larman A980) constructed a convex
body K 6 X^ almost all of whose two-dimensional plane sections are
polygons but for which ext K has Hausdorff dimension one. (This answered in the
negative a question posed by Schneider; see Problem 77 in [36]). These
authors also showed that a convex body K e X" has exi(K C\ E) of
(Hausdorff) dimension at most r for almost all fe-flats E if and only if the
dimension of ext„_^ K is at most n - k + r. Burton A979b) proved that
nr(K-) < « (where 1 g r g « - 2) implies ^'"(ext,-, K) = 0. He also obtained
results on the facial structure of a convex body K having T]r(K)<^.
exhibiting some resemblance to polytopal behaviour.
Burton A980a) has found integral-geometric formulae for the functions t],.
The simplest of these says that t],(K) is, up to a factor depending only on n
and r. the integral of t]a(K n E) over all (n - r)-flats E, where the
integration is with respect to the rigid motion invariant measure on the space
of (« - r)-flats.
A number of inequalities for the functions r(, are known. Schneider
A978b) proved
UK) >(n- r + l)Vr{K)
for K 6 Xq and re {1,.. . , n - 2}, where V,(K) is the intrinsic volume,
defined in Section 4.2. This inequality is sharp, that is, the quotient
t]r(K)/Vr{K) comes arbitrarily close to « - r + 1 by proper choice o( K. A
weaker form of the inequality was obtained earlier by Firey & Schneider
A979). For non-negative integers r and s with r + s = n. Burton A979b)
showed
(r + 1)(.9 + l)r,,^AK) g r,,(K)UK),
which strengthens an inequality obtained by Eggleston, Grunbaum & Klee
A964) for polytopes.
A general problem of isoperimetric type can be formulated as follows: For
1 g i- < r g « determine the least number q(n, r, s) such that
n,(K) ^ q(n, r, s)(UK)y''
for all K 6 JC". For polytopes this problem was first studied by Eggleston,
Grunbaum & Klee A964). It is not known whether q(n, r, s) < » in general.
The known results, coming from various sources, are listed in Burton &
Larman.A981): The isoperimetric inequality gives the value of q(n, n, n - 1).
Explicit upper bounds are known for q(n, r, s) when s divides r, and
furthermore for q(n, n - \,s). with a better value for (^C, 2, 1), and for
<yC,3, 1). Burton & Larman A981) found an upper bound for
q(n, n -2, « - 3) if « S 4.
The following inequality of the Brunn-Minkow,ski type was established by

70 Boundary structure
Dalla A987): For K,Le X" and X g [0,1],
r,,((l - l)K + AL) S A - X)ii,i,K) + Ar,,(L).
One may speculate whether rj,((l - A)A^ + AL)''' is in general a concave
function of A (it is for r = n and r = « - 1; see Sections 6.1 and 6.5).
Hausdorff measures of finite-dimensional skeletons of infinite-dimensional
convex sets were investigated by Dalla A985).
2.2. Singularities
Through each boundary point of a convex body there is a supporting
hyperplane, but not necessarily a unique one. Non-uniqueness gives rise
to singularities, and these will be studied in the present section.
First we introduce some convex cones that describe the behaviour of
a convex body at one of its points. Let K e 7JC" be a convex body and
\ti X e K. Define
P{K, X) := {X{y - x)\y e A:, A > 0} = [JX{K - x).
Then P{K, x) is a convex cone, and P{K, x) = eiff K - x if and only if
xeTeVintK. The closed convex cone SiK,x) '■=c\P{K,x) can
evidently be represented as
SiK,x)= n H-iK,u)-x,
x€H(.K,u)
where the intersection extends over all u e lE"\(o} (or u e 5""^) for
which X lies in the supporting hyperplane H{K, u). (For x e intK, we
adopt the usual convention that the intersection of an empty family of
subsets of E" is equal to E".) S{K, x) is called the support cone of K at
X. (In Bonnesen & Fenchel [8], S{K, jc) -*- jc is called the projection cone
of K at X.) Further, we define the normal cone of X at jc by
NiK, x) := {u e 1E''\(o}|jc e HiK, u)} U {o}.
li X e K D H{K, u), then u is called an outward (or outer, or exterior)
normal vector of K at x. Thus N{K, x) consists of all outward normal
vectors of X at jc together with the zero vector. Clearly N{K, x) is a
closed convex cone and
N{K, x) = SiK, x)*, B,2.1)
that is, N{K,x) and S{K, x) are a pair of dual convex cones. The
normal cone is closely related to the metric projection, in fact, it follows
from Lemma 1.3.1 that
NiK, X) = p-^\x) - X B.2.2)
for jc e K.
If K lies in a proper linear subspace E of E" and if Se{K.,x) and

2.2 Singularities 71
Ne{K, x) are, respectively, the support cone and normal cone of K at
X, taken with respect to the subspace E, then obviously
SiK, X) = SsiK, X),
while
NiK, X) = NsiK, x) + E\ B.2.3)
where £-"- is the orthogonal complement of E.
The following theorem describes the behaviour of the support cone
and normal cone under addition and intersection.
Theorem 2.2.1. Let K, Le.X".
(a) If X e K and yet, then
SiK + L,x + y) = SiK, x) + SiL, y),
NiK + L, jc + y) = NiK, x) n Nil, y).
(b) IfxeKDL and relint K n relint L + 0, then
SiK n L,x) = SiK, x) n SiL, x),
NiK n L,x) = NiK, x) + NiL, x).
Proof of (a). Performing translations if necessary, we may clearly
assume that x = y = o. Let u e NiK + L, o)\(o}. Then K + L C Hu.o-
From K = K + {o} Q H~o and o e K \t follows that u e NiK,o).
Similarly we get u e NiL, o).
Let we NiK,o) n A'(L, o)\{o).From w e N iK, o) weha\c K C H~si\
similarly L C //^o and thus K^ + L C W^a. Since oe K -^ L, we get
u e NiK + L,o). This proves NiK + L,oj = NiK,o) n N(L,o).
From B.2.1) and Theorem L6,3 we now get SiK •\- L,o) =
SiK,o) + SiL,o).
Proof of (b). Let xeKDL and >> e relint A: n relint L. The first
equality of (b) is clear if x = y; hence we assume that x^y. Let
z e SiK,x)n SiL,x). Since y-jc e relint A: - jc C relint P(A:, jc) and
zec\PiK,x), it follows from Lemma LI,8 that [y-x,z)C
relint PiK.x). Hence, for w e [y - x, z) there is a number A > 0 such
that [x,x + /iw]cK. Similarly, there is a number /ot > 0 such that
[x, X + ixw] G L, It follows that w e PiK n L, x). Since w €[y - x,z)
was arbitrary, we deduce that zed PiK A L,x) = SiK D L, x). Thus
SiK,x) r\ SiL,x)c: SiK f) L,x). The opposite inclusion is trivial.
This proves the first equality of (b). The second equality is again
obtained by applying B.2.1) and Theorem 1.6.3, ■

72 Boundary structure
For a point jc e bd K^ we denote by .V, the smallest exposed face of K
containing x.
Lemma2.2.2. Let K e X" and x, y e bd K. Then N{K, x) = NiK, y) if
and only if S^ = Sy (in particular, if y e relint 5^),
Proof. If NiK, x) = N{K, y), then each exposed face containing x
contains v also, and each exposed face containing y contains x. It
follows that 6'v = Sy. Suppose that 5^^ = Sy (which is clearly satisfied if
y e relint5J.Letw eNiK, jc),Then5, C K D H{K, w),hence>- e H{K u)
and thus u 6 N{K, y). The assertion follows. ■
We can use this to extend the definition of the normal cone. Let F be
a nonempty convex subset of K. Then we define N{K, F) ;= N{K, x),
where x e relint F. This does not depend on the choice of x: if
F n int K =F 0. then /V(A:, x) = {o}. Otherwise, let x,y e relint F. Then
Sj is the smallest exposed face of K containing F, hence 5v = Sy and
Lcmma2.2.2gi\es NiK. x) = NiK, y).
The normal cone NiK, F) is closely related to the conjugate face F,
as shown in the following lemma.
Lemma 2.2.3. Suppose that o e int AlT and let F he a nonempty convex
.subset of K. Then
NiK, F) = posF U {o}.
Proof Let u e NiK, F)\{o}. Then //,, i with v = hiK,u)-^u is a
support plane of K satisfying x e K d //„ i for some point x e relint F.
It follows that F C Wi,.i; hence (>, y) ^ 1 for y e Af and (y, v) = I for
y e F. This shows that ve F and hence u e posF. Thus NiK, F) C
pos f U {o}. The argument can be reversed. ■
It follows from Lemma 2.2.3, B.2.3) and Theorem 2.L2 that distinct
normal cones of a convex body have disjoint relative interiors. In
particular, we have
int/V'(A:,;t) n intA'(A:, y) = 0 for NiK, x) ^ NiK, y). B.2.4)
We note that
(J NiK,x) = I", B.2.5)
since each support set of K contains an extreme point of K.
The normal cones lead, in a natural way, to a classification of the
singular boundary points of a convex body. Let K e X^ be a convex

2.2 Singularities 73
body with interior points and let xebdK. We say that x is an
r-singular point of K if dim N{K,x) ^ n - r. Bonnesen & Fenchel [8]
(p. 14) call X a p-Kantenpunkt (ridge point of order p) if
dim N{K,x) = n - p. Thus x is r-singular if and only if jc is a p-
Kantenpunkt for some p ^ r. An (n - 2)-singular point is briefly called
singular. If the supporting hyperplane to X at jc is unique, i.e., if jc is
not singular, then jc is called a regular or smooth point of K. The
convex body K (of dimension n) is called regular or smooth if all its
boundary points are regular. We denote the set of regular boundary
points of K by legK.
The following theorem shows that a convex body cannot have large
sets of r-singular boundary points.
Theorem 2.2.4. Let K e X") and r e {0, . . . , n - 1). The set of
r-singular points of K can he covered by countably many compact sets of finite
r-dimensional Hausdorff measure.
Proof. We use the metric projection pi^. Let Mr denote the intersection
of a fixed closed ball containing K in its interior with the union of all
r-dimensional affine subspaces of E" that are spanned by points with
rational coordinates. Let jc be an r-singular point of K. Then the
translated normal cone
N{K,x) + x = pl'ix)
is of dimension at least n - r. It is, therefore, easy to see that this set
meets M^; hence jc e p/dMr). As Mr is a union of countably many
compact sets of finite r-dimensional Hausdorff measure and as pn is a
Lipschitz map by Theorem 1.2,2, the assertion follows. ■
Theorem 2.2.4 implies, in particular, that the set of r-singular points
has (r -*- l)-dimensional Hausdorff measure zero, and furthermore that
the set of singular points has (n - l)-dimensional Hausdorff measure
zero. It also implies that K has at most countably many 0-singular
points. This assertion can be generalized as follows. By a perfect face of
K e Xq one understands a face F of K for which
dimf -*- dim NiK, F) = n.
Theorem 2J.5. A convex body K e Xo has at most countably many
perfect faces.
Proof. If f is a perfect face of K, then the set p^^{TelintF) is of
dimension n and hence contains a point z with rational coordinates in

74 Boundary structure
its interior. Since pxiz)€Te\intF and different faces have disjoint
relative interiors by Theorem 2.1.2, the assertion follows. ■
The classification of support planes of a convex body, which we now
describe, will in later chapters turn out to be of greater importance than
the classification of boundary points. Let K e JC" be a convex body. Let
u e lE"\(o}. Then u is an outer normal vector of K at each point of the
support set FiK, u) = K D H{K, u). The normal cone NiK, F{K, u))
(which is equal to N{K,x) for any point x ^ leWni F{K, u)) has a
unique face, denoted by T{K,u), that contains u in its relative interior
(cf. Theorem 2.L2). We call T{K,u) the touching cone of K at u,
since for every vector d e reUnt r(K^, w) the support plane H{K,v)
touches K at the same set; choose x ei&Wni F{K,u) and let D be the
flat through x totally orthogonal to lin T{K, u); then
FiK, v) = D n K
for each d e relint T{K, u). In fact, D D K Q F{K, v) is clear. On the
other hand, let y e F{K, v); then (y - x,v} = 0 and (y - jc, r) ^ 0 for
each t e TiK, u). Let t e T{K, u). Since d e relint T{K, u), there is a
vector w e T{K,u) and A>0 such that kv = t + w. We get
0 = (y - AT, Ad) = (y - jc, r + w) and hence (y - x, t) = 0. This shows
that y e D.
The vector u will be called an r-extreme normal vector of K, and
H{K,u) an r-extreme support plane of K, if dimT{K,u)^ r + 1
(r = 0, . , . , n - \). Thus u is an r-extreme normal vector of K if and
only if there do not exist r + 2 linearly independent normal vectors Wj,
. . . , Wr+2 at one and the same boundary point of K such that
w = Wi + . . . + Wr+2- For 0-extreme we say extreme.
Remark 2.2.6. The set D is the affine hull of the set which in Bonnesen
& Fenchel [8] (p. 16) is denoted by ^D„ and is found in a different way.
(The construction described there can be interpreted as a dual way of
finding the minimal face T{K,u) of N{K, F{K,u)) containing u by
successively determining the minimal exposed face containing u.)
Bonnesen and Fenchel call H{K,u) a p-Kantenstiitzebene if dimD = /j.
Thus H{K,u) is a p-Kantenstiitzebene if and only if it is
an {n - p - l)-extreme, but not an {n - p - 2)-extreme, support plane
oiK.
Further, the vector u is called an r-exposed normal vector of K, and
H{K,u) an r-exposed support plane of K, if A\mN{K, F{K,u))^
r-f 1.

2.2 Singularities 75
The choice of notation suggests an underlying duality. In fact, let
K e -1^0 and o e int K^. A given support plane H of K can be written as
H =//„ , with u e E"\{o}. Writing FiK, u) = F, we have NiK, F) =
pos F by Lemma 2.2.3 and u e F. Hence, the touching cone T{K, u) is
the smallest face of pos f containing u, and this is the positive hull of
the smallest face of F containing u. It follows that u is an r-extreme
point of X* if and only if u is an r-extreme normal vector of X. If w is
an r-exposed normal vector of K, then dimpos F =^ dim N{K, F)
^ r + 1, hence dimf^r. Thus u is contained in the exposed face
F of dimension at most r and is, therefore, an r-exposed point of K*.
Vice versa, let u be an r-exposed point of K*. Since u = F{K,u)
= F (where the circumflex refers to K*), F= u is, by Theorem 2.1.4,
the smallest exposed face of K* containing u. Hence, dimf ^ r and
thus dim N{K. F)'^ r + \, that is, u is an r-exposed normal vector
oi K.
The preceding observation leads to a dual version of Theorem 2.1.7:
Theorem2.2.7. Let K e X" and r e @,. . . , n - 1}. Then each r-
extreme support plane of K is a limit of r-exposed support planes of K.
Proof. If d\mK-n, we may assume that o e int K^ and then apply
Theorem 2.1.7 to K* to get the assertion. Suppose that dim K = k< n.
Let w e lE"\(o} be a given vector. We may assume that o e K; let
E = \\nK and write w = wq -*- "i with u^e E and u■^ e E^. If w is an
r-extreme normal vector of K and Wq = o, then u is an r-exposed
normal vector of K, since T{K, u) = N{K, F{K, u)) = E^ in this case.
Suppose that Uq ^ o. Then F{K, u) = FiK, uq) = F, say. Let NeIK, F)
C £ be the normal cone of K^ at f and Te{K, Uq)C E the touching
cone of K^ at Wo, both with respect to E. Then
N{K,F)= Ne{K, F)+ E\
T{K,u)= Te{K,uo)+ EK
It is now clear that the vector u is an r-extreme (r-exposed) normal
vector of K if and only if Wq is an (r - n -*- A:)-extreme {{r - n + k)-
exposed) normal vector of K. Thus the assertion of the theorem is
obtained in an obvious way if the result already proved is applied in E.
The classification of support planes plays a role since certain
problems in later chapters lead to pairs of convex bodies that have some, but
not all, support planes in common. In particular, the convex body
K e .K" containing the convex body L e J^" is called a p-tangential

76 Boundary structure
body of L if each (n - p - l)-extreme support plane of Af is a support
plane of L (p e {0, . . . , n - 1}). Thus a 0-tangential body of L is the
body L itself, and each p-tangential body of L is also a ^-tangential
body for p < q ^ n - I. An (n - l)-tangential body is briefly called a
tangential body. The following theorem characterizes p-tangential
bodies in a slightly different way.
Theorem2.2.8. Let K, L e X", L C K and p e {0, . . . , n - I}. Then K
is a p-tangential body of L if and only if the following condition holds.
(*) Each support plane of K that is not a support plane of L contains
only (p - \)-singular points of K.
Proof. Suppose that K^ is a p-tangential body of L. Let // be a support
plane of K and assume that H contains a point x oi K that is not
(p - l)-singular, which means that dimN{K,x) = n — p. If u is the
outer normal vector of W, then N{K, F{K, u)) C N{K,x), hence
dim T{K,u)^ n - py so that w is an (n - p - l)-extreme normal vector
of K. Therefore, // is a support plane of L. Thus condition (*) holds.
Assume that (*) holds. Let H be an (n - p - l)-extreme support
plane oi K. By Theorem 2.2.7, H is the limit of a sequence (//,),ef^ of
{n - p - l)-exposed support planes of K. Each W, contains a point jc,
of K with dim N{K,Xi) ^ n - p, that is, Xj is not (p - l)-singular. By
condition (*), H, is a support plane of L, Since this is true for all /, H
is a support plane of L. This proves that K is a p-tangential body of L.
If K IS a 1-tangential body of L, then each point of K^ in a support
plane of K that does not support L is a 0-singular point of K. There
are at most countably many such points, and K is clearly the convex
hull of L and these points. If x and y are any two of these 0-singular
points, then [x, y] meets L, since otherwise one finds a support plane of
K not supporting L but containing a point of K that is not 0-singular.
For intuitive reasons that are now evident, 1-tangential bodies are also
called cap-bodies.
The notion of the p-extreme normal vector of a convex body admits
a generalization that will be useful in Section 6.6. Let K^,...,
K„^i e X" be n - 1 convex bodies. We say that the vector u e E"\(o}
is (K-i, ..., K„_i)-extreme if there exist (n - l)-dimensional Hnear
subspaces £i, . . . , £„_i of E" such that
T(Ki, w) C E, for / = 1, . . . , n - 1
and
dim El n ... n E„_, = 1.

2.2 Singularities 77
In particular, w is a p-extreme normal vector of the convex body K if
and only if u is
(K, ..., K, B", ... , B")-extreme.
n - I - p p
Here, of course, the p-tuple of unit balls B" may be replaced by any
other p-tuple of regular convex bodies.
We could also dualize the notion of r-singular boundary points in a
similar way to that in which we dualized the notions of p-extreme and
p-exposed boundary points. We need only the following case. Let
K e X" and u e 5"~'. The vector u is called a regular normal vector of
K, and H{K,u) is a regular support plane of K. if dim FiK,u) = 0;
otherwise u is called singular. Thus u e S"~^ is a singular normal vector
of K if and only if H{K, u)n K contains a segment. A convex body is
called strictly convex if its boundary does not contain a segment. We
denote the set of regular normal vectors of K by regn K.
Theorem 2.2,9. The set of singular unit normal vectors of a convex body
K e X" has (n — \)-dimensional Hausdorff measure zero.
Proof. Obviously we may assume that dim K - n and o e'xniK. Let u
be a singular unit normal vector of K. Then H{K,u) contains a
segment F. The point x := h{K, w)~'w is a boundary point of the polar
body K*, and F Cx (where the circumflex refers to K*). From Lemma
2.2.3 it follows that N{K*, x) D F, hence x is a singular boundary point
of K*. Thus u is contained in the image of M of the set of singular
boundary points of K* under the radial projection from bd K* to 5"~'.
Since this radial projection is obviously a Lipschitz map, it follows from
Theorem 2.2.4 that M has (n - l)-dimensional Hausdorff measure zero.
It is now in order to define spherical images and reverse spherical
images. Let K e X". For a subset /3 C E" we denote by o{K, P) the set
of all outer unit normal vectors of K at points of /3, thus
oiK,p) := U NiK,x) nS"-\
and call this the spherical image of K at p. Similarly, for a subset
(oCS"~^ O-e denote by T{K,aj) the set of all boundary points of K at
which there exists a normal vector of K belonging to w, thus
TiK,io) := [JFiK,u).

78 Boundary structure
We may call this the reverse spherical image of K at co. The spherical
image map of K is the map
defined by letting Ok{x), for x e regK^, be the unique outer unit normal
vector of Af at jc. The reverse spherical image map of K (not the inverse
of Ok, which does not exist in general) is the map
Tf^: regn K ^ bd K
for which Tk{u), for u e regn K, is the unique point in FiK, u). Let Xj,
xeiegK and suppose that Xj—^x for ;—»3o. Every convergent
subsequence of the sequence (ok{Xi))j^f^ converges to a normal vector of K
at jc, hence to Ok{x). Therefore the map a^ is continuous. Similarly,
the map r^ is continuous. In particular, if K is smooth and strictly
convex, then o^ and t^ are homeomorphisms and are inverse to each
other.
Since we want to be able to speak of the Lebesgue measure of the
spherical image o{K.P), we show that it exists for a sufficiently large
family of sets /3.
Lemma2.2.10. Let Ke X" and let /3cE" be a Borel set. Then the
spherical image o{K, P) of K at p is Lebesgue measurable on S"~^ ■
Proof. Let (^ be the family of all subsets /3C1E" for which o{K,P) is
Lebesgue measurable. First let /3 be closed. Let {Uj)j^^ be a sequence in
a{K,ft) for which Uj^u for ;^«», For ye N, we can choose
Xje K np such that Uj e N{K,Xi). Since K n P is compact, a
subsequence of ixj)j^f^ converges to a point x e K A p, and from Uj^> u it
follows that u e NiK.x), hence ue o{K, P). Thus a{K,P) is closed
and hence Ped. Next let p e <^ and P'=:='£"\p. Suppose that
ue oiK,P)no{K,P^). Then FiK,u) contains a point of P and a
point of /?*^, hence w is a singular normal vector of K. From Theorem
2.2,9 we deduce that
:^>"-\oiK, p) n a{K, p')) = 0.
Hence the set oiK,P') differs from the set S"~hoiK,P), which is
Lebesgue measurable, only by a set of (spherical Lebesgue) measure
zero. It follows that o{K,P'^) is Lebesgue measurable and thus P"^ e <-4.
Finally, it is clear that a{K, U,/3,) = [}iO{K,Pi) for any family (/3,). We
have shown that < i* is a a-algebra containing the closed sets. Hence, < /
contains all Borel subsets of E", as we set out to prove. ■

Notes for Section 2.2 79
To formulate an analogous result for the reverse spherical image
r( K, cd) we define the radial map
f: lE"\{o} ^ 5"-' by /(x) := ^.
Lemma2.2.11. Let K e Xq with o e int A: and let loCS"'^ be a Borel
set. Then f{T{K,oj)) is Lebesgue measurable on 5"~'.
The proof, which uses Theorem 2.2.4, is so similar to that of Lemma
2.2.10 that we can omit it.
Notes for Section 2.2
1. Some considerations using the first part of Theorem 2.2.1 can be found in
Jongmans A981a); for instance, the remark that the sum of a polytope and a
non-smooth convex body cannot be smooth, and some related results.
2. Theorem 2.2.4 is essentially due to Anderson & Klee A952). For « = 3, a
different proof was given by Besicovitch A963a). A generalization to certain
non-convex sets can be found in Federer A959, 4.15C) on p. 447). A weaker
form of Theorem 2.2.4 was claimed without proof by Favard A933a, p. 228).
That the set of singular boundary points of a convex body X € ^J is of
(n - l)-dimensional Hausdorff measure zero was first proved by Reidemeister
A921); see also Aleksandrov [1], (Chapter V, §2) for an equivalent result.
A stronger, and in a sense optimal, version of Theorem 2.2.4 was obtained
by ZajiCek A979), who showed that the countably many sets covering the
r-singular points can be chosen to be of a more special type.
Theorem 2.2.4 implies, in particular, that the set of regular points of a
closed convex set with interior points is dense in the boundary of the set. This
is also true in arbitrary separable Banach spaces (Mazur 1933; see also Marti
[25], p. 112).
3. The special case of Theorem 2.2.5 for one-dimensional faces was first proved
by Fujiwara A916). The general case appears (with different proofs) in
Bourbaki [10] (Chapter IV, §1, Exercise 2e) and in Br0ndsted A966). The
latter author introduced the term 'perfect face', while Bourbaki speaks of
regular' faces.
Fedotov A979a) called a face F oi K e X" isolated at a point .r e F if
there exists a neighbourhood U oi x for which
u n ext<j,™f a: = u r\ f.
If this holds, then he showed that x is an internal point of F, that F is
isolated at each of its internal points and that F is a perfect face.
4. Fedotov A978a) made a study of faces, and of extreme and exposed points in
relation to normal vectors, and obtained, for example, the following results.
Let K 6 ;^ci' and /?C E". If .r e exp, K and /3 is a neighbourhood of x. then
the spherical image o(K,(i) of X at /3 contains an (n - i - l)-dimensional
submanifold. Vice versa, if /? is open and the spherical image of /3 contains an
(n - i - l)-dimensional Lipschitz submanifold, then /? n exp, X =f 0-
5. Lemma 2.2.10 appears (essentially) in Aleksandrov [1], (Chapter V, §2) and
Lemma 2.2.11 in Aleksandrov A937a), §2. A more detailed study of spherical
images can be found in Aleksandrov A939c), §5.

80 Boundary structure
6. Minkowski's characterization of extreme support planes. Let K e X'd and
u 6 E"\{o}. For small 6>0, let rF) be the radius of the largest (n - 1)-
dimensional ball contained in K Ci H„^f:,u)-fi- Then u is an extreme normal
ve.ctor of K if and only if r(^)/(S—>oc for d-*0. This was proved by
Minkowski A911) for n=3 and extended to higher dimensions bv Favard
A933a).
7. Tangential bodies. The definition of a p-tangential body given above is
essentially the one in Bonnesen & Fenchel [8], p. 14. Favard A933). p. 273,
used property (*) of Theorem 2.2.8 as a definition (and talked of a tangential
body of order n - p). The equivalence expressed by Theorem 2.2.8 was
proved in Schneider A978c).
8. The convex cone P(K,x) defined at the beginning of Section 2.2 is often
denoted by cone (x, K) and called the cone of K at x. Above we were only
interested in its closure, but of course the cone P(K,x) itself carries more
information about the behaviour of K at its- point x. Let CC E" be a convex
cone. There exists a convex body K e X" and a point x e K such that
P(K, x) = C if and only if C is an F„-set. This was proved by Sung & Tam
A987). Their work is also related to earlier work by Larman A974) and
Br0ndsted A977) on certain cones (inner aperture, barrier cone) associated
with an unbounded convex set and to a certain classification of boundary
points of convex bodies by Waksman & Epelman A976).
2.3. Segments in the boundary
We have called a convex body strictly convex if its boundary does not
contain a segment. As the example of a polytope shows, the boundary
of a convex body can contain many segments. The directions of these
segments, however, make up a relatively small set; in the example it is
confined to finitely many great spheres (where directions are
represented by elements of the unit sphere S"~^). It is easy to construct
convex bodies where the set of directions of segments in the boundary is
dense in 5"~', but still of measure zero. The latter holds true in general,
but no easy proof seems to be known. With some effort, however, the
following stronger result can be shown.
Theorem 2.3.1 (Ewald-Larman-Rogers). Let K e J(" be a convex body
and let U{K) C 5""' be the set of all unit vectors that are parallel to a
segment in the boundary of K. Then U{K) has a-finite (n -2)-dimen-
sional Hausdorff measure.
Since the set of unit vectors parallel to a segment in a facet of K has
positive (n - 2)-dimensJonal Hausdorff measure and since a convex
body can have infinitely many facets, the result of Theorem 2.3.1 is best
possible.
Theorem 2.3.1 can be considered as one of the most remarkable
results on the boundary structure of general convex bodies. For this

2.3 Segments in the boundary 81
reason, and since a generalization to be treated below will be applied in
Chapter 4, we shall give the full proof, essentially following Ewald,
Larman & Rogers A970), with a variant at the end of the proof that
was proposed by Zalgaller A972).
An important tool, which is useful in other contexts, too, is the
following cap-covering theorem due to Ewald, Larman and Rogers.
Recall that a cap of a convex body K e X^ is an n-dimensional set of
the form K D H'^ where //"^ is a closed halfspace. For given r, R >0
we denote by TC'oir, R) the set of all convex bodies K e IC" satisfying
rB" + X C K C RB" + x
for some x e K.
Theorem 2.3.2 (Ewald-Larman-Rogers). For Ke Xu{r, R) there exist
positive constants C], . . - , C4 depending only on n, r, R such that the
following holds. For 0 < e < C] there exist caps Ki, . . ., K^ of K whose
union covers' the boundary of K and whose widths A{Kj) and volumes
V„iKd satisfy
CiE < A{Ki) < Cjf for / = 1, . . ., w,
m
2v„iK^)<c,e.
1=1
The proof will be split into a sequence of lemmas. The first of these
is well known (Bonnesen & Fenchel [8], p. 53) and quoted here without
proof.
Lemma2.3.3. If K e .Xq ^nd o is the centroid of K, then -K C nK.
For K € X", X e K and k>0 the convex body
Mix, A) := jc + k[iK - jc) n (jc - K)]
is called a Macbeath region.
We assume in the following that a convex body K € .Xnir, R) is given
and, without loss of generality, that
rB" C K C RB".
In particular, the origin o is a point of K.
Lemma 2.3.4. If x, y e K and M{x, \) n M{y, \) 4= 0, then Miy, 1) C
.Mix, 5). '
Proof. A point z e M{x, j) n M{y, \) is of the form
2
z = x + \{x-k,)^y ^ i(/c2 - y) B.3.1)

82 Boundary structure
with ki, ki e K. Let w e M{y, 1). Then
w = y + {ki - y) = y + {y - k^)
with suitable /cj, k^e K\ hence
w = ki = X + 5[{\x + ^fcj) - x]e X + 5(K - x)
and, using B.3.1),
tv = V + (y - ki) = 6jc - 2/ci - 2/c2 - k^
^X + i[x-{\k, + §/C2 + 1^4)]
€ jc + 5(jc - K),
hence w e A/(jc, 5). ■
Lemma2.3.5. Let H* - H'^„ be a closed halfspace meeting K but not
rB" and such that the distance between its bounding hyperplane H and
the parallel support plane H{K, u) is at most jr. Let c be the centre of
gravity (w.r.t. H) of K D H. Then K n H* C M(c,3n).
Proof. We suppose that the assertion is false. Since K C c + 3n{K - c),
there must exist a point k e K D H"^ with k ^ c + 3n(c - K). Writing
r)= l/Cn), we have
c - nik - c) ^ K. B.3.2)
We may assume that |wl = 1. The point k e K n H* can be written in
the form k = h + tu with h e H and 0 ^ r S jr. We can also write
c^su + z with (z, w) = 0. Since r5"ClE"\// + , we have s^r^lt.
Define
(; := —-— (h - c) = z;
s + t s + t
then ((;, w) = 0 and
c + i; = k € K n H. B.3.3)
s + t
On the other hand, using B.3.3),
rijs + t) s ,
c V = [c - ri{k - c)]
s - r}t s - r)'
belongs to H but not to K, by B.3.2) and r)t < \r<s. Since c is the
centre of gravity oi K n H, Lemma 2.3.3 now implies
ilis + t) ^ 1 ^1
s — rjt n — 1 n
Using rjr ^ r ^ jr ^ 55, we arrive at
1 . 1 s -r]t ^ 1 s - \s
3n ^ n s + t n s + \s
a contradiction.
1
3n'

2.3 Segments in the boundary 83
For 0 < e < r we put
A:_, := n [ff'iK, u) - eu]
ueS"-'
(this is the inner parallel body of K at distance e, cf. Section 3.1).
Lemma2.3.6. If 0 < e < \r-R-\ then
A - 4r-^Re)K C K.,C K
and
V„iK\K_,) < 4nr-^RV„iK)e.
Proof. Let xehdK^f. The non-negative function h{K-x,-)-e
attains a minimum on S"~\ say at u. This minimum is equal to 0, since
otherwise x e int K-f. Hence x e H{K, u) - eu and, therefore,
\x\'^ {x, u) = h{K, u) - e^ r - e> jr (observe that e<^r since
r < R). Since |jc| < R, the angle 0 between u and the vector x satisfies
cose^r/{2R). Let x'= ^x with ^>0 and x'ebdK. Then
ijc'-Jc|cos6>^eandhence |jc'- jc|/|jc| ^4r-^Re. Thus
A - 4r-^Re)K C K-,C K
and
V„iK\K.,) ^ V„iK\il - 4r-'Re)K)
= [l-il-4r-'Re)"]V„iK)
<4nr-^-ReV„iK). ■
Lemma 2.3.7. Let iX e < r and jc e bd K-^. Then
Bix, e) C Mix. 1) and A{Mix, 1)) = 2e.
If e<{Rln)[rl{2R)Y", there is a cap K" of K, bounded by a hyper-
plane through X, so that
K" C Mix, 3n).
Proof From the definition of K.^ it is clear that B{x, e)C K and thus
B(x, e) C Mix, 1). This implies zi(M(jc. 1)) ^2e. With u chosen as in
the proof of the preceding lemma, we have
Mix, 1) C H'iK, u)n[x - iH'iK, u) - x)].
The right-hand side is a strip bounded by two parallel hyperplanes at a
distance 2e apart, hence AiMix, 1)) = 2e.
Among all caps of K bounded by hyperplanes through x, there is one
with minimal volume, say K". The special cap K f\[H*iK,u) - eu].

84 Boundary structure
with u as above, can be covered by a cylinder of height e whose base is
an (n - l)-ball of radius R, hence
V„(A:")gK„_i/?"-'£. B.3.4)
There is a point y e K'^ lying in a support hyperplane H oi K parallel
to the bounding hyperplane H of K'^ through x. Let t be the distance
of X from H and t the distance of o from H. The convex body K
contains the convex hull of y and the intersection of rB" with the
hyperplane through o parallel to H. The cap K° contains a set similar
to this convex hull, with similarity ratio t/r. Hence,
which together with B.3.4) and t^ R yields
„ ^ nR^"-^
t" ^ 7— e.
^1-1
If now f <(/?/n)[r/B/?)]^", this yields t < \r. In particular, the
hyperplane H does not meet the ball ^rB". From the minimum property of
/C"* it follows that X is the centre of gravity of K n H; otherwise a
suitable small rotation of H around x would yield a cap with smaller
volume. We can now apply Lemma 2.3.5, with r replaced by \r, to
conclude that /C" C Mix, 3n). ■
We are now in a position to prove the cap covering theorem.
Proof of Theorem 2.3.2. Put Ci = {R/n)[r/i2R)]^". the constant
appearing in Lemma 2.3.7, and let 0<e<ci. Choose points x^,...,
x„, ebd K -f with
Mix,, }) n M(jcy, ^) = 0 for / =f / B.3.5)
and such that m is maximal (clearly a finite m exists, since
V„iM{Xi, i)) S V„E(a:„ if)) by Lemma 2.3.7).
Let z ehd K, and let y 6 [o, z] n bd K -f. As in the proof of Lemma
2.3.6 and by the choice of e,
\z - y\ < 4r~^Re\y\ < 4r~^R^e < \r,
which implies that y - (z- y) e K and hence zeM{y,l). By the
maximality property of JCi, . . . , jc^, there is an index i e {1, .... m)
for which
Mix,, 1) n Miy, i) + 0.
Lemma 2.3.4 now shows that z e Miy, 1) C M(jc,, 5). Since z ebd K
was arbitrary, we have proved that
m
bdK C Um(jc„5). B,3,6)

2.3 Segments in the boundary 85
Let ie(l,...,m}. The point jc, lies in the boundary of a supporting
halfspace H~ of K-^. Clearly
M{xi, \)\HJ C K\K-,
and hence, by B.3.5),
m
Since M(jc;, \) is symmetric with respect to the point jc,, we have
V„{M{Xi, \)\H~) = jV„(M(jc,, 2)) and conclude from Lemma 2.3.6 that
m
Xv„(M(a:„ \)) < Snr-'RV„iK)e. B.3.7)
1=1
By Lemma 2.3.7, for each / e {1, . . . , w} a cap K° of K, bounded
by a hyperplane through jc,, can be found such that
K'^ C MiXi, 3n). B.3,8)
There is a supporting halfspace Hu^a of K so that
K° = K D Hu^_a,-',
with suitable r, > 0. We define the cap Ki of K by
Ki := K D Hu„a,-6i/
From
Xj - 5{K - Xj) C Xi - 5{Hu,,a, - ^1) = fiu,.a,-6i,
it follows that K n M(jc,, 5) C K, and thus
bdK C \JK,
1=1
by B.3.6).
Choose a point y, e AlT n H„ „. From the convexity of AlT and from
B.3.8) we infer that
Ki C y, + 6(A:? - yi) C y, + 6[M(a:„ 3n) - y,]
= M(jc,, 18n) + 5jc, - Sy,.
From Lemma 2.3.7 it follows that A{Ki)^36ne, and since B(jc,, e)C
MiXi, 1) C a:,, we have AiK^) ^ 2e. Finally,
m m m
^VniKi) ^ 2v„(M(a:„ 18n)) = C6n)V,.(A/(a:„ i))
1=1 1=1 1=1
< {36n)"?,nr-'^RV„{K)e
by B.3.7), which completes the proof of Theorem 2.3.2. ■
The proof of Theorem 2.3.1 requires two more lemmas. For the first
one, a rectangular parallelepiped is referred to as a box.

86 Boundary structure
Lemma2.3.8. A convex body K e DC") can be covered by a box P with
smallest edge length A{K) and with volume V„{P) g n\ V„{K).
Proof. We use induction on n. The case n = I being trivial, assume that
n^2 and the assertion has been proved in dimension n - I. The
convex body of width A{K) has two parallel supporting hyperplanes
Hq, Hi of mutual distance A{K). Let n denote the orthogonal
projection from E" onto //q. There must be a point jco e
K n H^n Tr{K n //j), since otherwise K n Hq and n{K n Hi) could
be strongly separated by an (n - 2)-plane E in Hq, and small rotations
of //() and //[ around £, respectively H^ nn~^{E), would yield a
parallel strip containing K, of width strictly less than A{K).
Let Xi e K D HI be the point for which ttjc, = jco and let C be the
convex hull of ttK and jcj. For yenK, let y^e velbdnK be
such that y e [jCo, >'i]. If z e 7T~\yi) n K, then con\ {zyXo,Xi} C K,
and the intersections of the line ir'^iy) with this triangle and with
C have the same length. By Cavalieri's principle, V„{K)^V„{C) =
l/nAiK)V„-i{7TK), where V„-i denotes (n - l)-dimensional volume.
By the induction hypothesis, nK can be covered by an (n - l)-dimen-
sional box P' of smallest edge length A{TrKi)^ A{K) and (n-1)-
volume V„_i(P') ^ (n - 1)! V„^i{7tK). Hence, K can be covered by a
box P of smallest edge length A{K) and volume V„(P) =
AiK)V„_iiP') S AiK)in - 1)! V„_ii7rK) S „! i/„(A:). ■
Lemma2.3.9. Let Ko,Ki€ Xq. Let A{Ko, K^) be the set of all
differences Xi - Xq of boundary points Xq of Kq and x^ of Ki lying in parallel
support planes, thus
AiK^, A:,)= U [FiKi,u)- F{K,,u)].
If Ci C,„ are caps of Ko+ K^ covering the boundary of Ko + Ki,
then
m
AiKo. Kt)C U(DC, + a,),
1=1
where DC, is the difference body of C, and a, is a suitable translation
vector.
Proof, For / e {1, . . . , m) we can write C, = (Ko + Ki) n Hu,.a,-t,>
where //„„«, is a support plane of AlTo + K\, with exterior unit normal
vector u„ and r, > 0. Let a] := h^K^, w,) and C] := /C,, n Hu^_a)-r, for
v=0, L A given element of A{Kq,Ki) is of the form x^-Xq
with jCve fC/Cv, u) for suitable veS"~\ Then JCo + Jtie

2.3 Segments in the boundary 87
FiKo + Ki, v) ChdiKo + K^), hence there is an index ;' e A, . .. , w)
for which JCo + jcj e Cy. We have
a/ - T;^ {xo + x^, Uj) ^ ory,
(jCv, Uj) ^ a) for V = 0, 1.
From (jCv, Wy) < a) - Tj for v = 0 or 1 it would follow that
{Xq + Xu Uj) < OTy + or) - Tj = (Xj - Tj,
a contradiction. Thus orj - ty ^ (jCv, Uy) and, therefore, XyeCj for
v= 0, 1. This shows that
^1 - ^0 6 [jiC] - C?). B.3.9)
1=1
If b- € FiKy, w,), then C? + b] C C, and C,' + fe? C C,, hence
C} - C? C DC, + bj - bt
which together with B.3.9) proves the assertion. ■
Proof of Theorem 2.3.1. Let K e X" be given. Since each boundary
segment of K is contained in a support set F{K,u) and since
FiK + B", w) = F(Ar, w) + w, we may assume from the beginning that
K = K' + B" with a convex body K'; thus K^ is 1-smooth. (A convex
body L e .X" is called rj-smooth if to each jc e bd L there is a vector r
such that JC e rjB" + tC L.)
Let /7o, Hi be a pair of parallel hyperplanes in E". We say that a
vector u e S"~^ is associated with (K^./Tq, Hi) if the boundary of K
contains a segment parallel to u that intersects /Tq and /T]. Let ^^
denote the set of all hyperplanes in E" that are spanned by points with
rational coordinates. We fix a pair Uq, /7i of parallel hyperplanes in ''S
meeting the interior of K. Let U* be the set of all u e S"~^ that are
associated with (K, Flf^, FIi). Since there are only countably many
possible choices for 77,), /7i and since the union of the sets U*
corresponding to these different choices covers the set U{K) of
Theorem 2.3.L it suffices to show that U* has finite (n - 2)-dimen-
sional Hausdorff measure.
Since directions of boundary segments remain unchanged under
homotheties, we may assume that /7o = H„ _i and /7i = //„, with
weS"-K Let Ko^KDHo and Ki = Knni. In the following, Cj,
C2, . . . denote constants depending only on n, Ko, Ki, but not on e.
By A we denote the set of all differences Xi — x^ where Xq e Ko,
xi e Ki and x^ and .tj lie in a support plane of K. Then
U* = (y/lylly e A U(-A)}. Since U* is the image of A U (-A) under a
Lipschitz map, it suffices to show that A has finite (n - 2)-dimensional
Hausdorff measure.

88 Boundary structure
We apply Theorem 2.3.2 to the (n - l)-dimensional convex body
Kn + K^, with E" replaced by aff(/C,, + Ki) = //„„. Let Cj, . . . , C4 be
the constants appearing in that theorem (they now depend only on n,
Kq. K]). Let 0< e < vc] be given. There exist (n - l)-dimensional
caps Ci, ■ ■ ■ , Cm of Ko+ Ki covering the boundary of Kq+ Ki and
such that the widths A'{Kj) and volumes V„_i{Ki), computed in H„o,
satisfv
C2f^ < A'id) < C3e^ B.3.10)
m
2v„_,(C,)<C4e^ B.3.11)
;=i
Let JCi - JC() e A, with Xq e Kq and x^ e Ki lying in a support plane of
K. Then x^ — w and jcq + w lie in parallel support planes (with respect
to //.vo) of K^- w and Kq+ w, respectively. From Lemma 2.3.9 we
now infer that
m
A C U(DC, + r,) B.3.12)
;=i
with suitable translation vectors t,.
Let /e{l, ..., m). The difference body DC, has width A'iDd)
= lA'id) and (n - l)-volume
since DC;=C,-C, is contained in a translate of C, + (n - l)Ci,
according to Lemma 2.3.3. By Lemma 2.3.8, DC, can be covered by an
(n - l)-dimensional box Pj with smallest edge length wj = A'{DCi) and
with volume V„_i(P,) g (n - 1)! V„_i(DCi).
We have assumed that K is 1-smooth. Since /Tq, /7i meet the interior
of AlT, it follows that Xq and K^, and thus also ^0 + ^1. are r;-smooth
for some r; > 0. Hence, each cap C, contains a cap, of the same width,
cut from an (n - l)-ball of radius r). Now it follows from B.3.10) that
the edge lengths w, ^ ^2 = . . . ^ ^n-i of the box P, satisfy W2 > c^e.
The box P, can be covered by N, (n - l)-dimensional cubes of edge
length e, where
g C6e-''V„_,(Pi) g C7e-"V„_i(DCi) ^ Cge"" V„_i(C,).
By B,3.12), the set A can be covered by
m
Ngc8e-v„-,(C,)^c,e2-'.
1=1
cubes of edge length e, using B.3.11), Since this holds for all sufficiently
small numbers e > 0, the set A has finite (n - 2)-dimensional Hausdorff
measure. This completes the proof of Theorem 23.1, ■

2.3 Segments in the boundary 89
Theorem 2.3.1 can be considered as a special case of a more general
theorem referring to corresponding segments in the boundaries of a pair
of convex bodies. We denote by SO{n) the group of proper rotations of
E", equipped with its usual topology and differentiable structure. The
normalized Haar measure on SO(n) is denoted by v. On the compact
Lie group SO{n) we can choose a bi-invariant Riemannian metric. This
induces, in the usual way, a metric (distance function), which we denote
by di. Hausdorff measures on SO{n) refer to this distance function,
(The a-finiteness of Hausdorff measures, in which we are interested,
does not depend on the choice of the Riemannian metric, since any two
continuous Riemannian metrics on SO{n) induce equivalent distance
functions.)
Theorem2.3.10. Let K, K' e X" be convex bodies. The set U{K, K') of
all rotations pe SO{n) for which K and pK' contain parallel segments
lying in parallel supporting hyperplanes has o-finite [\n{n - 1) - 1]-
dimensional Hausdorff measure.
It is not difficult to obtain Theorem 2,3.1 from this result by taking
K' equal to a segment. Theorem 2.3.1 is interesting in itself as a strong
result on the boundary structure of convex bodies. The following
corollary of Theorem 2.3.10 will be needed in Section 4,5,
Corollary2.3.11. Let K e X". For K'eX", the set U{K,K') of
Theorem 2,3.10 has v-measure zero.
Let E be a k-dimensional linear subspace o/lE"(/ce{l, ,..,n-l}).
The set of all rotations peSO{n) for which pE is parallel to some
supporting k-flat of K that contains more than one point of K is of
v-measure zero.
Recall that a supporting flat of K is an affine subspace of E"
contained in a supporting hyperplane of K and having nonempty
intersection with K. The first part of the corollary is true since Theorem
2.3.10 implies that U{K, K') has \n{n - l)-dimensional Hausdorff
measure zero and hence v-measure zero, because ^n{n — 1) is the
dimension of SO{n). The second part is obtained by taking
K' = En B".
Although Corollary 2.3.11 is much weaker than Theorem 2.3.10, no
simpler proof of the former seems to be known. The proof of Theorem
2.3.10 requires some preHminaries.
A pair of orthogonal unit vectors in E" will be called a 1-frame. We

90 Boundary structure
denote by V„ 2 the set of all 2-frames in E". For {u.v)e V„ 2 and
p e SO{n) we write p(u, v) = (pw, pv). On V„ 2 a metric r is introduced
by
r((wi, i;i), (W2, i'2)) ■•= l«i - I + \vi - V2\.
In the following, c^, cn, ... denote positive constants independent
of e (but Ck may depend on those Cj with j < k). The assertions below
are understood to hold for all sufficiently small e > 0.
Lemma2.3.12. Let A,BCV„2 be sets of diameter cioe. The set
XiA,B) of all rotations pe SO{n) for which A D pB ^ 0 can be
covered by less than c^e"^""^'^"'-''''^ balls, in {SO{n), di). of diameter e.
Proof. First we assume that a, b e V„ 2 are 2-frames such that
T{a, b) ^ Ciof, and we assert that there exists a rotation pe SO{n) in a
(i2t-neighbourhood of the neutral element e such that b = pa.
We embed SO{n) into IR"" by associating with each p e SO{n) the
entries, in the usual order, of its matrix with respect to a given
orthonormal basis {bi, . . . , b„) of E". The standard Euclidean metric
on IR"" induces a second metric di on SO{n). It also induces a
Riemannian metric on the smooth embedded submanifold SO{n), and
this induces another metric d^, on SO{n). Since the submanifold SO{n)
is compact, the metrics di and ^3 are equivalent, that is, there exist
constants yi, 72 such that yi^2(P. <^) = ^3(p. <^) = y2^2(P-<^) for p,
oe SO{n). Also, the metrics di and dj, are equivalent, since both are
induced by continuous Riemannian metrics.
Since the metric r is invariant under the operation of SO{n) and
since d\ is bi-invariant, we may, without loss of generality, assume that
b = {bi, bi). Let a = {a^, ai). Applying the Gram-Schmidt orthonorm-
alization process to the n-tuple (ai, 02,^3, ..., b„) (which is linearly
independent if e is sufficiently small) and using \a\ - bi\ + \a2 - ^21 =
c'loE, we get an orthonormalized n-tuple {a^, ..., a„) satisfying
|a, - /);! ^ Cue for i = 1, . . . , n. If we take the coordinate vectors (with
respect to the basis bi, . . . , b„) of ai, . . . , a„ as the row vectors of a
matrix, then this matrix represents a rotation peSO{n) for which
pa = b.By definition of the metric ^2 we have
^2(P, e) = (la, - /)i|2 + _ + \a„- /,„|2)i,'2 g V7c„e
and hence diip, e) ^ c^e.
To prove the lemma, now let a e A,h e B. The set M of all
o e SO(n) for which a = ob is a left coset of the isotropy subgroup of
b, which can be identified with SO{n - 2), hence M is a submanifold of
dimension j(n - 2){n - 3). Let pe X{A, B). Then there are elements

2.3 Segments in the boundary 91
a'€ A and b'e B for which a'= pb'. By assumption we have
r(a, a') § CioE and rib, b')^ Ci^e. As shown above, there exist
rotations pi, p2 in a Ci2e-neighbourhood of e such that a'= p^a and
b' = P2''- It follows that a = p\^pf>ib, hence /ot •= P\^pf>i e M. Since
the metric di is bi-invariant, we get
d\{p, m) = diipifxp2\ m) = ^i(PiM. MP2)
= ^i(PiM, m) + '^iCm. MP2) = ^i(Pi- ^) + '^iC^. P2)
^ 2cue.
Thus A'C/l, i3) is contained in a 2ci2e-neighbourhood of M. The
smooth, compact submanifold M of dimension 5(n - 2){n — 3) can be
covered by less than cj^e"^""^'^""'*'^ balls of diameter e (for a proof,
see Flatto & Newman A977), Theorem 2.1). Then XiA, B) can be
covered by less than c^e''-"'^^'-"'^^^ balls of diameter e. ■
The proof of Theorem 2.3.10 is an extension of the one given for
Theorem 2.3.1, and we use the same notation, as well as some results
established in the course of that proof.
Proof of Theorem 2.3.10. Let K, K' e J^" be given. Both bodies can be
assumed to be 1-smooth. Let /Tq, /7i be a pair of parallel hyperplanes in
E". The 2-frame iu,v) is said to be associated with (K^,/7o,/7i) if the
support set F{K,v) contains a segment parallel to u that intersects Hq
and /7i and if (w, w) ^ ^, where w is the unit vector orthogonal to /7o
and pointing from Flo to 77,. Let pe U{K, K'). Then there exists a
2-frame (w, v) such that F{K, v) and F{pK', v) each contain a segment
parallel to u. Hence, there exist four hyperplanes /7o, Fix, FFq, FI[ in ^^
such that /7o and Fix are parallel, say at distance a, FIq and n[ are
parallel and at the same distance a, (w, d) is associated with (K, FIq, /7i)
and (p~'w, p'^v) is associated with (K', Ho, FI[).
We fix a pair /7o, /7i of parallel hyperplanes in ^P that meet the
interior of K, and a pair /7[), n[ of parallel hyperplanes in ^j? at the
same distance that meet the interior of K'. Let U*{K, K') be the set of
all rotations pe SO{n) for which there exists a 2-frame (w, o) associated
with (/C,/7o, Hi) and such that (p~^w,p~'d) is associated with
iK',n'o,rTi). Clearly it suffices to show that U*iK,K') has finite
[\n{n - 1) - l]-dimensional Hausdorff measure.
We may assume that /7o = //„._i and H, = H^^. In the following,
^15. ^16,. • . denote positive constants depending only on n, on Ky-
K n Hy and on K'J'= K'D n'^ (v=0, 1). Let V* be the set of all
2-frames associated with (/C,/To,/7i). If {u,v)eV*, then there are
points jcq, Xi satisfying

92
Boundary structure
xn,xie FiK, v), B.3.13)
" = (^1 - ^oVUi - ^ol
The point jcq + jcj lies in one of the caps Ci, . . . , C,„ used in the proof
of Theorem 2.3.1, say jcq + Xi e Q. Then jcj - jcq e DC, + t, with some
vector /; (compare the proof of Lemma 2.3.9). The set DC, was covered
by N, (n - l)-dimensional cubes of edge length e, say by W\'\ . . . ,
W*v,- We denote by V,* the set of all (w, d) e V* for which points jcq, a:,
satisfying B.3.13) exist such that jcq + Xj e C, and .Vi - xq e W ■' + tj.
Then V* is the union of all these V^.
For jc e E", we denote by x the orthogonal projection of x onto the
linear subspace //„.o- Let Wi, W2 e 5"~\ (Uv» ►^) = ^ and JCve //„ o for
v= L 2. Then
X Y X Y
w
and hence
Ixi - X2\
\X\ - X^
(^1, )
(jfl - ^2, W2)
(w, Wi) (w, W2)
= 3|xi - X2I + 6lx,||wi - W2l-
Now let (wi, Di), (w2, i'2) e V*. Then
(w, W2)
B.3.14)
|u, - U2I ^ diam wf S V« - le-
The vectors V\, vi are exterior normal vectors of Ki)+ Ki at points
belonging to the same cap C,. As in the proof of Theorem 2.3.1, this
cap contains a cap, of the same width A'id) < c^e^, that is cut, by the
same (n - 2)-plane, from an (n - l)-ball of some positive radius r). It
follows that the angle between 0] and O2 is less than Ci^e. Let Aj, A2 > 0
be such that AiUj, A2t32 are unit vectors, then
|A,t;, - A2t;2l = ChE.
Since Aj, A2 = 1 and Di, V2 are unit vectors, we have
|i;i - vil S |AiDi - A2iJl-
From B.3.14) we obtain
lAjDi - A2i.'2l = 3|Ait;i - A2i52l + 6|wi - W2I
and thus
\vi - V2\ ^ Ci6e.

Notes for Section 2.3 93
It follows that, in terms of the metric r introduced on V„ 2.
diam Vfj ^ cpe.
By the proof of Theorem 2.3.1, the total number of the sets V,*
is N ^ ci^e'^~". Thus V* can be covered by N = ci^e^'" subsets of
Ai, .. . , A,\ of V„2 of diameter Cne.
Similarly, the set of all 2-frames associated with (K', /Tq, TTi) can be
covered by N subsets A'l, . . . . A'^ of V„ 2 of diameter cne.
Now let p e U*{K, K'). Then there exists a 2-frame (w, v) associated
with (A:,/7o,/7i) such that p~\u,v) is associated with iK',n'o,n\).
For suitable i, j we have (w,i')e/l, and p~^{u.v) e A'j, hence
{u,v) e AjH pA'i and therefore pe X{A,, A'j) (as defined in Lemma
2.3.12). Thus
.s
U*{K, K') C U XiA„ A'l).
By Lemma 2.3.12, each set XiAj.A'j) can be covered by less than
c,9£~(""^>("~'*>''^ balls of diameter e, hence U*iK, K') can be covered
by less than
balls of diameter e. Since this is true for all sufficiently small e > 0, the
set U*iK,K') has finite [n(n - l)/2 - l]-dimensional Hausdorff
measure. This completes the proof of Theorem 2.3.10. ■
Notes for Section 2.3
1. The question of whether the set U(K) in Theorem 2.3.1 has (n - l)-dimen-
sional measure zero was first asked by Klee A9$7a). (Ewald, Larman &
Rogers A970) mention an example due to Bing and Klee of a topological
2-sphere in E' containing segments of all directions; thus some assumption
Such as convexity is necessary.) For « = 3 (the two-dimensional case is
trivial), McMinn (I960) showed that U(K) can be covered by the ranges of
countably many Lipschitz mappings from [-1,1] to 5^. This implies that
U(K) is of a-finite one-dimensional Hausdorff measure. A shorter proof was
given by Besicovitch A963b). For « §4 Klee A969a) posed the problem of
showing that U(K) ^ S""'. This was proved in the stronger form given in
Theorem 2.3.1 by Ewald, Larman & Rogers A970). In fact, these authors
obtained a more general result. To formulate it, let G(n, r) be the Grass-
mann manifold of all r-dimensional linear subspaces of E". It is a compact
differentiable manifold of dimension r(n - r) and can be equipped with a
rotation invariant Riemannian metric. This induces a distance function and
thus Hausdorff measures. The following general result can be stated.
Theorem. Let K e X" and lg5grg«-l. The set of all r-dimensional
linear subspaces of E" parallel to some supporting r-flat of K that contains an
5-dimensional convex subset of K has <7-finite ['■(n - r) - 5)-dimensional
Hausdorff measure.

94 Boundary structure
The case s =r of this theorem was proved by Ewald. Larman & Rogers
A970). By extending their methods, Zalgaller A972) treated the case 5 = 1,
and briefly sketched a proof of the general theorem above.
Some of these results can be interpreted in terms of the existence of
r-dimensional Chebyshev subspaces of ^-dimensional Banach spaces; see
Klee A969a) and Zalgaller A972). The case * = 1 implies, in particular, that
for almost all r-dimensionai linear subspaces E, the shadow boundary of K
in direction E is sharp. This was stated, without a complete proof, bv Ewald
A964).
By further extending the methods of Zalgaller, Schneider A978d) obtained
Theorem 2.3.10.
As mentioned after Theorem 2.3.1. the possible presence of facets is an
obstruction to improving the result, and a similar remark concerns the
generalizations mentioned above. However, Ewald, Larman & Rogers A970)
showed the following. If2gr§«-2 and K e '<^;; has no facets, then the
set of ail r-dimensional linear subspaces of E" parallel to some r-dimensional
convex subset in the boundary of K has r{n - r - l)-dimensional Hausdorff
measure zero, Further, the following was proved by Larman & Rogers
A971). Let H be a hyperplane in E". The set of unit vectors parallel to
segments in the boundary of a convex body K that are parallel to H but not
lying in the support planes of K parallel to H has (n - 2)-dimensional
Hausdorff measure zero.
Using the result of Ewald, Larman & Rogers, Ivanov A973) deduced the
following result on segments in the boundary; The union of all lines in E"
that meet the boundary of the convex body K 6 X" in a segment has a-finite
(n - l)-dimensional Hausdorff measure. Consequently, almost all points
X 6 ^"\K have the property that the shadow boundary of K under central
projection from x is sharp.
For convex subsets in the boundary of a convex body cut out by concurrent
sections. Ivanov A976) showed the following. Let K e X'f, be a convex body
with o 6 int A^ and let 2=rg«-l. The set of all r-dimensional linear
Subspaces of E" containing an (r - l)-dimensional convex subset of bd K that
does not meet the relative interior of a facet of K has Haar measure zero in
G^n.r).
Methods related to the cap covering theorem of Ewald, Larman & Rogers,
which is Theorem 2.3.2 above, have proved useful in other contexts, too.
See, for instance, B^rdny & Larman A988) and BSriny A989).
2.4. Polytopes
The boundary structure of polytopes is particularly simple as far as the
facial structure and the classification of boundary points are concerned
(not, of course, from the combinatorial or metrical viewpoint). In this
book, polytopes are mainly considered as tools for obtaining results on
general convex bodies. We treat, therefore, only the most basic
properties of polytopes.
A polytope in E" is, by definition, the convex hull of a finite
(possibly empty) subset of E". If P = conv(jci, . . . , Xk), then clearly
the extreme points of P are among the x^ xi,: hence, by Minkow-

2.4 Polytopes 95
ski's theorem, the polytopes are precisely the convex bodies with finitely
many extreme points. If // is a support plane of P, then
H n P = conv(// n {jci, . . . , Xk}),
hence each support set of a polytope is itself a polytope. Since each
extreme point of a Minkowski sum P] + P2 's the sum of an extreme
point of Pi and an extreme point of P2, the sum of two polytopes is a
polytope.
First we show that for polytopes the distinction between faces and
exposed faces (support sets) is unnecessary.
Theorem 2.4.1. Let P C E" be a polytope, F] a support set of P and F a
support set of Fi. Then F is a support set of P.
Proof. We may assume that o e F, There is a support plane //^o to P
with Hu,qC\ P= F\ and PC H~_q. In //^o there is a support plane H
to F, with H n Fi = F, say
H = {x e //,,o|(a:, v) = 0},
FiC (jc e //,,o|(a:, v) ^ 0}.
Define
rjQ := max(-(A:, v}/{x, u}\x e extP\extFi}
and Hirf) := H^u+i:.o with r; > r/g. We have {x,u}<0 for xe
ext P\ext Fi, hence
{x, rju + v) < rjoix, u) + {x, v) ^ 0
by the definition of r)Q. For x e ext Fi\ext F we get
{x, r]u + v) = {x, v) < 0,
and for x e ext F, {x, rju + 1;) = 0, Thus ext F C H{r]), whereas ext P\
ext FC int//;j^y+„n. We see that Hirfj is a support plane to P with
H{r})n P = F and therefore F is a support set of P. ■
Corollary 2.4.2. Each proper face of a polytope is a support set.
Proof Assume that P is a polytope of dimension n and that the
assertion has been proved for polytopes of smaller dimension (for
zero-dimensional polytopes there is nothing to prove). Let F be a
proper face of P. As remarked earlier (before Theorem 2.1.3), F is
contained in a support set Fi of P and it is a face of F]. Now either
F = Fi and then F is a support set of P, or F is a proper face of F]
and then a support set of F], by the induction hypothesis. By Theorem
2.4.1, F is also a support set of P. ■

96 Boundary structure
In particular, each extreme point of a polytope P is an exposed point
of P The extreme points of a polytope are called its vertices; the set of
vertices of P is often denoted by vert P. The one-dimensional faces of a
polytope P are its edges, and the (dim P - l)-dimensional faces are its
facets, li x\, ■ .. , Xk are the vertices of the polytope P and F is a
proper face of P, then there is a hyperplane H with
F = H n P = conv(// n (jci, . . . , Xk))-
Thus each face is the convex hull of a subset of the vertices, and 'TiP)
is finite.
A polytope was defined as the convex hull of finitely many points.
Alternatively, it can be represented as the intersection of finitely many
halfspaces.
Theorem 2.4.3. Each polytope is the intersection of finitely many closed
halfspaces.
Proof Let P C E" be a polytope. Since flats and half-flats can be
represented as intersections of finitely many closed halfspaces, it suffices
to assume that dim P = n. Let Fi, . . . , F^ be the facets of P; then
f, - HiC\ P where H^ is a unique support plane of P. Let //"be the
closed halfspace bounded by //, and containing P (/ = 1, . . . , k). We
assert that
P = Hi n ...n HI. (*)
The inclusion P C H\ r\ ... r\ HI is trivial. Let x e 1E"\P. Let A be
the union of the affine hulls of x and any n - 1 vertices of P. We can
choose a point y e (int P)\A. There is a point z e bd P n [jc, y\, it lies
in some support plane to P and hence in some face F of P. Suppose
that dim F := j'^ n - 2. By Caratheodory's theorem, z belongs to the
convex hull of some / + 1 S n - 1 vertices of P and hence to A. But
then y € A, a contradiction. This shows that f is a facet, hence F = F;
for suitable / e A, ..., k). From y e int P C int//^ we deduce that
a: ^//~. This proves (*). ■
u.a
Let Pe Xo he a polytope and F a facet of P. Then F = P n H,
and P C Hu^a for suitable u and a. We may assume that u e S""^; then
u is called the outer unit normal vector of the facet F. The outer unit
normal vectors of the facets of P are called, in brief, the normal vectors
of P.
Corollary 2.4.4. If P e Xo is a polytope and Wi, . . . , u^ are its normal
vectors, then

2,4 Polytopes 97
^ - I \^u„h{P.u,)
;=i
In particular, the numbers h{P, Wj), . , , , h{P, u,,) determine P uniquely.
Proof. If f, is the facet of P with normal vector u,, then
F, = Pn H
u„h(F.u,)- The assertion is now clear from the proof of
Theorem 2,4.3. ■
At this point, it is convenient to use polarity, and this requires the
following lemma. It shows, in particular, that the polar body of a
polytope (with respect to an interior point) is again a polytope.
Lemma2.4.5. Let Ui, . . . , u^ e IE"\{o}. //
k
p ■■= n^:,.!
1=1
is bounded, then
P* = conv{t<i, Uk).
Proof. (We remark that o e int P, hence P* is well defined.) Let
Q := conv {ui, . . . , W/t}. For jc e P we have {x, w,) ^ 1, hence w, e P*.
Thus QQ P*.
Let i; e 1E"\B. Then there is a hyperplane //,.„ strongly separating Q
and V, and by an appropriate choice of signs we may assume that
{z,Ui}<a for / = 1, .... k and {z,i>)>a. Since P is bounded,
clearly pos{wi, ..., Uk} = ^" (otherwise the closed convex cone
pos{wi, ..., Uk) has a boundary point and hence a supporting
hyperplane, necessarily through o, each outer normal vector of which is
an element of P); hence there is a representation
k k
o = S-^i"/ with A, ^ 0 and ^-^i > 0.
1=1 1=1
This yields 0 < X'^/* ^nd thus a> 0; without loss of generality, or = 1.
But then z e P, hence /; ^ P*. This proves the equality Q = P*. ■
Theorem 2.4.6. Each bounded intersection of finitely many closed half-
spaces is a polytope.
Proof Let P = C\i=iH; be bounded, where //" C E" is a closed
halfspace. We may assume that dim P = n (otherwise we work in aff P)
and, after a translation, that o e int P. Then o e int H,\ hence H~ can
be represented in the form H[ = Hu,a. Lemma 2.4.5 shows that P* is a

98 Boundary structure
polytope. By Theorem 2.4.3, P* is the intersection of finitely many
closed halfspaces (and o e int P* , as always). Hence P** = P is a
polytope. ■
Theorem 2.4.6 implies, in particular, that the intersection of finitely
many polytopes is a polytope and that the intersection of a polytope
with a flat is a polytope.
More information on the faces of a polytope is now easy to obtain.
Theorem 2.4.7. Each proper face of a polytope P is contained in some
facet of P.
Proof. We may assume that dim P = n. In the proof of Theorem 2.4.3
it has been shown that P has a representation
1=1
where //~ is a closed half space and f, := P n //, is a facet of P (/ = 1,
...,k).
Since bd P C Uf=i(P H //,), each boundary point of P is contained in
some facet f,.
Now let f be a proper face of P and choose x e relint f, Then
X e Fj= P r\ Hj for some j e {\, . . . , k). Since x e relint F and Hj is a
support plane of P, we deduce F C Hj, hence F C Fj. ■
Corollary2.4.8. Let P be a polytope, let F> e 'T;(P) and f* e 'J^iP) be
faces such that F' C F*. Then there are faces F' 6 'IfiiP) (/ = ; + 1, . . . ,
k-i) such that F' C F'+^ C . . . C F*"' C F*.
Proof. Clearly F' is a face of F* (being a support set of P and hence of
F*). If F' is a proper face of F*, then F' is contained in a facet F*"' of
F*, and by Theorem 2.1.1, F*"^ is a face of P. By repeating this
argument, the assertion is obtained. ■
By Lemma 2.2.3, the normal cones of an n-dimensional polytope can
be related to the faces of its polar polytope (with respect to an interior
point). This yields:
Theorem2.4.9. Let P e .Xg he a polytope, let F^, . . . , Fk be its facets,
and let Ui he the outer unit normal vector of F, (i = I, . . . , k). If F is a
proper face of P, then
NiP, F) = pos(u,|FC F,}.

2.4 Polytopes 99
Proof. We may assume that 0 e int P. From Lemma 2.2.3 we know that
N{P, F) = pos F. Writing v, = uJhiP, Ut), we have
k
i=\
by Corollary 2.4.4, hence Lemma 2.4.5 yields
P* = conv(i;i, . . . , D;^}.
The face F of P* is the convex hull of those d, that it contains, and
pos F is the positive hull of these v, and hence of the corresponding w,.
Now D, e F is equivalent to F C //„.!, which is equivalent to F C F/. ■
We note some consequences of Theorem 2.4.9. Let P e IC^ be a
polytope and F a proper face of P. Then F is the intersection of the
facets of P containing F; this follows from Theorem 2.4.7 and the fact
that the intersection of all facets containing F is a face of the same
dimension as F. Let G be anotker proper face of P. From Theorem
2.4.9 it follows that
F C Go NiP, F)D NiP,G) B.4.1)
and that
dim N{P, F) = n - dim F. B.4.2)
As a consequence, a point jc of P is r-singular if and only if it belongs
to some face of dimension Sr. Thus for n-polytopes, the notions of
r-extreme, r-exposed and r-singular boundary points all coincide. In
particular, the extreme points (vertices) of P are precisely the points x
with dim N{P,x) = n.
From B.4.1) one further deduces for v e lE"\{o} that
F = F(P, v)ov e relint N{P, F) B.4.3)
and that
NiP, F)= f] ^iP' ^)- B.4.4)
jteext F
If o e int P, so that the polar polytope P* is defined, then F C G for
proper faces F and G of P implies F D G by B.4.1) and Lemma 2.2.3.
Thus the map F >-^ F from the face lattice '7(F) to the face lattice
7(P*) (observe that 0 = F* and P = 0) is an antimorphism, that is, a
bisection that reverses the inclusion relation.
A polytope is called simplicial if all its proper faces (equivalently, all
its facets) are simplices. An n-polytope P is called simple if each of its
vertices is contained'in exactly n facets. Clearly this holds if and only if
P* (taken with respect to some interior point of P as origin) is
simplicial.

1(X) Boundary structure
The normal cones of a polytope give rise to a series of important
metric invariants: Let P e X" be a polytope and F e 7;fe(P) a /c-face of
P. Then the number
y{f, P) :- z.''-'^-i(N(P, F) n 5''-')K_,.
(with the convention yiF, P) = I \i k = n) is called the external angle of
P at its face F. Thus yiP^ P) 's the (n - k - l)-dimensional measure of
the spherical image a{P, relint F) of P at relint F, divided by the total
measure of the {n ~ k - l)-dimensional unit sphere. If dim P < n, the
number yi^^^ P) remains the same if computed in any affine subspace
containing P; this follows from B.2.3) and Fubini's theorem. From
B.2.4) and B.2.5) one deduces that
S YiF, F) = 1. B.4.5)
We shall now establish a special result dealing with approximation by
polytopes, which will be useful in later chapters. Some preparations are
required.
Two polytopes Fi, F2 are called strongly isomorphic if
dimF(Fi, u) = dimF(F2, u)
for all we 5""'. Obviously this is an equivalence relation; the
corresponding equivalence class of a polytope F will be called its a-type.
Lemma 2.4.10. If Fj, Pj are strongly isomorphic polytopes then, for each
u e 5""', the support sets F{P\, u), F(F2, u) are strongly isomorphic.
Proof Let w 6 5"~' be given. Let u e 5""^ and put FiFiPt, u),v) :=
F', {i = 1,2). We may assume that o e Fi n F2. As shown in the proof
of Theorem 2.4.1, for t} sufficiently large the hyperplane H,^u+v.<i
supports Fj and //;,„+„,o H Fj = F; for / = 1, 2. Since Fi, F2 are
strongly isomorphic, we get dim F\ = dim F2 and thus the assertion. ■
Lemma2.4.11. The polytopes Pi, PjeX" are strongly isomorphic if
and only if
(yV(F„ x)\x € ext Pi) = {NiP2, y)\y e ext F2}. (*)
Proof Suppose that Fj, F2 are strongly isomorphic. Let JceextFi.
Choose w eintN(Fi,Jc) and >> e F(F2, w), then >'eextF2. Let
veNiPi,x) and A e [0,1); then v^:= A - X)u + kv elm NiPi,x),
hence dim F(F2, ua) = dim F(Fi, j^a) = dim (x} = 0 and, therefore,
P(P2> vx) = {yx) for some y^ e ext P2, which implies 1;;, 6 int N(F2, yO-

2.4 Polytopes 101
Since this holds for all Ae[0,1), we conclude that y;^ = y. It follows
that veNiP2,y), thus NiPi,x)C. NiPj, y). Since xeextP^ was
arbitrary and the roles of Pi and P2 can be interchanged, it follows that
NiPux) = N{P2, y). Thus (*) holds.
Suppose, conversely, that (*) is true. Let u e 5""' and choose a
vertex Xy of F(Pi,w). There is a vertex X2 of P2 for which
N{P2,X2) = N{Pi,xi). The vector u lies in the relative interior of a
face of N{Pi,Xi) of dimension n - dim F(P,, u) (as follows from B.4.2)
and B.4.3)), hence dim F(Pi, u) = dim F(P2. ")• Since u was arbitrary.
Pi and P2 are strongly isomorphic. ■
Let Pi, P2 be strongly isomorphic polytopes. Let F be a proper face
of Py. The normal cone N{Pi, F) is a face of N(Pi,x) for some vertex
X of F, hence by Lemma 2.4.11 a face of N{P2,y) for some vertex y of
P2 and thus the normal cone of a unique face <p(F) of P2. It is clear
(using B.4.1)) that qp (extended to improper faces) is a bijective map
from 'TiPi) to ''7{P2) that preserves inclusions, that is, a combinatorial
isomorphism. This explains the term strongly isomorphic. In the
literature, strongly isomorphic polytopes have also been called
analogous (for this reason, we speak of the a-type of a polytope), strongly
combinatorially isomorphic {strongly combinatorially equivalent), or
locally similar.
Lemma2.4.12. Let P be a simple n-polytope with normal vectors W],
. . . , Uk. Then there is a number j8 > 0 such that each polytope of the
form
k
P' ■■= \\Hu^,h{P,u,)+a, (*)
1 = 1
with \ai\^ p is simple and strongly isomorphic to P.
Proof. We can choose pairwise disjoint balls B{x, p) for jc e ext P such
that each ball B{x, p) meets only those facets of P that contain x. Next,
we can choose a number /3 > 0 with the following property: If jc is a
vertex of P and if, say, U\, . . . , u„ are the outer normal vectors of the
facets containing x, then for |a,|^/3 the intersection point of the
hyperplanes //„/,(p„)+a (/ = 1, .., n) lies in B{x,p), and H„ /,.
(p.u)+a does not meet B{x,p) for ;' ^ A, . . . , n). It is then clear tliat
P' defined by (*) with |a,| ^ /3 is a polytope having a vertex x' in each
of the balls B{x'p) (jc e ext P) such that N{P',x') = N{P,x). We
deduce that P' has no further vertices and that it is simple and strongly
isomorphic to P, by Lemma 2.4.11. ■

102 Boundary structure
Lemma2.4.13. Let P be an n-polytope with normal vectors ui, . . . , Uk-
Then to each j8 > 0 numbers aj, . . . , a^ with jad S jS can be chosen
such that
k
1=1
is a simple n-polytope with normal vectors Wj, . . . , u^ and the property
that each normal cone of a vertex of P' is contained in some normal cone
of P.
Proof. We choose balls B{x,p) (jc e ext P) as in the proof of
Lemma2.4.12. If |ai|, ..., \(x,^\ are sufficiently small, then P' is an
n-polytope with normal vectors u^, . . . , u^ and such that any vertex x'
of P' is contained in some ball B{x, p). By Theorem 2.4.9, N{P',x') is
then the positive hull of only those normal vectors that belong to facets
of P containing x; hence N{P',x') C N{P,x). By choosing the or,-
successively in an appropriate way, we can clearly achieve that the
resulting polytope P' is simple. ■
We are now in a position to prove the following result on
simultaneous approximation of convex bodies by strongly isomorphic poly topes.
Theorem2.4.14. Let Ki, . . . , K„ e K" be convex bodies. To each e>0
there exist simple strongly isomorphic poly topes Py,. . . , P„e X"
satisfying d{Ki, Pi) < e /or / = 1, . . . , m.
Proof Let e > 0 be given. By Theorem 1.8.13 there are poly topes Qi,
. . . , <2„ e X" with SiKi, Qi) < e/2 for / = 1, . . . , m. Let P-=Qi +
■ ■ ■ + Qm (which is also a polytope) and let Wi, . . . , w^ be the normal
vectors of P. Starting with P, we construct P' as in Lemma 2.4.13 and
put P, := B,-f aP', where a > 0 is sufficiently small that C(P,, Q,)
< e/2 and hence 6{Ki, P^ < e for / = 1, . . . , m. Let jc be a vertex of
some P,. Then x = q + ap with suitable g 6 ext Q, and peextP'.
From Theorem 2.2.1 we have
N{Pi,x) = N{Qi,q)nNiP\p).
By the construction of P', the normal cone N{P',p) is contained in
some normal cone of P, and by Theorem 2.2.1 the latter is contained in
some normal cone of Q,, necessarily N(Qi, q), because N(P', p) n
NiQi,q) is n-dimensional. Thus N(P,, jc) = N{P',p). Since x was an
arbitrary vertex of P,, we deduce from Lemma 2.4.11 that P, is strongly
isomorphic to P'. This implies also that P, is simple. ■

2.5 Higher regularity and curvature 103
Notes for Section 2.4
Standard references on convex polytopes, from which we have taken some of
the basic arguments, are the books by Griinbaum [15], McMullen &
Shephard [26] and BrOndsted [11).
Strongly isomorphic polytopes. The approximation theorem, 2.4.14, is due to
Aleksandrov A937b). He defined strongly isomorphic polytopes in a
different, but equivalent way (and called them analogous). Our definition appears
in Shephard A963), where the term "locally similar' was used. Still another
term was introduced by Griinbaum [15], p50.
Lemma 2.4.11 can also be found, in essence, in W. Meyer A969, Corollary
2.7), and in Kallay A982).
Monotypic polytopes. McMullen, Schneider & Shephard A974) studied the
n-polytopes P, called monotypic, with the property that the a-type of P is
uniquely determined by the system of normal vectors of P (see also Section
3.2, Note 3).
Approximation by polytopes. While in this book the approximation of convex
bodies by polytopes is only used as a tool, it is an interesting topic in its own
right and can be studied under various aspects. We refer the reader to the
survey article by Gruber [33], which contains many references. Later
contributions to this subject are by Bokowski & Mani-Levitska A984, 1987),
Florian A986), Gruber A988, 1990+, 1991b) and Schneider A986a, 1987b).
Schneider A988c) gives a survey on the approximation of convex bodies by
random polytopes.
2.5. Higher regularity and curvature
Another special class of convex bodies, besides the polytopes, with an
accessible boundary structure are the convex bodies whose boundaries
are regular submanifolds of E", in the sense of differential geometry,
and which satisfy suitable differentiability properties. The present
section is mainly concerned with the second-order boundary structure of
such convex bodies. The central notions are those of curvature, principal
curvatures and radii of curvature. The last part is devoted to the
curvature properties of general convex bodies.
First we define curvatures for a general convex body K € JCq with
interior points. We assume that jc is a smooth boundary point of K. Let
u be the unique outward unit normal vector to K at jc. The linear
subspace H{K, w) - jc of E" is called the tangent space of AT at jc and is
denoted by T^K. We can choose a number e >0 and a neighbourhood
C/ of JC such that U C\bAK can be described in the form
U r\hAK ^ {x + t - f{t)u\t e T^K n B(o, e)}, B.5.1)
where /: T^K n B(o,e)—»IR is a convex function satisfying /^O and
/(o) = 0.
Let t e T^K n 5" ' be a given unit tangent vector of K at jc. For
0 < T < e, let c(r) be the centre of the circle lying in the plane through
JC spanned by u and t, going through the point jc + rr-/(rr)w and

1(L Boundary struciure
touching H{K,u) at x. If {Tj)j^n is a sequence such that ty^O for
;■—»00 and (c(ry))y6f^ converges to some point x - pu, and thus
. ^/
y^« 2/(ryr)'
then p is called a radius of curvature of X at jc in direction t. The set of
all numbers p (ao admitted) arising in this way is an interval
[Pj(A-, f), p^{x, t)] with 0 ^ pi(jc, r) g PsCx, t) ^ 00. We call
p,(jc, r) = liminf
rio 2/(rr)
the /ower radius of curvature and
Ps(jc, r) = limsup
r^
rio'^ 2/(t0
the upper radius of curvature oi K &i x in direction t. The reciprocal
values (possibly »)
Ki(A:, r) := \Ip,{x, t), K,ix, t) := \/p,ix, t)
are, respectively, the lower curvature and upper curvature oi K al x in
direction r. If both values coincide and are finite, their common value
k{x, t) is the curvature oi K al x in direction t.
Now we introduce differentiability assumptions. A convex body
K e Xq is said to be of class C*, for some Ac e M, if its boundary
hypersurface is a regular submanifold of E", in the sense of differential
geometry, that is /c-times continuously differentiate. K is of class C" if
it is of class C* for each k eN. Since locally bd K can be represented as
the graph of a convex function, as in B.5.1), it follows from Theorems
1.5.12 and 1.5.11 that K is of class C^ if and only if it is smooth (i.e., it
has a unique support plane at each boundary point). Here one has to
use the property that a differentiable convex function is continuously
differentiable, as follows from Theorem 1.5.2, applied to the partial
derivatives.
We assume that the reader is familiar with the elementary differential
geometry of hypersurfaces of class C^. We recall some of the basic
definitions and facts and then concentrate on some special aspects of the
convex case.
Let K € Xq be of class C^. We identify, in the canonical way, the
abstract tangent spaces of E" with the vector space E" itself and thus
also the abstract tangent space of the submanifold hd K at jc with the
linear subspace T^/C, as introduced above. For jc € bd AT, let v(jc) be the
outward unit normal vector of AT at jc. The map v: bd /C —» 5""' C E"
thus defined (also denoted by Ok in Section 2.2) is called the spherical
image map (or Gauss map) of K; it is of class C'. Its differential

2.5 Higher regularity and curvature 105
dv^ := W^ maps T^K into itself. The linear map W^.T^K^ T^K is
called the Weingarten map. The bilinear form defined on T^^K by
Ilxiv, w) := (W^v, w) for v, w e T^K
is the second fundamental form oi bd K at x. The first fundamental
form is just the restriction of the scalar product to Tj^K,
/^(d, w) := (d, w) for v, w e T^K.
In local coordinates, the foregoing can be expressed as follows. Let
X: M —» E", where M C R""' is open, be a parametrization of class C^
of a neighbourhood of jc on bd K^ where, say, x = X{y) with y e M.
Put N ■= V o A", so that N: M—»5""^ is a map of class C^ We have
dNj, = dv^ o dAy and hence Wj, = dNy « dA^V For a differentiable
function «jp on M we denote by (pt the partial derivative with respect to
the ith (Cartesian) coordinate. Then dX{ei) = A",, dN(e,) = N, (where
e, is the ith basis vector of R""') and hence W^A, = N, (as usual, we
omit the argument y). The matrix (/,y)";^\ of the bilinear form //^ with
respect to the standard basis (A], . . ., A„_i) of Tj,K has entries
/;,= //,(A„ Ay)= (W,A„ A,)
= (yV,-, Ay) = -(yV, A,y), B.5.2)
the latter because (N,Xj) =0. Thus //^ is symmetric, and the
Weingarten map is self-adjoint (with respect to ( •, • )). The matrix (g,y)",^'i
of the first fundamental form is given by g,y = (A,, Ay). For the matrix
n-'
(';)iV=i of the Weingarten map, which is defined by
Ni= 2 li^r i = 1, . . . , n - 1, B.5.3)
one obtains /,* = (N„ A*) = S/'K^/- ^k) = 'Zjligik, hence
l\ = 2 likS"', B.5.4)
where (g'').,; denotes the inverse matrix of (gjy),,/.
The eigenvalues of the Weingarten map W^ are, by definition, the
principal curvatures oi K at jc. The equality Wj,v = Ad for v e Tj,K\{o}
and A e IR is equivalent to //^(d, w) = A/^(d, w) for all w e T^/C. Thus
the principal curvatures are also the eigenvalues of the second
fundamental form relative to the first fundamental form. In local coordinates
as above, d = 2i=iv'Xi is an eigenvector of the Weingarten map and A
is a corresponding principal curvature if and only if
' S Ci - A<3>' = 0,
.=1
or equivalently

106 Boundary structure
n-l
1=1
for y= 1, ...,«- 1.
For the special representation given by B.5.1), that is, X{t) =
X + t ~ f{t)u for t e T^K n Bo(o. f). one obtains (at the point x)
lii ~ fij ^"*^ 8ii ~ ^ij ^"*^ hence that the principal curvatures at x are the
eigenvalues of the Hessian matrix of /. This provides a simple
interpretation of the principal curvatures and shows that they are non-negative,
by Theorem 1.5.10.
By Hj we denote that yth normalized elementary symmetric function
of the principal curvatures; thus //q = 1 and
//, = (" 7^]"^ 2 k,r-k; B.5.5)
for y = 1, . . . , n — \. We refer to H\ as the mean curvature and to
//„_! as the Gauss-Kronecker curvature-
As is well known from differential geometry, under our present
assumptions the curvature k{x, t) defined initially exists and is given by
11 At, t)
Kix, t) = -r^ for t e T,K\{o}. B.5.6)
Let (d,, . . . , D„_i) be an orthonormal basis of Tj,K consisting of
eigenvectors of the Weingarten map W^, and let /c, be the principal
curvature corresponding to Dj. Then
Kix, t) = ki(t, v^y + ... + k„.i(t, i;„_,J B.5.7)
for each unit vector t 6 T^K. This result is known as Euler's theorem.
The following assumption, stronger than C*, will often be important.
We say that K is of class C+ (for k^2) if K is of class C* and the
spherical image map v:bdK^—»5"~^ is a diffeomorphism (of class C',
and hence C*"^). This is equivalent to the assumption that its
differential, the Weingarten map, is everywhere of maximal rank, and thus to
the assumption that all principal curvatures are non-zero, or equivalently
that //„_i =^ 0.
Let K be of class C+. Then the map v has an inverse v~' of class C',
and for the support function Hk oi K we get
hKiu)- (v-Hu),u};
hence hf( is differentiable on S"~^ and thus on lE"\(o}. From Corollary
1.7.3 we have
grad h^iu) = v M-^J
for welE"\{o}. It follows that the support function hn is, in fact, of
class C^.

2.5 Higher regularity and curvature 107
For several investigations on convex bodies, particularly in connection
with Minkowski addition, it appears more natural to impose
differentiability assumptions on the support function than on the boundary.
Let K 6 ^0- By Corollary 1.7.3, the support function hfc is differenti-
able on E"\(o} if and only if K is strictly convex, and if this is satisfied,
Hk is continuously differentiable. Let Hk be of class C^ (note that K
may well have singularities, so that hd K need not be of class C'). For
u e lE"\{o}, let ^(u) be the unique point of hd K at which u is attained
as an outward normal vector. By Corollary L7.3,
^(u) = gvad HkIu). B.5.8)
The map ^ thus defined on lE"\{o} is positively homogeneous of degree
zero.
Now we assume that h^ is of class C^. Then ^ is of class C'. Its
restriction to 5""', the map |: 5"~' -^hdK, will be called the reverse
spherical image map. For u e S"~^, let r„ be the (n - l)-dimensional
linear subspace of E" that is orthogonal to w; thus r„ is the tangent
space of S"~^ at u. The differential d|„ := W„ maps r„ into r„. We
call the linear map W„: r„—» r„ the reverse Weingarten map and the
bilinear form
//„(d, w) := (W„D, w), 11,^6 r„,
the reverse second fundamental form of K at w.
Let N: M-» E", where M C IR"~' is open, be a parametrization of
class C^ of a neighbourhood of u on 5""' and let u = N{y). Put
A':=_|oA^. We have dXy = d^o dNy,_hcnce Wu = dXyodN~^ and
thus WuNi = Xi. The matrix {bij)"~lx of //„ with respect tothe standard
basis {Ni, . . . , Nn-i) of r„ is given by
br, = /7„(yv„ yvy) = (w„yv,, yv^) = (J^,-, n^)
n n
A = l m=l
by B.5.8), where N = («', . . . , n"). Thus /7„ is symmetric and W„ is
self-adjoinL The first fundamental form /„ of the sphere 5""^ at u is
given by /„(i', w) := (v, w) for d, w 6 r„. Its matriix with respect to
(yv,,..., yv„_i) is denoted by (e;y);;^',; thus e,y = 7„(yv„ Ny) = (yv,. yvy).
Introducing the matrix (fci)"/l\ of the reverse Weingarten map by
n-l
A'i = S t>'iNi, / = 1, . . . , n - 1, B.5.10)
we find bik = (Xi, Nk) = 2/''i(^;. ^a) = S/''i«/* and hence
n-l
/>i= S''.*^*'. B.5.11)
where (e'O;,/ is the matrix inverse to (ey),,/.

108 Boundary structure
The eigenvalues ri, . . . , r„_i of the reverse Weingarten map are
called the principal radii of curvature oi K at w. It should be kept in
mind that, while the principal curvatures are functions on the boundary
of K, the principal radii of curvature are considered as functions of the
outer unit normal vector, in other words, as functions on the spherical
image. We denote by Sj the /th normalized elementary symmetric
function of the principal radii of curvature:
\ J / lgi,<...<;,§n-l
The vector d 6 r„\(o} is an eigenvector of the rever^ Weingarten
map and A is a corresponding radius of curvature if W^d = Ad, or,
equivalently, //uCu, w) = aT^Cd, w) for all w e r„. In terms of the local
parametrization N of 5""' used above and with v = "^"^iv'Ni, this is
equivalent to
n-l
2 (b'i - Xd'i)v' =0, ;• = 1, . . . , n - 1 B.5.13)
i=l
and to
n-l
2 it>i, - ^e^y =0, ;• = 1, . . ., n - 1. B.5.14)
1=1
In particular, the product of the principal radii of curvature is given by
det (/),,)
s„.,oN = dctib'd=~-~^. B.5.15)
det (e;y)
Since the principal radii of curvature, ri, . . . , r„_i, are the eigenvalues
of the matrix (fc;),,y, the matrix {b] + pd'i)ij with p6 R has the
eigenvalues ri + p, . . ., r„_i + p, hence
delibi + pd'd = S^p* f" ~ M ^„_,-, o N. B,5.16)
It may sometimes be desirable to express the foregoing notions more
directly in terms of the support function. This can be achieved by means
of the following lemma. Let d^/i„ be the second differential oi h = h^
at u, considered as a bilinear form on E". For u e lE"\(o}, let tTu be the
orthogonal projection onto the orthogonal subspace r„.
LemmaZ.S.l. For ue^"\{o),
d'h.ia, b) = \u\-'(W,fl,,nua, n,h} B.5.17)
for all a, be I".
Proof. Let u e IE"\(o} be given. By homogeneity, we may assume that
|w| = 1. Let (ei, . . . , e„) be an orthonormal basis of E" with e„ = u. We

2.5 Higher regularity and curvature 109
write (>>', . . . , y") for the coordinates of >> e E" and /ly for the second
partial derivatives of h with respect to this basis. The homogeneity
relation
n
2hiiiy)y' =0 for/= !,,...« B,5.18)
yields /i,„(u) = 0 and hence
n-l
d2/i„(a, /)) = 2 hijiu)a'b> for a, b e E".
From B.5.8) we get for /, j ^ n - 1,
hiiiu) = i^iiu), ei) = (dUed, e,)
= (d|„(e,), e,) = (W,e;, e,),
hence
n-l
■./=i
which completes the proof of the lemma. ■
Immediately from B.5.18) and B.5.17) we see the following.
Corollary 2.5.2. The eigenvectors (with respect to (■,•}) of the second
differential of hn at u e S"~^ are u, with corresponding eigenvalue 0, and
the eigenvectors of the reverse Weingarten map at u, with corresponding
eigenvalues rj,. . . , r„_i, the principal radii of curvature.
It now follows from Theorem 1.5.10 that the principal radii of
curvature are non-negative.
Corollary 2.5.3. For j e {1, ..., n-l}, {"~^)Sj{u), the jth elementary
symmetric function of the principal radii of curvature at u e 5"~^ is
equal to the sum of the principal minors of order j of the Hessian matrix
(with respect to any orthonormal basis of 1E"J of hK at u.
In particular
(n - l)si = n + ... + r„_i = AhK on 5""', B.5.19)
where A denotes the Laplace operator on E".
For n = 2, there is only one principal radius of curvature, and we call
it the radius of curvature, denoted by r.
For K as before, let u e 5""' and a unit vector r e r„ be given. We
denote by r{u, t) the radius of curvature, at u, of the image proJ£ K of

110 Boundary structure
K under orthogonal projection onto the linear subspace E spanned by u
and t. To compute r(w, r), we can choose an orthonormal basis (ei,
. . . , e„) of E" such that ei = t and e„ = u. The support function of
proj;r K is just the restriction oi h^ to £, hence B.5.19) gives
riu, t) = huiu) + h„„iu) = huiu),
where the notation is as in the proof of Lemma 2.5.1. By that lemma,
//„(r, t) = (W„r. t) = d^^it, t)
n-l
Thus
iluit, t)
r(u, t) = " for t e r„\{o}, B.5.20)
in analogy to B.5.6), and in analogy to Euler's theorem B.5.7) we have
riu, t) = ri(r, v^y + ... + r„_,{t, v„-^)^ B.5.21)
for each unit vector t e T^, where d; is the unit eigenvector of W„
corresponding to r,.
In the planar case, n = 2, it is often more convenient not to use the
homogeneous support function Hk, but the function h: [0,2n]^fU
defined by hia) = hidua), where (ei, €2) is an orthonormal basis of E^
and Ua = eicosa + e2sina. Using the homogeneity relations, one then
computes Ahidua) = ih" + h)ia), hence the radius of curvature is given
by
riua) = ih" + h)ia). B.5.22)
To extend B.5.22) to n>2, we introduce the spherical Laplace
operator Ag on 5""'. A simple way to do this is as follows (see, e.g.,
Seeley 1966). If /: 5""' ^ R is a function of class C^ one extends it to
IE"\(o} as a positively homogeneous function of degree zero; then Ag/ is
the restriction of A/ to 5""'. If /: lE"\{o}-»IR is positively
homogeneous of degree k, then
A/ = As/ + kik + n - l)f on 5""'.
Hence, B.5.19) gives
s,^hK^-~-^l!^shK- B.5.23)
All this can be applied to convex bodies K e Xq that are of class C+.
As seen before, the support function /i/^ of such a body is of class C^,
and we have ^(w) = v~'(w) for u e 5""'. This yields
W, = d|„ = (dvg(,))-' = Wl^,), B.5.24)

2.5 Higher regularity and curvature 111
SO that the reverse Weingarten map at u coincides with the inverse
Weingarten map taken at ^(u). In particular, the eigenvectors of W|(„)
and Wu are the same, all principal radii of curvature are positive and
'"'(") = TTJru B.5.26)
if the ordering is chosen properly. This implies
^/(") = -%^ (§(")). B-5.27)
Hiix) = ^^^^^ (vix)), B.5.28)
for / = 1, . . . , n - 1.
To have complete symmetry, it remains to show that a convex body
K e ^0 with a support function of class C^ and with everywhere
positive radii of curvature is necessarily of class C+. For the proof, we
may assume that o e intK^. By Remark 1.7.7, the radius function of the
polar body K* is given by p{K*, •) = h{K, ■)'' on 5""', hence K* is
of class C^. As before, we use a local parametrization N: M ^^" of
class C^ of the sphere 5"~' around u and put X = ^oN. From
W„Ni = Xi and the assumption that all eigenvalues of W„ are non-zero,
it follows that A"], . . . , X„-i are linearly independent. Hence, A" is a
local parametrization of class C' of hdK. In particular, it follows that
K is of class C^ and that 1 is a diffeomorphism of class C' from 5""'
onto bd K. The spherical image map v* of X* can be described by
V*: p{A:*, u)u ^ i^, ueS"-\
\^{u)\
(see Remark 1.7.7) and thus is a composition of C'-diffeomorphisms.
Since v* is a diffeomorphism, K^* is of class C+ and hence has a support
function of class C^ and nowhere vanishing radii of curvature.
Repetition of the argument, with the roles of K and K* interchanged, shows
that K is of class C+.
The elementary symmetric functions Sj and Hj appear in some
formulae related to surface area and volume computations. Let K e Xq
be a convex body of class C+. Let X. M^E", where M C R""' is
open, be a local C^-parametrization of bd K. As is well known from
differential geometry, the 3^"~'-measure of X{M) can be computed
from
a5''-i(A(M)) = /^d^,
with the differential geometric surface area element given by

112 Boundary structure
dA = Vdet(gy) dy' • • • dy"-\
where yS . . . , y"~^ are Cartesian coordinates in M. Since /T is of class
C+, the map N = vo A" is a local C'-parametrization of 5""', hence
z.''-'(yv(M)) = [ do
J M
with
do = Vdet(e;y) dy' • • • dy"-\
Now B.5.10) gives
hence det(gy) = [det(/)')l^ det(e,y) and thus
dA = |det(&{)|da = i„_, o A^da
by B.5.15). We deduce that for any integrable real function / on bd K^
we have
j^^J{x)dX"-\x) = j^^J{^{u))s„.,{u)d^."-\u). B.5.29)
Similarly, for any integrable real function g on 5""' we have
\,„.,8iu)<iX"-\u) = j^^^g{vix))H„.,{x)dX"-\x). B.5.30)
Next, we compute a 'local parallel volume', which will play an
essential role in Chapter 4. For a relatively open subset p oibd K, let
Ap{K, p) := {x e E"|0 < d{K, x) ^ p and p{K, x) e p)
for p>0. Thus Ap{K,P) is the set of all points in E" at positive
distance at most p from K and with the nearest point in K falling in p.
Similarly, for an open subset co of 5""' we define
Bp{K, CO) := {x e E"|0 < d{K, x) ^ p and u{K, x) e o)
for p>0. If K is smooth and strictly convex, we clearly have
Ap{K, P) = Bp{K, v(/3)). For K of class C+, the measure of this set can
be computed as follows. It suffices to assume that p= X{M) for some
local parametrization X as above. Then, obviously,
Ap{K, p) = {X{y) + AyV(>')|>' e M, 0 < A ^ p}.
Thus Ap{K, P) is the image of the region M x @, p] C IR"~' x R under
the injective map {y. A) ^ X{y) + ^N{y), which has Jacobian
\X^ + ANi, ..., A'„_, + AN„_i, N\,
where | •, . . , -I denotes the determinant. For this Jacobian, we obtain
from B.5.10) and B.5.16) that it is equal to

2.5 Higher regularity and curvature 113
i i
= del{b'i + Xdl)\Ni, ..., N„-i, N\
= ±SA* l"~ ^]s„.,.,oNVdct{eif).
Here we have used the fact that \Ni, . . . , N„_i, N\^ is equal to the
square of the (n - 1)-volume of the parallelepiped spanned by Ni, . . . ,
N„-i, which is equal to the Gram determinant det((N,, Nj)).
Integration of the absolute value now yields, with at = v(/3), or f} = ^{co),
^>"{Ap{K,p))=X''{B,{K,co))
^-Ip-^His^dX"-^
B.5.31)
where B.5.30) and B.5.27) have been used.
So far, we have considered only one convex body K and have,
therefore, not indicated that the notation introduced depends on K. In
the following, we shall write K{K,x,t), U^{K,v,w), Ux{K),
r{K,u,t), respectively, for k<jc, r), IIj,{v,w), 11^, r{u,t), and so on,
where necessary.
The following theorem, on pairs of convex bodies, allows us to
deduce a global inclusion result (up to translations) from local curvature
comparisons. Here 11^ = Il'y means /^(r, t) = U'y{t, t) for all t.
Theorem2.5.4. Let K, L e DC" be convex bodies of class C\. Then the
following conditions are equivalent.
(a) IIx{L) ^ IIy{K) for all pairs of points jc e bd L and y ebd K at
which the outer unit normal vectors are the same;
(b) r{L, u, t) = r{K, u, t) for each orthonormal pair of vectors u, t;
(c) h/i - hi is a support function.
Proof. Of course, (a) is also equivalent to
(a') K<L, JC, t)^ KiK, y, t) for jc, y as in (a) and all t e T^L = TyK
and (b) is equivalent to
(b') /7„(L) ^ riu{K) for u e 5"'^
By Theorems 1.5.10 and 1.7.1, h^ - /zl is a support function if and

114 Boundary structure
only if d^ihfc - hi) is positive semidefinite. By Lemma 2.5.1, this is
equivalent to (b') and thus, by B.5.20), to (b). Using B.5.24), we have,
for teT^nS"-\
riM.t)= (Wj,t) = {W^^i^t,t),
ihiu)i'' 0= (W'«(«)'' ')■
The equivalence of (a) and (b') now follows, since A^ B for self-
adjoint positive endomorphisms A, B of T^ implies B~'^i4~'.
{A^ B means {At, t) ^ {Bt, t) for all t e T^ and hence is equivalent to
the fact that each eigenvalue of A relative to B is = 1. But ACj = kiBei
is equivalent to B~'(Bei) = A,i4~\Be,); thus the eigenvalues of A
relative to B coincide with the eigenvalues of B~^ relative to ^4"^) ■
The assertion (c) in Theorem 2.5.4 implies that, for suitable z € E",
^K ~ ^L+z — 0 ^nd thus L + zC K (see also Theorem 3.2.9).
The reverse Weingarten map and the notions derived from it have the
advantage that they show simple behaviour under Minkowski addition.
Let Ki, ... , K„ € Xq be convex bodies with support functions of class
C^, and let
K = kiKi + . . . + ^-mKm
with Ai, . . . , A„ ^ 0. Then B.5.8) implies that
^iK,u)= 2^rmr,u),
r = l
which in turn yields
W,iK)= SA.W„(/C,),
r=l
lluiK)= iKIIuiKr).
r = l
If N:M—»E" is a local C^-parametrization of 5""' and if
X^'''> ■■= g{K,,N{-)), then the matrix of HuiK,) with respect to the
basis (N,, . . . , N„_,) is given by fe/y(/Cr) = {xY\ Nj), hence
m
biiiK) = 2Kbii{Kr).
r = l
In particular, for the product of the principal radii of curvature of the
Minkowski linear combination K we get, by B.5.15),
5„-,(/C, N{-)) = [det(e,y)]-idet(Ai6,(A:i) + . . . + X„b^j{K„)).
Generally, if A^, . . . , A„ are symmetric real k x k matrices, the
determinant of XiAi + . . . + X„A„ is a homogeneous polynomial of
degree fc in A], . . . , A„. It can be written as

2.5 Higher regularity and curvature 115
m
det(A,yli + ... + A„yl„)= 2 A,-, • ■ • A,,D(A,, ■ ■ ■ , ^/,),
'i '■* = !
B.5.32)
since the coefficient of A,^ ■ • • A,^ does, in fact, depend only on .4;^, . . . ,
Ai^. One may assume that the coefficients are symmetric in their
arguments, and then they are uniquely determined. D(Ai, . . . , .4^) is
called the mixed discriminant oi Ax, . . . , Aj^.
We deduce that we have an identity
m
.9„_i(AiA:, + ... + a„a:„, «)= 2^ a,, • • ■ Ai„.,5(A:„,..., a:,„.,, «)
'i '»-i=i
B.5.33)
for u e 5""', where s is symmetric in its first n - 1 arguments. We call
s{Ki, . . . , Kn-i, •), which is defined by B.5.33), the mixed curvature
function oi K\, . . . , K„-i. In local coordinates as above,
siK„ . . . , A:„_„ N{-)) = [det(e,y)]-'D(F,v{A:i)), • • ■ , (MA^n-i)))-
B.5.34)
In particular, we see from this and B.5.16) that
SjiK, ■) = siK,...,K, B",...,B", •) B.5.35)
i I n - I-J
for y = 0, . . . , n - 1 and any convex body K with a support function of
class C^.
If we want to express s{Ki, ..., K„-i,u) by means of support
functions, we may, for fixed u € 5""', choose an orthonormai basis (ei,
. . . , e„) of E" with e„ = u. li K € JC" has a support function h of class
C^ then hi„{u) = 0 by B.5.18), hence Corollary 2.5.3 gives
s„_dK,u) = det{hi-{u))^-}i.
We deduce that
siKu . . . , A:„_i, u) = DimiK,, «)r.-Ji, . . . , (V^n-i. «))"/"=!)■
B.5.36)
Finally, in this section, we come back to general convex bodies and
describe, without proofs, their curvature behaviour. Let K € Xq, let x
be a smooth boundary point of K and let u be the outer unit normal
vector oi K a\ X. We may assume that bd K is represented around x by
a convex function /, as in B.5.1). For /i > 0, define
D{h) := Bh)-^l^{t € T,K D B(o, E)\f{t) g h).
If the Hausdorff closed limit lim/,(oO(/i) := D exists, this set is called

116 Boundary structure
the (Dupin) indicatrix of K at x. li K is of class C^, then differential
geometry tells us that D exists and is given by
D = ^iy'eilkiiy'y + ... + k„_,{y"-'f g l| B.5.37)
if (ei,. . . , e„-i) is an orthonormal basis of eigenvectors of the Weingar-
ten map at x and A:, is the principal curvature corresponding to e,. For a
general convex body K, the (smooth) point x is called a normal point
or Euler point of K ii ax x the indicatrix exists and its boundary (if any)
is a quadric in T-^K with centre x. \i x is a normal point, then the
indicatrix at x can be represented in the form B.5.37), with a suitable
orthonormal basis (ej, . . . , e„_|), and this defines the principal
curvatures ki,. . . ,k„-x. They are non-negative, since the indicatrix is
always a convex set. Furthermore, it follows from the definition of D
that the curvature k{x, t) exists for each unit vector t e T^ and is given
by
Kix, t) = ki{t, e,J + . . . + k„-i{t, e„-i>^;
thus Euler's theorem holds.
It follows from work of Aleksandrov A939c) that ;t is a normal point
of K if and only if the representing function / is twice differentiable at
o, in the sense explained in Section 1.5, Note 2. The almost everywhere
twice differentiability of convex functions then yields the following
theorem due to Busemann & Feller A936a) for n S 3 and to
Aleksandrov A939c) for arbitrary n.
Theorem 2.5.5. For each convex body K € jETo. ^'"'^-almost all
boundary points are normal.
Although this is a strong theorem, it should be kept in mind that
essential information on the shape of a general convex body may be
carried by those points that are not normal. This is shown clearly by the
example of a poly tope.
Notes for Section 2.5
Jensen's radii of curvature. The definitions of lower and upper radii of
curvature given at the beginning of Section 2.5 are not the only natural
ones. To describe a different possibility, we assume that n =2 and that
K 6 Xo is represented in a neighbourhood of the smooth boundary point x
in the way given by B.5.1). Let f € T^K Pi S"-^. For 0< t, < e let'o, be an
outer unit normal vector of K at x(t,) := j: + T,f — f{xit)u, and let z(t,, o,)
be the intersection point of the normals {j:-Am|ASO} and

Notes for Section 2.5 117
{j:(Ty) - AyylASO}. If t,—»0 for /—»oc and z(T,,i),) converges to some
point X - ru, thus
r = lim —, ,
,-«Vi - {u,,>,y
then r is called a Jessen radius of curvature o( K &t x in the direction t. The
set of all numbers r (<» admitted) obtainable in this way is an interval [r^, r^]
with OS ri g rs^ oo. It contains the interval lp„ps]- Jessen A929) showed
that each Jessen radius of curvature, say r, satisfies
Ps - Va(A - P.) S r S A + Va(P. - a),
and that these inequalities comprise the only general restrictions for p-,, p^,
r^, r^, In particular, pi = p^ implies r^ = r,. Connections with the second-
order differentiability properties of / at o are described in Busemann [12],
Section 2.
Complete Riemannian submanifolds with non-negative sectional curvatures. If
K € .Tg is of class C°° (say), then bd K with the induced Riemannian metric
is a Riemannian manifold with non-negative sectional curvatures. This
assertion has an inverse that is not restricted to the compact case:
Theorem (Sacksteder). If A/ is a C (n - l)-dimensional complete orient-
able Riemannian manifold of non-negative sectional curvature that is not
identically zero, and if X: A/—»E" is an isometric immersion, then X is an
embedding and X(M) is the boundary of a convex set.
This theorem is due to Sacksteder A960), with special cases proved
before by Hadamard A897), Stoker A936), van Heijenoort A952) and
Chern & Lashof A958).
The Gauss map of such submanifolds was investigated by Wu A974).
More generally, let K CE" be a closed convex set with interior points,
K ^ E", and let a{K) be the set of all outer unit normal vectors at boundary
points of K. Then Wu showed that there exists a ^-dimensional great
subsphere 5* of 5""' (OS k^ n - 1) and a unique subset oo of 5* that is
relatively open and (geodesically) convex such that aoCa{K)Cc\ao. Wu
also showed by an example that a{K) itself need not be convex.
Boundaries and support functions of class C^. That the support function of a
convex surface of class C^ with positive curvatures is of class C^ was noted
by Wintner A952, Appendix). Here the assumption of positive curvatures
cannot be omitted. Hartman & Wintner A953) gave an example showing
that the support function of a convex body need not be of class C^ even if
the boundary is real-analytic and also treated some related questions.
Differentiability properties of projections and Minkowski sums. Let K e Xq
be a convex body of class C+, and let K' be its projection onto a
Ac-dimensional linear subspace E. TTie support function h^ of K is of class
C^, and the support function of ^C' is obtained from hic by restriction; hence
it is also of class C^. Since all radii of curvature of K are positive, the same
holds for K', as can be deduced from B.5.20) in connection with Lemma
2.5.1. Thus /C' is of class C^.
Without the assumption of positive curvatures, no such argument is
possible. The case n = 3, k = 2 was investigated by Kiselman A986). He
showed that K' need not be of class C^ even if K is of class C. Further,
he proved: If the boundary of K is of class C^ with Lipschitz continuous
second derivatives, then K' has a twice differentiate boundary. If bdK is
real-analytic, then the boundary of K' is of Holder class C'^*' for some
e > 0 (but it may be exactly of class C^*^''* for any odd integer <? S 3).

118 Boundary structure
Even more unexpected are the results that Kiselman A987) obtained for
Minkowski sums. He posed the question of when it is true that K + L is of
class C' if K, L € X^ are of class C*? He showed that for « = 2 the answer
is in the affirmative for k = I, 2, 3, 4 but not for k = 7. Further, the
boundary of K + L, where K, L € T^l have real-analytic boundaries, is of
Holder class C**^'', and this is best possible. He further mentions an
example of convex bodies K, LeJ^l with C* boundaries such that K + L is
not of class C^.
5. A weaker form of Theorem 2.5.4 is due to Rauch A974). He proved, in a
different way, that condition (a) of Theorem 2.5.4 implies that some
translate of L is contained in K. For the proof given above, see Schneider
A988b); compare also Note 6 in Section 3.2.
6. Differential-geometric properties of general convex surfaces. Theorem 2.5.5
is, of course, closely related to second-order differentiability properties of
convex functions, and we refer to Section 1.5, Note 2, for relevant
references. A detailed investigation of the differential-geometric properties
of convex surfaces without differentiability assumptions was undertaken by
Busemann & Feller A936a, 1935a,b, 1936b). Among the topics that they
treated, for convex surfaces for which no regularity was assumed, were
versions of the theorems of Meusnier, Olinde Rodrigues and Euler, um-
bilics, shortest curves, upper and lower indicatrices, Gauss curvature and
spherical images; see also Busemann [12]. The extension of Theorem 2.5.5
from n = 3 (Busemann & Feller 1936a) to general n by Aleksandrov A939c)
is not straightforward. Among several other related results, Aleksandrov
also investigated consequences of the twice differentiability almost
everywhere of support functions.
7. e-smooth points. If KeX'a and j: € bd ^C is a normal point of K, then j: is an
e-smooth point of K for some e > 0, that is, there exists a ball B of radius e
such that X & Be K. Hence, it follows from Theorem 2.5.5 that the set of
all boundary points of K that are not e-smooth for any e > 0 is of
^""'-measure zero. For this result, McMullen A974b) has given a simple
direct proof, thus answering a question posed by Sallee A972).
8. Limit sections. The concept of the Dupin indicatrix can be generalized as
follows, say, for « = 3. Let K € JLl, x ^hAK, H & support plane io K dX x,
and (Hi)ifi^ a sequence of planes meeting K and converging to H. If there
is a sequence (a,),cij of similarities such that the sequence {Oj^Hi n K))ifi^
converges to some two-dimensional convex body C, then C is called a limit
section of K at x. The point x is called universal if every plane convex body
is a limit section of K at x, for suitable choices of //, and a,; it is called
p-universal if an analogous statement is true where the planes Hj have to be
parallel to H. Melzak A958) proved the existence of a convex body with a
p-universal point (and otherwise of class C), the existence of a convex
body for which every boundary point is universal and related results.
9. Approximation with converging radii of curvature. From the counterpart of
Theorem 2.5.5 for support functions (Aleksandrov 1939c) it follows that a
convex body K e X^ has principal radii of curvature ^(m), where
0 S r,(M) g r2(u) g . . . g r„_,(M) < »
for a-almost all u € 5""'; here a denotes the spherical Lebesgue measure on
5""'. The r,{u) can be defined as the eigenvalues, corresponding to
eigenvectors orthogonal to u, of the second differential of the support
function ^^ at m € 5""' (cf. Corollary 2.5.2); they are defined a-almost
everywhere on 5""' and are measurable and a-integrable. The following
approximation result was proved and applied by Weil A973); for an
extension, see Weil A974a).

2.6 Generic boundary structure 119
Theorem. For any convex body K € X" there exists a sequence iKi)i^m of
convex bodies of class C+ converging to K such that the radii of curvature
of Kj and K, denoted by r\''>, . .., r<,'L, and ri, ..., r„_,, respectively,
satisfy the following assertions:
(a) r|'' —» r, a-almost everywhere for i —»<»,/'= 1, ...,«- 1;
(b) ry—ri in the L>E"->) norm for i-» =o, /= 1, .. ., « - 1;
(c) the measures /'■>''dc' converge weakly to the measure jrjda for ('—»««,
/'= 1, ..., M -2.
10. Tangential radii of curvature. The number r(u, t) defined by B.5.20) is
sometimes called the tangential radius of curvature of K at §{u) in the
direction t. Formula B.5.21) was proved by Blaschke [5], p. 117, for n = 3;
see also Firey A967b), p. 12, for the general case.
2.6. Generic boundary structure
Due to their simple boundary structure, polytopes as well as C+-regular
convex bodies are favourite objects of investigation. On the other hand,
one may argue that these are very special convex bodies, and one may
ask whether there exists something like the boundary behaviour of a
■general' convex body. This question can be made precise if the notion
of Baire category is used, and then several affirmative answers, often
surprising ones, are possible.
The space X" of convex bodies with the Hausdorff metric is a
complete metric space, by Theorems 1.8.2 and 1.8.5, and thus a Baire
space. This means that the complement of every first category subset of
X" is dense in X". A subset is called of first category or meagre if it is
a countable union of nowhere dense subsets. A subset that is not of first
category is said to be of second category. The complement of a meagre
set is called residual. In particular, a dense open subset is residual, and
an intersection of countably many residual sets is residual. Meagre
subsets of a Baire space can be considered as 'small'. We say, therefore,
that a certain property E for elements of a Baire space is generic if the
subset of all elements having property E is residual. For the space X" it
has become customary to say instead that most convex bodies have the
property E, or that a typical convex body has this property.
Category arguments can be used to give (non-constructive) existence
proofs for objects with certain 'irregularity' properties, which otherwise
may be hard to obtain. When such an argument works it shows that, in
fact, most elements of the space have the property in question. For the
boundary structure of convex bodies, which is our concern in the
present chapter, there are several results showing that a typical convex
body has properties that, at first sight, one would be inclined to consider
as 'pathological'. This holds, at least, for second-order properties,
whereas the typical first-order boundary behaviour is not very exciting.

120 Boundary structure
The following theorem shows, in a slightly more precise form, that most
convex bodies are smooth and strictly convex.
Theorem2.6.1. In K.", the set of smooth convex bodies and the set of
strictly convex bodies are dense G^-sets; hence, most convex bodies are
smooth and strictly convex.
Proof For A: € M, let c^^ be the set of all K eJC" having a boundary
point at which there exist two outer normal vectors forming an angle
^ \jk, and let ^i^ be the set of all K eJi" for which bd K contains a
segment of length ^l/k. Since J^"\[Jkm''^k 's the set of all smooth
convex bodies in X" and J^"\[)keN'^k 's the set of all strictly convex
bodies, it suffices to show that cy/.^ and 'Ti^ are closed and nowhere
dense. Let (/C,),sf>j be a sequence in c^^ converging to K eJC". For
each i eN, we choose a point Xj e bd /C, at which there exist two
outward unit normal vectors «,, Vj of Kj, forming an angle ^ l/k. After
selecting subsequences and changing the notation, we may assume that
Xi^^x, Ui^» u and y,-*y for i-* ». Then x e K (by Theorem 1.8.7),
and from (;«:,, «,) = h{Ki, ui), (;t,, y,) = h{Ki, y,) and Lemma 1.8.10 it
follows that {x,u) = h{K, u) and {x, v) = h{K, v); thus x ehAK and
u, V e N{K,x). Since u and v form an angle S l/k, we conclude that
K e cA^. Thus <-4k is closed. In a similar way, one easily shows that 3^
is closed. Since the set of smooth and strictly convex bodies is dense in
JC" (this follows, e.g., from Section 3.3), the closed sets d^ and 3k
have empty interior and thus are nowhere dense. ■
While Theorem 2.6.1 shows that the boundary and the support
function of a typical convex body are differentiate, in general no
stronger differentiability properties are satisfied. This is shown by the
behaviour of the curvatures described in the theorems below. These
results were discovered by Zamfirescu A980a, b).
By J^"CJC" we denote the subset of smooth (= regular) convex
bodies. For these bodies, the lower and upper radii of curvature were
defined in Section 2.5. By Theorem 2.6.1, JC" is a dense G^-set in X".
In particular, it is itself a Baire space, and every residual subset of 5f" is
residual in 3J^". Hence, it causes no loss of generality if we restrict the
following considerations to the space 31".
Theorem2.6.2. For most convex bodies K in Ji", at every boundary
point X ehAK and for every tangent direction t at x,
Pi{x, t) -0 or p^(x, 0 = «>.

2.6 Generic boundary structure 121
Thus the typical convex body is smooth and strictly convex, and its
curvatures ic{x, t) are zero wherever they exist and are finite. By
Aleksandrov's theorem, 2.5.5, the latter holds for almost all boundary
points ;t of a convex body and for every tangent direction t at x.
Corollary 2.6.3. For most convex bodies K in IK", k{x, t) = 0 for ^"~'-
almost all x ehAK and every tangent direction t at x.
The preceding results do not exclude the possibility that
simultaneously p,(x, 0 = 0 and p^{x, t) = <x> for the points ;t in a boundary set of
K of ^""^-measure zero. Typically, in fact, this holds in a residual set:
Theorem 2.6.4. For most convex bodies K in X", at most boundary
points X of K and for every tangent direction t at x,
p,{x, 0 = 0 and p^{x, t) =■ ^.
The ideas involved in the proofs given below for Theorems 2.6.2 and
2.6.4 are taken from Zamfirescu A980a, 1985). We start with some
preparations. Let K e JC", x ehd K and u be the unique outer unit
normal vector oi K &t x; let t e T^K be a unit tangent vector. We
define a two-dimensional half-plane by
D{x, u, 0 := {x + Xu + fit\k eU,fji^O)
and a convex arc by
SiK,x, 0 := bd/C n Dix, u, t).
For or > 0, we say that K is a-flat at (x, t) if
S{K, X, 0 n int B{x - au, a) = 0,
and K is a-curved at {x, t) if
SiK, X, 0 C B{x - au, a).
It follows from the definitions of the lower and upper radii of curvature
that Pi{x, 0 > 0 if and only if K is a-flat at {x, t) for some or > 0, and
that Ps{x, t)< 'x> ii and only if K is or-curved at (x, t) for some a>0.
For k eN v/e define the following sets.
ii'^ := {K € JC"\ There exist x ehdK and t e T^K such that K is
A:"'-flat at {x, t) and A:-curved at {x, t)}-
Forgiven K €.X", let
■^lif^) '■= {x e bd A:| There exists t e T^K such that K is *"'-flat at
Ai(K) •■= {;t € bd A:| There exists t e T^K such that K is ;t-curved at
(^,0}-

122 Boundary structure
For m eN and f = 1, 2 we put
(gi.„, := {K€X"\ There exists xehdK such that Boix,m~^)n
bd KCA'kiK)}.
Lemma2.6.5. The sets A\{K), AJdK) are dosed in bd K. The sets d^,
6l,m, &\,m ore dosed in JC" (k, m € N).
Proof. We start with ct^^. Let (Kj)j^^ be a sequence in cTt.k converging
to a convex body K eJC". For each j e H, we choose a point Xj € bd Kj
and a unit tangent vector tj e T, Kj such that
SiKj, Xj, tj) n int Bixj - k-^Uj, k'^) = 0,
SiKj,Xj, tj)C B{Xj - kuj, k),
where Uj is the outer unit normal vector of Kj at Xj. After selecting
suitable subsequences and changing the notation, we may assume that
Xj —»X and tj-* f for y -* >». Then it is clear that x ehd K, the sequence
("/)/ef>j converges to the unique outer unit normal vector u ot K at x,
and t is a unit tangent vector ot K at x. We assert that
S{K, X, t) n int B{x - k-^u, ^t"') = 0, B.6.1)
S{K, X, t) C Bix - ku, k). B.6.2)
Suppose that B.6.1) were false. Then there is a point z e S{K,x, t) and
a ball B with centre z such that B C int Bix - k~^u, A:"'). Now
Kj n DiXj, Uj, tj) -* K n D{x, u, t) for y ^ «>.
This follows from Theorem 1.8.8, applied to Kj and the intersection Lj
of D(Xj,Uj, tj) with some ball containing all the bodies K\, K2, ■ ■ . ,
and from the fact that K and D{x,u,t) cannot be separated by a
hyperplane since K is smooth. It follows that
S{Kj, Xj, tj) n int BiXj - k-^Uj, k-') 4 0
for sufficiently large j, a contradiction. Thus B.6.1) holds. Suppose that
B.6.2) were false. Then there is a point z' e SiK,x, t) and a ball B'
with centre z' such that B' n B{x - ku, k) = 0. As above, we conclude
that
S{Kj, Xj, tj) (f BiXj - kUj, k)
for sufficiently large j, again a contradiction. Thus B.6.2) holds. Hence,
K is A:~^-flat and A:-curved at (x, t). This proves that cA^ is closed.
Let ixj)j^j^ be a sequence in AiiK) converging to a point x. For
each j, we choose a unit tangent vector tj e T^K such that
S{K, Xj, tj) n int B{Xj - k-^Uj, *-') = 0,
where Uj is the outer unit normal vector of K at Xj. Precisely as above.

2.6 Generic boundary structure 123
with Kj replaced by K, we find that K is A:"^-flat at {x, t) for suitable
t € Tx{K). Thus Ai{K) is closed. In a similar way, we may show that
aI{K) is closed.
Let {Kj)i^f^ be a sequence in (g^ „ converging to some K e JC". For
each / € N, we choose Xj e bd Kj such that
Boix/, m-^)nbdKiC AiiKj).
After selecting a subsequence and changing the notation, we may
assume that Xj--*x ehd K for /—» ». Let y e Boix, m~^) D bd K. For
each jeN, we can choose y/ehd Kj in such a way that yj~*y for
y—» 00. For j sufficiently large, we have \xj - yj\< m~^ and thus
yj c AliKj). Hence, there is a unit tangent vector tj e Ty Kj such that
Kj is A:~^-flat at iyj,tj). The argument used before shows that K is
A:~^-flat at (y, 0 for suitable teTyK. Thus y e AliK) and hence
Boix, m~^)nhdK C aI{K), which shows that &l^„ is closed. Similarly
one shows that &\,„ is closed. ■
Proof of Theorem 2.6.2. Let cf C Ji" be the subset of smooth convex
bodies having a boundary point x and a tangent direction t at x such
that Pi^x, 0 + 0 and p^{x, 0 + °°- Then
keH
Let keH be given. If PeJC" is a polytope and 0<f<A:"', then
P + eB" € JC"\cik. Since the set of bodies of this type is dense in :Z"
and oik is closed in Ji." by Lemma 2.6.5, c^t.^ is nowhere dense in JC".
Thus ci? is of first category in ;Z". ■
Proo/ of Theorem 2.6.4. Let 3:^* CJC" he, the set of smooth convex
bodies K for which the set of all boundary points x with some tangent
direction t at x satisfying
p^{x, 0 >0 or p^{x, t) <'x>
is of second category in hdK. We have to show that JC* is meagre in
'K"r. Clearly
Z* =
a: € .T^:?! U •^U/^) U U aI{K) is of second category).
k€^ keN )
Let /C € ;Z*. Then there are a number ie A,2} and a number k eH
such that A'k{K) is not nowhere dense. Since the set A'i,{K) is closed by
Lemma 2.6.5, it has nonempty interior relative to hdK. Hence
K € 6jt „ for some m'e N, and we conclude that
k.meN

124 Boundary structure
Let k, m eN and / € {1,2} be given. Let P e X" be a polytope with
faces of diameter less than m"' and choose 0 < e <{2km)~^. Then
P + eB" € X"V/k.m- Since the bodies of this type are dense in X" and
(!^'i,„ is closed in X" by Lemma 2.6.5, the set <&'^„ is nowhere dense in
x'^. Thus X* is meagre in Z^ ■
Notes for Section 2.6
1. Theorem 2.6.1 was first proved, in an infinite-dimensional version, by Klee
A959b); it was rediscovered by Gruber A977). Klee also proved that in each
infinite-dimensional Banach space most compact convex sets are the closures
of their sets of extreme points.
Theorem 2.6.1 can be strengthened as follows. In a metric space (X,p), a
set M is called porous if there is a number a>0 such that, for every x e X
and any ball B(x, e) with centre x and radius e, there exists a point
y € B{x, e) such that
Biy, ap(x, >■)) n M = 0.
A countable union of porous sets is a-porous. One says that nearly all
elements of a metric Baire space have a property E if the set of elements not
having property £ is a a-porous set. Zamfirescu A987a) proved that nearly
all convex bodies in X" are smooth and strictly convex.
2. Typical curvature properties. Gruber A977) proved that the set of convex
bodies in "?" that have a boundary of class C^ or which are, in a certain
sense, uniformly strictly convex, is of first category. Theorem 2.6.2 and
Corollary 2.6.3 (as well as some other corollaries) are due to Zamfirescu
A980a). In connection with Corollary 2.6.3, the following simple example in
E^ is of interest; Take a convex polygon P e Xl and let TP be the convex
hull of all the points dividing each side of P into three equal parts. Then
lim/fc^xT*P is a smooth, strictly convex body with curvature vanishing almost
everywhere on the boundary. This example is due to de Rham A957).
A weaker form of Theorem 2.6.4, in which the set of boundary points x of
K at which pi(x, f) = 0 and pi(x, f) = <» for every tangent direction t is
merely dense in bd K, was proved by Schneider A979b). The general form of
Theorem 2.6.4 was obtained by Zamfirescu A980b), and the simpler proof
given above appears in Zamfirescu A985). The curvature properties of typical
convex bodies were further investigated by Zamfirescu A988a). He proved,
among other results, that on the boundary of a typical convex t)ody K, the
set of points x with k{x, t) = x for some t € T^iK) and that of points in
which the lower curvature in some direction equals the upper curvature in the
opposite direction are dense. Further results on the boundary structure of
typical convex bodies can be found in Zamfirescu A985, 1988a).
3. Typical properties of normals. The curvature behaviour of a convex body is
closely related to the behaviour of its normals (that is, the lines through
boundary points spanned by corresponding normal vectors). The behaviour of
the normals of typical convex bodies was investigated by Zamfirescu A982a,
1984b). For instance, he proved that for most convex bodies K in X", most
points of E" lie on infinitely many normals of K.
4. Other topics treated under category aspects include geodesies (Zamfirescu
1982b, 1989+, Gruber 1991a), convex curves on convex surfaces (Zamfirescu
1987b), diameters (Zamfirescu 1984a, BSrSny & Zamfirescu 1990),
approximation (Schneider & Wieacker 1981, Gruber & Kenderov 1982, Gruber

Notes for Section 2.6 125
1983), shadow boundaries (Zamfirescu 1988b, 1991. Gruber & Sorger A989),
non-differentiability properties of the metric projection (Zamfirescu 1990);
see also Theorem 3.2.14.
There are several further category results in convexity, concerning
properties that are farther away from the main themes of this book. One can find
references in the survey articles by Zamfirescu A985), Gruber A985) and
Zamfirescu [45].
5. Convex functions. Smoothness properties of typical convex functions were
studied by Gruber A977) and Klima & Netuka A981); see also Howe A982).
6. Schwarz & Zamfirescu A987) considered typical convex sets of convex sets.
7. Typical compact sets. By Theorem 1.8.2, the space §" of nonempty compact
subsets of E" with the Hausdorff metric is also a Baire space. Some category
results are known for this space. Most compact sets in C^" have Hausdorff
measure zero, if any Hausdorff measure is given (see Gruber A983, 1989) for
more general results), and they are homeomorphic to Cantor's ternary set
(see Wieacker 1988b for references). Wieacker found some surprising results
on the (first-order) boundary structure of the convex hull of a typical compact
set. In the following theorem, which collects the main results of Wieacker
A988b), exp* K denotes the set of farthest points of the convex body K; thus
X € exp* K if and only if there is a ball B such that K <Z B and
xe Kf\hAB.
Theorem. For most-compact sets C € (y", the convex hull K = convC has the
following properties:
(a) bd K is of class C but not of class C^;
(b) exp* K and ext K\exp K are dense in ext K;
(c) ext K is homeomorphic to Cantor's ternary set;
(d) exp K is homeomorphic to the space of irrational numbers;
(e) exp* K is homemorphic to the topological product of the space of
rational numbers and Cantor's ternary set,
(f) The Ac-skeleton ext^ ^C of ^ is of (Hausdorff and topological)
dimension k (k =0, . ., n - I).

Minkowski addition
3.1. Minkowski addition and subtraction
The aim of the third chapter is a more systematic study of Minkowski
addition. In the present section we shall first collect some elementary
formal properties of this addition and then investigate its convexifying
effect. Finally, we shall consider a certain counterpart to Minkowski
addition, a useful operation that has been called Minkowski subtraction.
The sum of two subsets A, B C E" was defined by
A + B ■■= {a + b\a € A, b € B},
and the scalar multiple by
kA ■■= {Aa|a e A)
for real numbers A. One may rewrite the definition of the sum in the
form
A + B = \J{A + b).
This gives a 'kinematic' interpretation that may be helpful to the
intuition: ^4 + B is the set that is covered if A undergoes all translations
by vectors of B.
We recall some of the information on Minkowski sums that is
scattered in former sections. If A and B are convex, compact or
polytopes, then ^4 + B is, respectively, convex, compact or a polytope.
As a map from &" x (§" into 6", Minkowski addition is continuous. &."
and X" with Minkowski addition are commutative semigroups with unit
element {o}. X" satisfies the cancellation law (Section 1.7), that is, if
K, L, MeX" 2inA K + M = L + M, then K = L.
There are some trivial rules valid for arbitrary subsets .4, B, C of E"
and non-negative real numbers A, //, namely
126

3.1 Minkowski addition and subtraction 127
(A U B) + C = (A + C) V (B + C), C.1.1)
(An B) + CC{A + C)n{B + C), C.1.2)
kA + XB= X{A + B),
kA + fxAD {X + (i)A.
If A is convex, then XA + (lA = {X + (i)A (Remark 1.1.1). Special
relations hold for convex bodies with convex union;
Lemma 3.1.1. Let K, L e JC" be convex bodies such that K \J L is
convex. Then
{K n L) + C = (K + C) n {L + C) C.1.3)
for CeJC", and
{K U L) + (K n L) = K + L. C.1.4)
Proof. Let X e (K + C) r\ (L + C); then x = y + c = z + d with y e K,
z € L and c, d e C. Since K (J L h convex, there is a number A e [0,1]
such that A - A)y + Xz € K H L and hence
X = (l - X)iy + c) + Xiz + d)
= A - X)y + Xz + {I- X)c + Xd
€{K n L) + C.
Thus {K + C) n (L + C) C {K n L) + C, which together with C.1.2)
yields C.1.3).
For the proof of C.1.4), we note that (K U L) + (K n L) C K + L
holds trivially. Let x e K, y e L. Since K V L is convex, the segment
[x, y] contains a point x' = {I - X)x + Xy e K A L, where A € [0, 1].
Then y' := Xx + {I-X)y e K U L dind x + y = x' + y'. This shows that
/<: + L C (A: n L) + (A: U L) and thus proves C.1.4). ■
We remind the reader that .4 - B for .4, B C E" is the set
A - B = {a - b\a e A,b € B),
and -A = (-1)^4 = {-a\a e A) is the image of A under reflection in
the origin.
\i K eJC" \% & convex body, the set
DK ■= K - K = {x - y\x, y e K)
is a convex body centrally symmetric with respect to the origin; it is
called the difference body of K. Since h{-K, u) = h{K, -u), we have
h{DK, u) = h{K, u) + h{K, -u) = w{K, u)
for u € 5""'; thus the support function oiDK at the unit vector u is the

128 Minkowski addition
width of K in direction u. If the difference body of AT is a ball, then
w{K, •) is constant, and K is called a body of constant width.
The Minkowski addition of non-convex sets is, in more than one
sense, a convexifying operation. Roughly speaking, vector sums of many
bounded sets are approximately convex. This observation can be made
quantitively precise and can be connected with several possible ways of
measuring the non-convexity of a compact set. The following results of
this kind have proved useful in mathematical economics and in
stochastic geometry.
Theorem 3.1.2 (Shapley-Folkman). Let A^, . . . , AkCl" and let
k
X € 2conVi4,.
i=i
Then there is an index set I C {\, . . . , k) with card I = n such that
X € ^COnvAi + X'^r
iel it I
Proof. If
k
X € 2conVi4, = conv{i4i + . . . + ^4^)
1=1
(by Theorem 1.1.2), then x e con\{A[ + ... + ^4;^) with suitable finite
sets A'l C Aj. Hence, it suffices to prove the result under the assumption
that the sets ^4], .. ., ^4^ are finite.
Define /:(E")*-*E" by f{yi, ..., yk)-==yi+ •■• + y* and let
P C (E")* be the polytope defined by
P := conVi4i x . . . x conVi4^;
then X e f{P). Let z be an extreme point of /"'C^)'"' f^ and let F be
the face of P with z e relint F. Since f~\x) is of codimension n and z
is extreme, it follows that dimF^n. The face F is of the form
F=FiX...xFk, where F, is a face of conVi4,. Writing / •=
{i|dimF,>0}, we deduce that card I ^ n. For /€{l, ..., k}\I we
have F, = (a,} with an extreme point a, of conVi4,, hence with a, € Aj.
This yields
X = fiz) e /{Fi X . . . X F^) = F, + . . . + F^
C "^convAi + ^Ai. ■
iel it I
For convenience, we formulate the subsequent results only for
compact subsets (although some of them could be generalized).

3.1 Minkowski addition and subtraction 129
Corollary 3.1.3. If Ai, . . . , A,, efy" and A^C RB" for i = I, ... , k
with some real number R, then
k k
dlS^i.conv^A)^ nR.
Proof Let x €conv(i4i + . . . + .4^). By Theorem 3.1.2 we can
assume, after renumbering, that x = a + a„+i + ... + a^ with m^ n,
a € Xi^iConVi4, and a, € ^4, for i = m + 1, . . . , k. By the definition of
the Hausdorff distance, there is a point a' e Ai + . . . + A„ such that
\a - a'\ ^ 6{Ax + . . . + A„, conv(>l, + . . . + A„)) ^ mR,
the latter because ^4] + ... •¥ A„(Z mRB". Hence, the point a ■ =
a' + flm+i + ... + ak^ Ai + ...■¥ Aj, satisfies |;t - a| = \a - a'\ %
mR % nR. The assertion follows. ■
Remark 3.1.4. The essential feature of the estimate of Corollary 3.1.3 is
that the right-hand side is independent of k. In particular, if (i4i),6|!)j is a
sequence of uniformly bounded compact sets in E", it follows that
d j— {i4i + ... .4^), —conv(.Ai + . . . + AiM -» 0 for A: -* ».
Thus averaging of compact sets in the Minkowski sense is
'asymptotically convexifying'.
There are sharper estimates that demonstrate the convexifying effect
of Minkowski addition. They can, for instance, be deduced from the
following lemma. For a compact set A eft!" there is a unique ball of
smallest radius containing A, the circumball of A. Its radius, R{A), is
called the circumradius of A.
Lemma3.1.S. Let A^, ..., A„e&" and j:€conv(i4i+ ... + A„).
Then there is a point a e Ai + ... + A„ such that
m
\x - a|2 % S/?(^,)'-
1=1
Proof. We use induction with respect to m. If m = 1, let z be the
centre of the circumball of ^4]. Since x e convi4i, we can choose a e Ai
with {z - x,a - x) ^0 and then obtain
\x - a|^ = \x - a|^ + \z - x\^ + 2(z - x, x - a)
=.\x - a\^^ RiAif.
Thus the assumption holds for m = 1. Suppose it is true for some
m^l. Let j:€conv(i4i+ ... + i4„+i). Then x = y + z with ye

130 Minkowski addition
conv(i4i + .. . + A„) and z e conVi4„+i. By the inductive assumption,
there exists a point a e Ai+ ... + A„ such that
m
1=1
Let w := p{conv A„^.i, x - a); then
\x - a- w\^^\x - a- z\^ = \y - ap ^ S^(A)^-
1=1
Suppose for a moment that x - a - w ^ o. The hyperplane // through
w orthogonal to x - a - w separates x-a and conVi4„+i (Section
1.2), and w e H nconVi4„+i = conv{// D A„^.i). By the case m = 1 of
the lemma, there is a„+i € // n .4„+, such that
\w - a„+,P ^ R{H n yl„+iJ ^ /?{^m+i)'-
We have
{x - a - w, a„+i - vv) = 0. C.1.5)
If X - a- w = o, we choose a„+i € i4„+, with Iw-a^+il^^
/?(>l„+i)^, and C.1.5) holds trivially. Now a + a„+i € >li + . . . + >l„+,
and
\x - {a + a„+i)|^ = |(;c - a - vv) - (a„+i - h')!^
m + l
= |;c - a - w|2 + |a„+, - wM S R{A{)\
1=1
which completes the proof. ■
To formulate a consequence, we define, for a compact set ^4 €<S",
the inner radius of A by
p(/l) := sup inf (A ^ 0|a: € conv(»i D B(a:, A))}.
xsconv/l
Then p(i4) = 0 if and only if A is convex, and trivially p{A) ^ diam/l,
with equality if A is affinely independent.
Theorems.1.6 (Shapley-Folkman-Starr). Let A^,. . . , Ak e&". If
X € conv {Ai + . . . + Ak), then there exists a point a e A\ + . . . + Ak
such that
\x - a\-^ "S/n" max /o(>l,), C.1.6)
ISiS*
hence
o XA,, conv J^AA ^ VV max p(>li). C.1.7)
iTi i=i / is.s*
Proof. Let a: €conv(y4i + . .. + .4^). By Theorem 3.1.2, there is an
index set / C {1, . . . , k} with card I ^ n such that x = X.*=iai with
a, € conVi4, for i e I and a, e .4, for i ^ /. By Lemma 3.1.5, there exists

3.1 Minkowski addition and subtraction 131
a point b € 2is/'^i such that
la,-b
i€l
iel
Hence, the point a ■= b + 2i</''i satisfies a e Xi*=i^i ^n*^
\x - ap ^ J^RiAi)^ ^ n max RiAd^. C.1.8)
.-6/ iSiS*
We can write
k
X = 2^1 with Xi € conVi4i.
1=1
Put A'i ■= Aj n B{Xi, p{Ai)); then ;t,€ConVi4; by the definition of
p{Ai) (and by compactness), hence x e conv(.Ai + ... + .4;^). By the
result proved above, there is a point a e 2f=i^i C 2f=i^; such that
\x - a\ ^ \/~n max R{A'i) ^ a/TT max p{Ai),
which proves C.1.6). The estimate C.1.7) is an immediate consequence.
Both functions appearing in Theorem 3.1.6, d{A,con\A) and p{A),
are measures for the deviation of a compact set A from its convex hull.
As such, they also measure the non-convexity of the set, but in a way
that depends on the Euclidean metric and that is not invariant under
similarities. Since convexity is an affine notion, a geometrically
significant measure for the non-convexity of sets should be invariant under
affine transformations. Such a function can also be derived from the
theorem of Shapley and Folkman. Applying Theorem 3.1.2 with
k = n + I and Ai = A C E", we get
A + n conv A = conv A + n conv A.
Hence, if we define
c{A) := inf {A ^ 0\A + AconVi4 is convex},
then c{A) ^ n. It is clear that the function c is invariant under
non-singular affine maps and that c{A)=0 for convex sets A. The
following theorem (Schneider 1975d) shows that for compact sets c can
serve as a measure of non-convexity and that it describes the sets that
are least convex, in this sense.
Theorem 3.1.7. For every set A CE",
OS c{A) S n.
For compact sets A, equality holds on the left if and only if A is convex,
and on the right if and only if A consists of n + I affinely independent
points.

132 Minkowski addition
Proof. It is convenient to write
Ax ■= A + A)-'(»i + Aconv^i)
for A ^ 0; then ^4^ C conv A C conv Ax, hence
c{A) = inf {A ^ 0\Ax = conv^}.
The inequalities 0 ^ c{A) ^ n are clear. Now let A e^" be compact.
Assume that c{A) = 0. Let ;ic€conv.4, and let A>0. Then x e A^,
hence x = {I + k)~^{ax +^bx) with oa ^ ^^ and b^econv^. This
implies X - ax = AFa - ;«:) and thus d{A,x) ^2Adiami4. Since A>0 was
arbitrary, we see that x e A, hence A is convex.
Now suppose that c{A) = n. Then A cannot be contained in a
hyperplane (since otherwise c{A)^ n - I), hence conVi4 has interior
points.
Let ;«: € bd conv i4. Then x lies in a supporting hyperplane H of
conv A, hence
X e H n com A = conv(// n A) =^ {H A A)„.x C ^„_,.
Let A> n - 1 and put a := (k- n + 1)A + A)"'; then
X + ar(conVi4 - ;«:) C A - a)A„.i + or conv ^4 = A^.
Since or > 0 and this holds for all x e bdconv ^4, we see that the set
Rx '■= (conv i4)\i4;.
satisfies
clRx C intconv/1 for A > n - 1. C.L9)
If // > A, then
A^ = (l + ii)-^A + iii- A)(l + //)-i(l + A)-iconv^
+ A(l + X)-^con\A
D A + ii)-^A + (//- A)(l + //)-'(! + A)-'>1 + A(l + A)-iconv^
= Ax,
hence (i> X implies R^C Rx. Since the sets clRx are compact and
nonempty for A < n, there exists a point
z € n clRx.
0<A<n
By C.1.9), z € intconVi4. On the other hand,
z $ int(l + n)~^{a + nconv.4) for a e A, C.1.10)
since otherwise, for sufficiently large A< n,
z € int(l + Xy^{a + Aconv>l) c int^l^
and thus z ^ cl Rx, a contradiction.
By Carathdodory's theorem, there exists an affinely independent

3.1 Minkowski addition and subtraction 133
set Y C A such that z e conv Y, and some subset X C Y satisfies z e
relintconv A". Suppose that dimaff A" < n. Then
z € convZ = A + Xy^iX + AconvA) for A a n - 1
and hence z e relint(l + n)~'{^ + n conv A") for suitable x e X. Since
z€intconVi4, we must have relintconv A'C intconVi4 and hence
z€intil + n)-^ix + Aiconvj4). This contradicts C.1.10). Hence dim
aff A!" = n, and Y is the set of vertices of an n-simplex S. From C.1.10)
it follows that z is the centroid of 5, since this is the only point not
contained in Uxsy'nt(l + n)~\x + nS). If A contains a point a iY,
we can replace an appropriate point of 7 by a and still obtain an
affinely independent set 7 C ^4 satisfying z e conv Y. By the same
argument, Y is the set of vertices of an n-simplex of which z is the
centroid. But as Y and Y differ in precisely one point, this is
impossible. Hence A = Y, that is, A consists of n + 1 affinely
independent points. Vice versa, if A is of this kind and it A + AconVi4 is convex
for some A^O, then the barycentre of A belongs to A>,, which implies
A ^ n. This completes the proof. ■
Having studied some properties of Minkowski addition, we now
introduce a complementary operation called Minkowski subtraction
(although it was not introduced by Minkowski). While the sum of two
sets A, B C E" can be defined by
A + B = \J{A + b),
beB
the Minkowski difference of A and B is, by definition, the set
A~ B := Oi^-b).
biB
(If B is empty, ^4 ~ B is, by convention, equal to E".) Evidently we
may also write
A ~ B = {x e E"|B ■¥ x C A).
There are some trivial rules connecting addition and subtraction, namely
{A + B)~ B^ A, C.1.11)
(A ~ B) + B (Z A, {B ^ 0) C.1.12)
{A~ B) + CC{A + C)~ B, C.1.13)
{A ~ B) ~ C = A ~ {B ■¥ C), C.1.14)
A ■¥ B <Z Co Ad C ~ B. C.1.15)
The verification is immediate. Under convexity assumptions, a little
more is true. If A is convex, then A~B is an intersection of convex

134 Minkowski addition
sets and hence convex. For convex bodies K, L eJi" vit say that L is a
summand of K if there exists a convex body M such that K = L + M.
A closer look at summands is the aim of the next section. Here we
remark:
Lemma3.1.8. Let K, L e JC" be convex bodies. Then
{K + L) ~ L = K.
The relation
(K ~ L) + L = K
holds if and only if L is a summand of K.
Proof Let x €{K + L) ~ L; hence L + x C K + L and, therefore,
hi + /i{;t) = h^ + hi- Subtracting hi, we get x e K. Thus {K + L)~
LC K, which together with C.1.11) proves the first assertion. If
(K ~ L) + L = K, then L is a summand of K. Vice versa, suppose that
K=M + L for some MeX". Then K~L= {M + L)~L = M,
which proves the second assertion. ■
For convex bodies K, L e X", the Minkowski difference can also be
represented as an intersection of halfspaces, in the following form:
K ~ L= n [fi'iK, u) - h{L, u)u\
= A HuMK,u)-h(L,uy C.1.16)
In fact, X e K ~ L is equivalent to L + x C K, which is equivalent to
hiL +x,u)^ h{K, u) and hence to {x,u)^ h{K, u) - h{L, u) for all
M€5"-'.
In particular, C.1.16) implies
h{K ~ L,u)^ h{K, u) ~ hiL, u) for u e 5""'.
Let X ehd{K ~ L). Then L + x must contain a boundary point y oi K
(otherwise, L + x + vC K for all v in some neighbourhood of o and
thus X eint{K ~L)). If u is an outer unit normal vector of K at y,
then hiK,u) = h{L + x,u), hence h{K,u) - h{L, u)= {x,u)^
h{K ~ L,u). Thus, to each boundary point x of K~L there is a
normal vector u oi K ~ L dX x such that h{K ~ L,u)= h{K,u)-
h{L,u).
The following notion of parallel bodies is useful in several respects.
Let K, B eJC" be convex bodies and let A^0 be a real number. Then
the convex body K •¥ XB is called an {outer) parallel body of K, and
K~XB is an inner parallel body, both relative to B. The greatest

3.1 Minkowski addition and subtraction 135
number A for which K ~ XB is not empty is called the inradius of K
relative to B and denoted by r{K, B); thus
r{K, B) := max (A ^ 0\XB + t C K ior some t € E"}.
For fixed B, one defines the full system of relative parallel bodies of K
by
,^{K + pB for0^p<«>,
" '~ [K ~ -pB ioT -r{K, B)^p^O.
Then tor K, L e JC" and arbitrary p, a € IR the rule
Kp+ LaC{K + L)p^, C.1.17)
is valid.
Proof. Let p, a ^ 0. We have
Kp-¥ La = K •¥ pB •¥ L + aB = K •¥ L ■¥ {p ■¥ a)B
= iK + L)p^a.
Next,
K.p + L_<, + (p+ a)B
= (A: ~ pB) + pB + (L ~ aB) + aB C a: + L,
hence /C.p + L_<, C {/C + L) ~ (p + a)B = (/C + L)_p_<,.
If p ^ a, then
Kp+ L.o= K + {p- a)B + (L ~ oB) + oB C K + (p - o)B + L
= {K + L)p.a.
If p^ a, then
Kp + L_<, + (a - p)B = a: + pB + (a - p)B + (L ~ aB)
= a: + (L ~ aB) + aB C a: + L,
hence A:p + L_<, C (A: + L) ~ (a - p)B = (A: + L)p-a- ■
From C.1.17) we deduce the following lemma which later will turn
out to be very useful.
Lemma3.1.9. If K, B € X", then the full system p<-* Kp of parallel sets
of K relative to B is concave with respect to inclusion, that is,
A - A)A'p + XKg C A'(i_^)p+Aa
for X € [0,1] and p, o € [-r{K, B), «>),
Proof A - X)Kp + aa:,, = [A - A)a:](i_,)p + [aa:],,, c a:(i_,)p+,<, by
C.1.17). ■

136 Minkowski addition
When the preceding lemma is applied, it is important to know in
which cases the inner parallel bodies of K are homothetic to K. The
following lemma is a slightly improved version of a result of Bol
A943b). Recall (Section 2.2) that the convex body /C is a tangential
body of the body B eJC" if each extreme support plane of /C is a
support plane of B. For the following, an equivalent description is
convenient. Let /C be a tangential body of B. If ;«: is a boundary point
of K, then through x there is an extreme support plane of K, which is
hence a support plane of B. Vice versa, suppose that through each
boundary point of K there is a support plane of B. Then a support
plane of K that is not a support plane of B contains only singular points
of /C. By Theorem 2.2.8, /C is a tangential body of B. Thus a convex
body /C is a tangential body of B if and only if through each boundary
point of K there exists a support plane to K that also supports B.
Lemma 3.1.10. Let K, B e Xo be convex bodies, and let pe
@, r{K, B)). Then K ~ pB is homothetic to K if and only if K is
homothetic to a tangential body of B.
Proof. Suppose that K ~ pB is homothetic to K. After a suitable
choice of the origin we may assume that XK = K ~ pB, where
0<A<l.Then
X^K = X(XK) = X{K ~ pB) = XK ~ XpB = (K ~ pB) ~ XpB
= a: ~ A + X)pB
by C.1.14). By induction we get
X''K = K~(l + X+... + X''-^)pB
iox k eN, hence
{o} = a: ~ xB with T = ——,
because pu] x obviously implies K ~ pi,B -^ K ~ xB. Let x ehAK.
Then Xx eXK = K ~ pB, hence pB + Xx C K. Since Aj: is a boundary
point oi K ~ pB, the body pB + Xx contains a boundary point y ot K,
s&y y = pb + Xx with b € B. Then
• _ y ~ ^ _ pb
1 - A ~ 1 - A
The point y ebdK is a convex combination, with positive coefficients,
of the points xebdK and veK, hence any support plane to K
through y contains x and v and thus supports K at x and xB at v. Thus

Notes for Section 3.1 137
K K a tangential body of xB (and hence is homothetic to a tangential
body of B).
Conversely, suppose that K h a tangential body of xB, for some
T > 0. Then
K = r\H-ixB,u)= C\H-iK,u),
where ft)C5""' is the set of those normal vectors u for which the
support plane H{K, u) also supports xB. (To see this, observe that K is
the intersection of those of its supporting halfspaces that contain a
regular point of K in their boundaries.) Now
K ~ pB = I I Hu,h(K,u)-ph(B,u)^ \\Hu.h(K,u)-ph(B,uy
ueS" ' usto
For M € ft) we have
h{K, u) - ph{B, u) = ^^^ h{K, u),
hence K ~ pB C {\ - p/x)K. From Lemma 3.1.9 we get
and since K.-c = (o}, this gives K ~ pB = {I - p/x)K. ■
Notes for Section 3.1
1. Minkowski addition and subtraction. All the formal rules for Minkowski
addition and subtraction that we have mentioned, and many more, can be
found in Hadwiger [18] (who uses A y. B and A/B instead of .4 + J? and
A—B), with one exception. The exception is equation C.1.4), which
presumably first appeared in Sallee A966).
More recently, addition and subtraction (for not necessarily convex sets)
have found applications in mathematical morphology and image processing;
see Matheron A975), Serra A982, 1988) and Stoyan, Kendall & Mecke
A987); in the latter, note in particular the remarks in Section 1.4 on
quantitative image processing. A useful survey is given by Haralick, Stem-
berg & Zhuang A987). In these and related references, different notation is
used, namely
A@ B := A + B,
Ae B := A B.
TTiese operations are complementary to each other, in the sense that
AQ B = (A' @ By,
A@ B = (A' e By,
where .4' is the complement of the set A. In applications, B is usually a
fixed convex body, for instance in the plane a disc or a hexagon, which is
then called the structuring element. It is used to transform a given set A in
various ways, in particular by means of the following operations:

138 Minkowski addition
A @ -J?, the dilatation of .4 by J?;
A © -J?, the erosion of .4 by J?;
Ab -.^ (A Q -B) @ B, the opening of A by B,
A^ := (A@ -B)e B, the closing of A by B.
2. Analysis of X"-valued functions. It is not surprising that some parts of the
elementary analysis of real-valued functions can be carried over to X"-
valued functions, with Minkowski addition replacing the addition of real
numbers. Integrals of functions from an interval into DC" were constructed
by Dinghas A956, 1961). For later developments see, e.g., Artstein A974)
and the references given there. Artstein emphasizes the usefulness of the
support function. It appears that a consequent use of the support function,
together with Theorem 1.8.12 and some elementary estimates, would also
permit one to deduce many of the results in Ahrens A970, 1972) and in
Chapter IV of Meschkowski & Ahrens A974) directly from the
corresponding results for real-valued functions.
Vitale A979) has studied a Bernstein type of approximation for functions
from [0,1] into X", where Minkowski addition is used.
3. Vector spaces consisting of classes of convex bodies. The set X" of convex
bodies, equipped with Minkowski addition and multiplication by non-
negative real numbers, forms a commutative semigroup, satisfying the
cancellation law, with scalar operator. More algebraic structure can be
introduced if suitable equivalence classes of convex bodies are considered.
Vector spaces of classes of convex bodies, in addition endowed with norms
or scalar products, have been investigated by Ewald & Shephard A966),
Ewald A965), Shephard A966c), Schmitt A967), Lewis A975) and Geivaerts
A974b, 1976); see also Sorokin A968) and Zamfirescu A975).
In the background of part of these investigations is, roughly speaking, the
idea of decomposing a support function into its even and odd parts, and of
concentrating attention on one of these parts. Valette A971) treats a related
topic in this sense also.
4. Prolongation of linear series. Let ^o. Ki^ X" and Kx.= {I- k)Ko + ^-Ki
for A€ [0,1]. We say that the linear series (^a)/£|o,ii can be prolonged via
Ki if, for some A> 1, the function A - A)^(^o. •) + A^(^i, •) is still a
support function; in other words, if there exists a convex body Kx such that
^1 = A - A~')^o + ^'^Kj,. Evidently this is equivalent to the condition that
Kq be homothetic to a summand o( Ki. Known criteria for summands (see
Section 3.2) can therefore be used to study this prolongation. It was
investigated by Vincensini A935, 1936). He applied it to the 'construction' of
convex bodies with given difference body (domaine vectoriel). With his
notation, but using the support function with advantage, we can describe this
procedure as follows. Let D e X" be a given convex body, centrally
symmetric with respect to the origin. Let T be an arbitrary convex body with
the property that C := Dr= T- r is homothetic to a summand of D (this
is always the case if D and F are of class C+); say, aC is a summand of D
where a € @,1). Then
hiA, u) := J [hir, u) - h(r, -u)] + i h{D, u)
defines a support function, and the body A evidently satisfies DA = D. In a
series of papers, Vincensini A937a, b,c, 1938, 1956, 1957, 1965) studied
related questions and applications; see also Vincensini & Zamfirescu A967).
5. Extremal and facial structure of sums of convex sets. The following was
observed by Klee A959b, Proposition 6.5). Let ^, J? c E" be convex sets, A

Notes for Section 3.1 139
closed and B compact. Then for each point z e ext(.4 + B) there are unique
points Xi€ A and yz^ B (necessarily extreme points) such that z - Xi + y^.
Further, for each point j: € ext .4 there is a point y € B such that x + y e
ext (.4 + B). However, there may be points >» € ext J? such that x + y i
ext(A + B) for arbitrary x € A. This was further studied by Bair A979).
The extremal and facial structure of sums of closed convex sets in certain
infinite dimensional spaces was investigated by Husain & Tweddle A970),
A. K. Roy A972), Edelstein & Fesmire A975), Jongmans A976) and Bair,
Fourneau & Jongmans A977).
Some observations on smoothness properties of Minkowski sums are
found in Jongmans A979, 1981a, b).
6. The behaviour of certain cones associated with convex sets in real vector
spaces under Minkowski addition of the convex sets was studied by Dubois
& Jongmans A982).
7. Bodies of constant width. A comprehensive survey of bodies of constant
width was given by Chakerian & Groemer [31].
8. Pairs of constant width. As a generalization of bodies of constant width,
Maehara A984) introduced pairs of constant width. A pair {K, L) of convex
bodies K, L ^JC" '\s called a pair of constant width r if
h{K, u) + h{L, -u) = r for m € 5""'.
In particular, K is of constant width r if and only if (^, ^) is a pair of
constant width. If (^l, L) is a pair of constant width r, then ^ + L is of
constant width 2r. For .4 C E" and r > 0, let i2,(A) be the intersection of all
closed balls of radius r whose centres belong to A. Maehara proves the
following: A pair (K, L) is of constant width r if and only if Qr{K) = L and
Qr{L)-K. If A^0 is a set having circumradius at most r, then
{Qr{A), ii^riA)) is a pair of constant width.
Sallee A987) extended this to pairs of sets of constant relative width with
respect to a centrally symmetric convex body 5. This led to interesting
additional results and also to some open problems. (See also Note 4 in
Section 3.2.)
9. The theorem of Shapley, Folkman and Starr. Theorem 3.1.2, Lemma 3.1.5
and Theorem 3.1.6 are all from Starr A969), where credit for most of the
results (namely, up to C.1.8)) is given to unpublished work of Folkman and
Shapley; see also Arrow & Hahn A971), Appendix B, Ekeland & Temam
A976), Appendix I, and Starr A981). Our proof of Theorem 3.1.2
essentially follows Artstein A980), Theorem 5.1. Simple proofs of Theorems 3.1.2
and 3.1.6 were also given by Cassels A975). He introduced, for compact sets
A e€", the function
C}iA) := sup inf 2<*'U - ".PI >
j:6conv.4 \i=l I
where the infimum is taken over all representations x - 2" i<J^.ai with
w € N, a, € A, or, g 0 and 2<*. = 1. and he showed that
k
d 12^4. = \Ar max C3{A,)
\,=i / is.s*
for Ai, ..., Ake€". Like the functions defined by
Ci{A) := d(.4, conv.4),
• Ci{A) := p{A),
C3 measures the deviation of A from its convex hull. Since
Ci{A) g C3iA) S cM)<

140 Minkowski addition
the result of Cassels implies the inequality C.1.7), which can be written as
cJEAI = V^maxc4(^,).
W. Weil (private communication) asked whether also
CilE'^il = V^maxci(^,)-
Wegmann A980) investigated relations between Ci, c^, c^ and a further
function c: defined in a similar way. In particular, he showed that C} = d,
while simple examples A e&" exist with Cy{A) < C}{A).
The theorem of Shapley, Folkman and Starr has been applied in
mathematical economics; see Starr A969), Arrow & Hahn A971) and
Hildenbrand A974).
10. Minkowski addition in the theory of random sets. The theorem of Shapley,
Folkman and Starr has also found applications in stochastic geometry. For
these applications, the crude estimate of Corollary 3.1.3 is sufficient, since
one only needs Remark 3.1.4. The latter is used to deduce results on
sequences of independent random compact sets and their partial Minkowski
sums from results on the corresponding convex hulls, and for convex sets the
embedding into a Banach space via the suppport function can be utilized.
For such applications, see Artstein & Vitale A975), LjaSenko A979) (who
gives independent proofs for the required estimates), Weil A982b), Gin6,
Hahn & Zinn A983) and Artstein A987+).
Further applications of Minkowski addition in the theory of random sets
are found in Cressie A978), Mase A979), Schurger A983) and Gin6 & Hahn
A985a, b); for infinite dimensions see Gin6 & Hahn A985c) and the
references given there.
11. The main part of Theorem 3.1.7 was formulated and proved above for
compact sets only. Some more general and additional results were proved in
Schneider A975d). Let .4 C E" be an arbitrary set. If A is bounded and
c{A) = 0, then cl A is convex; if c{A) = n, then A consists of « + 1 affinely
independent points. If A is either unbounded or connected, then c{A) g
M - 1; in both cases the bound « - 1 is sharp.
The relation
A + Mconv.4 = (m + l)conv.4
for a compact set .4CE", which was noted before Theorem 3.1.7, as a
consequence of Theorem 3.1.2, appears also in Borwein & O'Brien A978).
These authors prove; A Banach space X is finite dimensional if and only if
for any compact set .4 C AT there exists or > 0 with
A + arclconv.4 = (or + l)clconv^.
The least such a that works for all A is the dimension of the space.
12. Computational aspects of Minkowski addition. Gritzmann & Stunmfels A990)
determine the computational complexity of computing the Minkowski sum of
k convex polytopes in E", which are presented either in terms of vertices or
in terms of facets. For fixed n, they obtain a polynomial time algorithm for
adding k polytopes with up to N vertices. Among other combinatorial
results, they have bounds for the number of vertices of the sum in terms of
the vertex numbers of the given polytopes.
13. Hammer's associated bodies and reducibility. For K € X", Hammer A951)
defined and studied the associated bodies

Notes for Section 3.1 141
K{p) := \
n [piK - b) + b] forp€ @, 1),
ebd/C
\J[piK-b) + b] forp€[l, oo).
ftebd/C
bebdK
This construction can also be described in terms of addition and subtraction.
In fact, it is not difficult to show (see also Voiculescu 1966) that
K{p) ^ K + ip- l)DK = pK + ip- 1)(- K) for p g 1,
K(,p) = K -~{l - p)DK = pK -~{l- p)(-K) for p g 1.
Hammer further proposed a notion of reducibility, as follows. There is a
number r^ S 1 such that
K = K(p)(p/Bp - 1)) for p g r^,
K 4= K(p)(plBp - 1)) for p < r^.
Then ^C is called reducible if r^ < 1, otherwise irreducible. This notion of
reducibility was studied in a series of papers by Zamfirescu A966a, b,c,
1967a, b,c,d, 1973, 1975). It appears that part of the results can be obtained
more easily using the following reformulation. Write
Uk '■- sup {a = 0\-aK is a summand of K)
or, equivalently, using support functions,
Uk = sup {a = 0\hK - ah(-K) is convex}.
Then it follows from the representation of K(p) above that r^ = A + cuk)'^-
Denoting by h^; the even part and by h'f^ the odd part of h^, we may also
write
rK = inf {p ^ 0|Bp - 1)^]^ + h^^ is convex}.
In particular, K is reducible if and only if it has a summand positively
homothetic to -^ B^mfirescu 1973). Now, elementary properties of
Minkowski addition and criteria for convex functions (Sections 1.5, 2.5) and
for summands (Section 3,2) can be used to obtain results on reducibility.
14. The Minkowski measure of symmetry. The number r(K, -K), the inradius
of K relative to -K, is an affine invariant of K which, under different
names, has often been discussed in the literature. By definition,
r{K,-K) = max{X^O\-XK + tc K for some teE"). Griinbaum [37],
Section 6.1, calls this Minkowski's measure of symmetry and denotes it by
Fi{K). Clearly Fi{K) = 1 if and only if K is centrally symmetric. If ^ € Xg
has its centroid at o, then
hiK, m) g — h(K, -u) = — h(-K, u) for u e S"-\
n n
which for special cases goes back to Minkowski A897) (see also Bonnesen &
Fenchel [81, §34); hence -KCnK and thus Fi(,K)^l/n. The equality
Fi{K) - l/n holds if and only if ^ is a simplex (see the references in
Griinbaum [37], 6.1). The set
C(K) := {a € l"\-FdK)iK - a) + a C K)
_ K^FdK)i-K) _ F,(K)
l+F^K) ~1 + Fi(/C)^''
is called the critical set of K. Klee A953) proved
—- +dim CiK)Sn,

142 Minkowski addition
which implies that dimC(^) = n -2 for K e JC". Dziechcinska-Halamoda
& Szwiec A985) for polytopes and Laget A987) in general proved that every
convex body in E" of dimension at most « - 2 is the critical set of some
convex body K eX".
15. Asymmetry in mathematical morphology. Inner parallel bodies of a convex
body K with respect to its difference body D^ and the Minkowski measure
of symmetry have also been discussed in mathematical morphology; see
Jourlin & Laget A988). Jourlin & Laget A987) asked for a characterization
of convex bodies K with the property that the inner parallel body
K — tDK, for some t>0, is positively homothetic to K. For the case of
the plane, an answer was given by Schneider A989a): If ^ € Xl has this
property and is not centrally symmetric, then K is a polygon with the
property that some homothet Q oi —K is properly contained in K and is
such that each edge of K contains a vertex of Q. Examples of such polygons
are the affine images of regular polygons with an odd number of edges, but
there are many others. The corresponding problem in higher dimensions is
open.
3.2. Summands and decomposition
Let K, L e JC" he convex bodies. As defined in Section 3.1, the body L
is a summand of K if there exists a convex body M eX." such that
K - L + M. In this section, we shall study different aspects of the
summand relation. After a remark on direct summands, we collect some
criteria for summands; then we investigate convex bodies having only
trivial summands.
We start with a result concerning direct summands, to which we shall
not come back. We write K = Ki@ . . . @ K„\i K = Ki+ . . . ■\- K„
for suitable /C, lying in linear subspaces £, of E" such that E" = £i ®
... ® E„. The convex body L is called a direct summand of K if there
exists a convex body M such that K = L® M. If a representation
K = L@ M is only possible with dim L = 0 or dim Af = 0, then K is
called directly indecomposable.
Theorem 3.2.1. Every convex body K e JC" with dim K ^l has a
representation
K = Ki@ ...@ K„
with m e {I,. . . , n} and directly indecomposable bodies Ki,. . . , K„
with dim K/^l. The representation is unique up to the order of the
summands.
Proof. The existence of the representation is obvious. To show the
essential uniqueness, suppose that
K = Ki ® . . . @ K„ = Li ® . . . ® Lr,

3.2 Sumroands and decomposition 143
where the bodies /C,, L, are directly indecomposable. Then m ^ 2,
r^2. Put K := K2® ... ® K„. "We can choose a vector m =t= o
orthogonal to Ki such that the support set F{K,u) contains only one
point, say x. Then FiK,u) = FiKi,u) + F{K,u)= Kt + x by
Theorem 1.7.5(c); thus
Ki+ X = F{K, u) = Fill, u)@ ...® FiLr, u).
Since Ki is directly indecomposable, the decomposition on the right-
hand side must be trivial, that is, at most one of the summands has
positive dimension. Thus Ki is a translate of a subset of some Lj. By a
similar argument, Lj is a translate of a subset of some /C,. Since
dim/C] SI and Ki, Ki for 1 =t=' are in complementary subspaces, we
must have / = 1 and thus Ki = Lj. Repeating the argument for K2 @
. . . @ Km, and so on, we arrive at the assertion. ■
We turn to summands in the general case and first prove a simple
intuitive criterion. For convex bodies K, L e JC" one says that L slides
freely inside K if to each boundary p>oint x ot K there exists a
translation vector t e E" such that x e L + t C K.
Theorem 3.2.2. Let K, L e X". Then L is a summand of K if and only
if L slides freely inside K.
Proof li K = L + M and x € K, there exist y e L and t e M such that
X = y ■¥ t, hence xeL + t<ZM = K.
Vice versa, suppose that L slides freely inside K. Let x ehdK be
given. By assumption, there exists t e E" such that x e L ■¥ t d K. Then
f € a: ~ L and a: € L + (A: ~ L). We deduce that bd /C C {/C ~ L) + L
and hence /C = {/C ~ L) + L. By Lemma 3.1.8, L is a summand of K.
As a consequence, we can show that every summand L of a polytope
P is itself a polytope: To a given vertex x of P, there is a unique vector
t, such that X e L-¥ t^C P. Clearly N{L,x - t^) D N(P, x). Thus L
has finitely many boundary points y\,...,yk such that Uf=iN(L,y,)
= E". If M is a normal vector of L at some exposed point a, then
u € N{L,yi) for some / and hence a = y,. Consequently, L has only
finitely many exposed points and thus is a polytope.
The possible summands of a polytope have a structure that is strongly
related to the polytope' itself. Let P, Qi, Q2 e'?>" and P= Qi + Q2.
Let F be a face of P. Choose u e 5""' such that F{P, u) = F and put
G. — ^{Qi.") for / = 1, 2. Then F= Gi + G2, hence G, is a summand

144 Minkowski addition
of F. In particular, the normal vectors (of the facets) of Qi are among
the normal vectors of P. Choose x e relint F. Then x has a
representation x = yi + y2 with y,€ relint G, (Lemma 1.3.12). By Theorem
2.2.1(a) we have
N(P, F) = N{P, X) = N(B„ yi) n NiQ2, y2)
= N(B„ G.) n NiQ2, G2),
hence N{P, F)C N{Qi,Gi). Since m € relint ^@1, G,), the face G,
does not depend on the choice of u. We say that Gi is the face of Qi
corresponding to F. If F' is a face of P containing F and G\ is the face
of Qi corresponding to F', then a vector v with F' = F{P, v) satisfies
V € N(P, F) C N(ei, Gi) and y € relint N(e,, Gi), hence ^@1, Gi) C
f^{Q\^ G\). Thus G, is contained in G',.
Particularly easy to describe are the summands of a polygon P e 'J)^.
Choose an orthonormal basis ^1,^2 of E^ and let ;to = ;to(P) be the
vertex of P with smallest e2-coordinate and, if there are two vertices
with this property, also with smallest ei-coordinate. Let ;to, JCi, .. ., Jc^t,
^*+i = ^0 be the vertices of P in cyclic order, in such a way that the
vectors
y, .= Xi - Xi-\, f = 1, . . . , A: + 1
satisfy
y, = L,(eiCosar, + 62 sin or,) C.2.1)
with L, > 0 and
0 ^ o-i < 0-2 < . . . < ock^x < 2n. C.2.2)
Then
2 y; = o. C.2.3)
1=1
Vice versa, if vectors v^,. . . , y^+i satisfying C.2.1), C.2.2) and C.2.3)
are given, then
i
jco (arbitrary), Xj := Xf^ ■\r ^Vi, j = \, ... , k
1=1
are the vertices of a convex polygon P. In fact, if y € {1, . . . , k ■¥ \) is
given, we can choose u e S^ with
{vi, u) > 0 tor i = j, j - I, . . . , m,
(y,-, m) < 0 for / = y + 1, . . . , m - 1,
where the indices / have to be understood as residue classes modulo
{k + 1). For r € {;■ - 1, y - 2, . . ., w - 1} we then have
{Xj - Xr, U) = (yr + l + t'r + 2 + • . . + Vj, u) > Q

3.2 Summands and decomposition 145
and for r € (y + 1, . .. , m - 1},
{Xj - Xr, U) = -{Vj+i + Vj+2 + . . . + Vr, U) > 0.
Thus each Xj is a vertex of F := conv {X(,, Xi, . . . , Xk}-
Now suppose that P = Q + Q'. Since each edge of Q is a summand
of an edge of P, the polygon Q has the vertices
xoiQ) and y^ = 2^,^/, j = I, ... , k C.2.4)
i=l
(not necessarily distinct), where O^A,^!. Clearly, Q' then has the
vertices
xo{P) - XoiQ) and y} = 2A - ^,>m y = 1, ..., A:. C.2.5)
1=1
Vice versa, if Q is a polygon whose vertices can be given by C.2.4),
then B is a summand of P, since C.2.5) can be used to define the
vertices of the other summand. Thus we see that Q is a summand of the
polygon P if and only if for each edge F{Q, u) of Q the support set
F{P, u) is an edge of P of at least the same length. This remark will
later be extended (Theorem 3.2.8).
We return to summands of general convex bodies and first observe
that there is a close relation to intersections of translates. Let
K, L eJC" be convex bodies such that L is a summand of K. Then
K = L + M for some MeJ^", and since K = {K~ M) + M by
Lemma 3.1.8, we get
L = K ~ M = f]iK - t).
Thus each summand of K is an intersection of a family of translates of
K. For n = 2, this necessary condition for a summand is also sufficient:
Theorem 3.2.3. For K e JC^, each nonempty intersection of translates of
K is a summand of K.
Proof Suppose that 0 =t= L = H/srCi^ + 0 with some set 0 =t= T C E^
First let /C be a polygon, say
i=\
Then
L = n(*: + 0 = n r\n:.,.Mu.,)
' k k

146 Minkowski addition
where /3, := inf,g7-(«,, f). It is now easy to see that the polygon L
satisfies the sufficient assumption stated above for summands of a
polygon K and hence that L is a summand of K.
Now let K e X^ be arbitrary. We can choose a sequence {PduN of
polygons such that
Pi D P2 D P3 . . . and f) ^i = f^^
hence lim P, = K (cf. Lemma 1.8.1). We have
L= r\iK + t)= f]f]ip, + t)= ciPi
teT leT ieN isM
with P; := OiMPi + ')• By the first part, P, = P- + Qi with Q, e JC^.
From P; D P2 D .. . we infer that lim P\= L, and since lim P, = K,
there exists limQ, ■= M e JO-, and K = L + M. Thus L is a summand
ot K. U
In E" with n ^ 3, an intersection of two translates of a convex body
K eJC" is in general not a summand of K. This is shown by the
example of a square pyramid in E^.
For further investigation of the intersections of translates, the
following may be remarked.
Lemma 3.2.4. Let K, L e .X". Then L is an intersection of translates of
K if and only if
K ~ (K ~ L) = L.
Proof It L = r\,^TiK - t) with some family T C E", then
L = C]{K - t)D f] {K- t)D L,
leT LCK-i
hence
L = f) iK-t)= f] (K- t)= K~{K~L).
LCK-t teK~L
The latter equality also shows that K~{K~ L) is an intersection of
translates oi K. ■
For K, LeX^ we deduce from Theorem 3.2.3 that K ~ L, li not
empty, is always a summand of K, and that L is a summand of K if
and only \i K ~ {K - L) = L.
Criteria for summands involving intersections of translates can also be
formulated in higher dimensions. Lemma 3.2.5 and Theorem 3.2.7
below are due, in more general forms, to Wieacker A988a).

3.2 Summands and decomposition 147
Lemma 3.2.5. Let K, L eJC". Then L is a summand of K if and only if,
for all X e K and yi, . . . , y„+i € L,
(;c - L) n (a: - yi) n ... n (a: - y„+i) + 0-
Proof By Lemma 3.1.8, L is a summand of K if and only if
{K ~ L) ■¥ L = K, hence if and only if, for each x e K, there is z € L
such that
x-zeK~L= r\{K- y)
or, equivalently,
{x-L)r\ {\{K-y)^<Z.
yeL
Now Helly's theorem, 1.1.6, applied to the finite subfamilies of
{{x - L)r\{K - y)\y € L}, together with the compactness of L and K,
yields the assertion. ■
Lemma 3.2.5 is improved by Theorem 3.2.7. Its proof uses the
following lemma.
Lemma 3.2.6. Let K, L eJC^. If projf L is a summand of pro]eK for all
two-dimensional linear subspaces E in some dense subset of G{n, 2), then
L is a summand of K.
Proof Define g{u) := h{K, u) - h{L, u) for m € E". Let Ee G{n, 2)
be a two-dimensional linear subspace for which ttK = ttL + Me with a
convex body Me C £; here n = projc- For u, v e E v/e have u + v e E
and hence
g{u + v) = h{K, u + v) - h{L, u + v)
= h{nK, u + v) - h{nL, u + v)
= h{ME, u + v) ^ h{ME, u) + h{ME, v)
= «■(") + giv).
The set of all pairs {u,v) for which the inequality g{u + v)^
Siu) + g{v) has thus been established is dense in E" x E"; therefore the
continuous, positively homogeneous function g is sublinear and hence,
by Theorem 1.7.1, is the support function of a convex body M e JC".
Then K = L + M. ■
Theorem3.2.7. Let K' L e X". Then L is a summand of K if and only
if, for all X e K and yi, y2 e L,
{X- L)r\{K- y.) n (A: - y^) + 0. C.2.6)

148 Minkowski addition
Proof. Only the sufficiency of condition C.2.6) has to be proved.
Suppose first that n = 2. We assert that
{K - yi) n (A: - y^) n (A: - y,) + 0 C.2.7)
for yi,y2, y-i€i L. Suppose this were false for some yi,y2>y-i^ L.
Then there is a line H that strongly separates {K - y{)f\{K - y2)
(which is not empty, by C.2.6)) and K - yj. For x e K, the set ;t - L
meets {K - y^)r\{K - yj), {K - y2)r\{K - y^) and {K-y^)r\
{K-y^) and, hence, also the disjoint segments H f\{K - y{) and
H r\{K - yi)- Thus x ~ L contains the open segment S between them.
From
5C (\{L - x) = L ~ K
xeK
we deduce that K is strictly contained in a translate of L, say in L.
Then h{K, ■)^h{L, •), while h(K,u)< h{L,u) for some u e SK For
y,Z€L with (y, u) = h{L, u) and (z, «) = h{L, -u) we get
h{K - y,u) = h(K,u)~ {y,u) < 0,
hiK - z, -u) = h{K, -u) - {z, -u) SO
and therefore (K - y)n{K - z) = 0. This contradicts C.2.6); thus
C.2.7) must hold.
From C.2.6), C.2.7) and Helly's theorem we now get
(x ~ L)n{K - yy) n (a: - y^) n (a: - yj) + 0
whenever x e K and yj, yj, yi e L. Lemma 3.2.5 then shows that L is
a summand of K. Thus the theorem is proved for n = 2.
Now let n^3. Let EeG{n,2) and n^ = proj^. For xenK and
yi,y2e ttL we choose x e K and yi, y2 e L with x = nx, yi = nyi and
find
(X- L)r\{K- n) n (a: - y2) + 0;
hence
{x - ttL) n {ttK - yi) n {ttK - y2) + 0-
By the first part of the proof, nL is a summand of nK. By Lemma
3.2.6, L is a summand of K. ■
For special convex bodies in X", such as polytopes or sufficiently
smooth bodies, more effective criteria for summands can be proved. The
case of polytopal summands is easy to check by means of the following
result.
Theorem 3.2.8. Let P,KeJC", where P is a poly tope. Then P is a
summand of K if and only if the support set F{K,u) contains a translate
of F{P, u) whenever F(P, u) is an edge of P (u € S"~^).

3.2 Summands and decomposition 149
Proof, li K = P+ M with M € JC", then f (/C, «) = F(P, «) + F(M, «)
D F{P,u) + t for some teM, so that the condition is necessary.
Suppose now that the condition is satisfied. First let n = 2. We can
easily construct a sequence (P,),^!^ of polygons such that Pj^ K for
i^. 00 and FiK, Uj) C F(P,, My) for each normal vector Uj of an edge of
P. By the criterion for summands of polygons described earlier, we have
P, = P + B, with some polygon Q,. From P, ^ /C we get Q, -* Q for
some B e 3^^ and /C = P + Q; thus P is a summand of K.
Now let n ^ 3. Let E e G{n, 2) be a two-dimensional linear subspace
with the property that dimf(P, w) ^ 1 for each w e E. Let rr= projg.
If w € E\{o} is such that F{nP, w) is an edge of the polygon nP, then,
by the choice of £, n~^F{nP, w) D P = F{P, w) is an edge of P. By
the assumption of the theorem the support set F{K,w) contains a
translate of the edge f(P, w), hence F{nK, w) contains a translate of
the edge F{nP, w). We have proved that nP and nK satisfy the
assumptions of the theorem. By the first part of the proof, n^P is a
summand of nK. Since the set of all 2-subspaces E that could have
been chosen above is dense in G{n,2), it follows from Lemma 3.2.6
that P is a summand ot K. ■
For a convex body K e J(" we denote by <^{K) the set of all convex
bodies that are homothetic to a summand of K. Thus <^iK) is a convex
cone in J^", in the sense that L, M €S{K) and A^O imply
L + M €S{K) and XL eSiK). Further, for convex bodies L,K eJC"
we write L ^ /C if dim F{L, u) ^ dim F{K, u) for all u e 5""'. Then the
.special case of Theorem 3.2.8 where K is also a polytope (this is due to
Shephard 1963), evidently implies: If P, K e ^" are polytopes, then
P€c^(A:)ifandonlyif P^ a:.
Another class of convex bodies for which criteria for summands are
easily formulated are the bodies of class C+. The following result is just
a reformulation of Theorem 2.5.4.
Theorem 3.2.9. Let K, L e X" be convex bodies of class C\. Then L is
a summand of K if and only if the radii of curvature satisfy
r{L, u, v) ^ r{K, u, v)
for each orthogonal pair of vectors u,veS"~^; equivalently, if the
curvatures satisfy
k(L, X, t) g KiK, y, t)
for xe F{L, u), y € F{K, u), u € S"'^ and t ± u.

ISO Minkowski addition
Consequently, if /C is a convex body of class C\, then there are
positive numbers r, R such that rK is a summand of the unit ball B"
and B" is a summand of RK.
Let K, L € X". One says that the convex body L rolls freely inside K
if for each rotation p e SO(n) and each boundary point x oi K there is
a translation vector t e E" such that x e pL ■¥ t <Z K. ^y Theorem 3.2.2,
this is equivalent to the condition that each congruent image of L is a
summand of K. Hence, Theorem 3.2.9 yields the following.
Corollary 3.2.10. Let K, L e X" be convex bodies of class C\. Then L
rolls freely inside K if and only if
max r{L, u, v) ^ min r{K, u, v).
This is a generalization of Blaschke's 'rolling theorem'. Blaschke [5],
§24, considered the case where n ^ 3 and L is a ball.
We turn now to the investigation of convex bodies having only trivial
summands. The convex body K e X" of positive dimension is called
indecomposable if a representation K = Ki + K2 with Ki, K2€X" is
only possible with K^, K2 homothetic to K; otherwise K is called
decomposable. Thus K is indecomposable precisely if S{K) =
{XK + t\X^O, f €E"}.
In dealing with indecomposable convex bodies (and elsewhere too) it
is convenient to introduce the class
X"s ■■= {K € X"\s{K) = o, biK) = 1},
where s is the Steiner point and b is the mean width (see Section 1.7).
We call X" the class of normalized convex bodies. Evidently it contains
precisely one element from each homothety class of convex bodies with
more than one point. From the linearity properties of Steiner point and
mean width it follows that X" is a convex subset of X", by which
we mean, of course, that K,LeX" and A €[0,1] implies
A - X)K + XLe X". By the continuity of s and b, X" is closed. If
K e X" and xeK, then [o,x]CK and hence BK-,_i/nK-„)|;ic| =
6([o,;ic]) ^ b{K) = 1. Thus X" is bounded, and by the Blaschke
selection theorem it is compact. We see that the map cp (see A.7.1) and
Theorem 1.8.11) maps X" bijectively onto a compact convex set in the
Banach space €{8"'^). Under this map, the indecomposable bodies in
X" correspond precisely to the extreme points of (p{X"). (The
definition of an extreme point is the same as in Section 1.4: x e (p{X") is
an extreme point of q>{X") if and only if it cannot be represented
in the form x = {\ - X)y + Xz with y,z€q>{X") and A €@,1).) In

3.2 Summands and decomposition ISl
fact, let K eX" be decomposable. Then K = L + M with suitable
L, M eX" that are not homothetic to K. We have b{L) + b{M) =
b_iL + M) = b{K)=l _and similarly s{L) + s{M) = o. Writing
L-=l[L- s{L)]/b{L), Mj= [A^- s{M)]/b{M) and X= b{M), we_get
L,M_€X"s and A - A)L + AM = a:, hence (p{K) = {I - X)q>{L) +
Xq){M) with <)c<L) =t= <P(/C), so that <)c</C) is not an extreme point of
(p{X"). The converse conclusion is obvious.
In the plane, the indecomposable convex bodies are easily
determined.
Theorem 3.2.11. The indecomposable bodies in JO- are the segments and
the triangles.
Proof. That segments and triangles are indecomposable is clear (a
summand of a triangle must be a polygon with the same normal
vectors). Vice versa, let K e JC^ he indecomposable and not a segment.
By Theorem 3.2.3, each nonempty intersection K r\{K + t) must be
homothetic to K. Let U denote the set of all exterior unit normal
vectors at regular boundary points of K. Choose u, v e U, u ^ ± v, and
let X, y be regular points of hd K where these normal vectors are
attained. Then K := K D (K + x - y) has interior points. Since K is
homothetic to K, the body K must have a boundary point where both u
and V, are normal vectors. This implies that no vector w e intpos {u, v}
can be a normal vector at a regular point of K. Thus, U must either
consist of three vectors or of four vectors in opposite pairs. Since the
support lines at regular p>oints (which are dense in bd K) determine K
uniquely, K is either a triangle or a parallelogram, but the latter is a
Sum of two segments and thus decomposable. ■
Remark. A regular hexagon in the plane can be written as the
Minkowski sum of two triangles and also as the sum of three segments. Thus a
representation of a convex body as a finite sum of indecomposable
bodies, if possible, is in general not unique.
The assertion of Theorem 3.2.11 is typical for the plane; in higher
dimensions the situation changes drastically. In fact, most convex bodies
in E", for n ^ 3, are indecomposable. We first show that there are many
indecomposable polytopes.
There are several criteria from which the indecomposability of a
polytope can be deduced; we choose one due to McMullen A987). Let
P e<J)" be a polytope. A strong chain of faces of P is a sequence
Fo, f 1, ■ • ■ , Fk of faces of P such that dim(Fy_, n F^) S 1 for y = 1,

1S2 Minkowski addition
. . . , k. A family <7^ of faces of P is called strongly connected if for each
F,G e(J- there exists a strong chain F = Fq, f i, ..., Fk= G with
Fj e(J for 7=1, ... , k. The family (J- touches the face F if
f n G =1= 0 for some G e(J.
Theorem 3.2.12. Let Pe^" be a polytope having a strongly connected
family '7 of indecomposable faces that touches each facet of P. Then P is
indecomposable.
Proof. We may assume that dim P = n. Let Q be a summand of P. Let
fo, f 1, ■ • ■ , Pk^^ form a strong chain. Let Gy be the face of Q
corresponding to f, (in the sense explained after Theorem 3.2.2). Then
Gj is a summand of f, and Fj is indecomposable, hence Gy = Ayf, + tj
with suitable Ay^O and t, e E". Because dim{Fy_i n Fy) = 1 for each /,
we conclude that Ay_i = Ay and fy_i = tj for y = 1, . . . , k. Since the
family <7^ is strongly connected, there are a number A ^ 0 and a vector
t € I" such that F{Q, v) = Af (P, v) + t whenever F{P, v) e (J.
Let Ml, . . . , M„ be the outer unit normal vectors of the facets of
P; let /€ {1, . . . , m). By assumption, F{P,Ui) has a vertex x lying
in some face F{P,v)e(~f. The corresponding vertex y oi Q lies
in F{Q,v), hence y = Xx + t. This yields h{Q,U;)= {y,Ui) =
{Xx + t, Uj) = Xh{P, Ui) + {t, Uj) = h{XP + t, Uj) for j = 1,. . . , m and
hence Q = XP + t by Corollary 2.4.4. Since Q was an arbitrary
summand of P, the polytope P is indecomposable. ■
Corollary 3.2.13. If all two-dimensional faces of the polytope P are
triangles, then P is indecomposable.
This can be used to show that, for n ^ 3, most convex bodies in JC"
are indecomposable.
Theorem 3.2.14. For n ^ 3, the set of indecomposable bodies in JC" is a
dense Gyset.
Proof. Since the simplicial polytopes are dense in JC" \i n"^ 3, it follows
from Corollary 3.2.13 that the set (^ of indecomposable polytopes is a
dense subset oi Ji". Let KeJC" be decomposable. Then one easily
checks that K can be written in the form /C = (L + M)/2 with convex
bodies L,MeJC" such that s{K) = s{L) = s{M), b{K) = b{L) =
b{M) and L^ M. For ke N, let "Sk be the set of all K e X" having
such a representation with d(L, M)^ l/k. Using the Blaschke selection

Notes for Section 3.2 153
theorem, one shows that ^^ is closed. Since cyf. = 3^"\U*6n'^*> the
assertion follows. ■
Since in a Baire space the intersection of two dense G^-sets is a dense
G^-set, the last result together with Theorem 2.6.1 shows that most
convex bodies in X", n^3, are smooth, strictly convex, and
indecomposable. It appears that no concrete example of such a body is explicitly
known. This is not too surprising, since it is hard to imagine how such a
body should be described. By a result of Sallee A972), to be
indecomposable it cannot be too smooth. More precisely, a neighbourhood U in
the boundary of a convex body K e J(" is called e-smooth, for some
£ > 0, if for each x e U there exists t e E" such that x e eB" + t C K.
Sallee proved: If the boundary of a strictly convex body K contains a
neighbourhood U that is e-smooth for some e > 0, then K is
decomposable.
Notes for Section 3.2
Direct decompositions. The proof of Theorem 3.2.1 can be extended to the
case of line-free closed convex sets. In this form, the theorem was proved by
Gruber A970a), although in a less direct way. For an approach involving
matroids, see Kinczes A987), Theorem 2.5. From a general and abstract
viewpoint, such decompositions were studied by Gale & Klee A975).
Theorem 3.2.2 appears at several places in the literature (sometimes
described as 'obvious' or 'folklore'); see Geivaerts A972), Sallee A974),
Weil A974b) and Firey A979).
Intersections of translates. Theorem 3.2.3 was proved in a special case (the
intersections of two translates) by W. Meyer A972). His proof can be
extended to the general case, which appears in Geivaerts A972). The
theorem does not extend verbally to higher dimensions; in fact, higher-
dimensional analogues lead to interesting problems. In [36] (Problem 71) it
was conjectured that a smooth convex body KeX", m = 3, with the
property that each nonempty intersection of K with a translate of ^C is a
summand of K, is necessarily an ellipsoid. This is still unknown. More
information is available in the case of polytopes. McMullen, Schneider &
Shephard A974) called a polytope P € 'P" monotypic if every polytope Q
with the same system of unit normal vectors (of facets) as P is strongly
isomorphic to P or, equivalently, if every polytope with the same system of
normal vectors as P is simple. They showed, among other equivalent
conditions, that a polytope is monotypic if and only if, for all / € E",
P n (P + f) is either empty or homothetic to a summand of P. They further
proved that a centrally symmetric monotypic polytope is the direct sum of
centrally symmetric polygons and line segments, and they classified the
three-dimensional monotypic polytopes as well as the monotypic w-polytopes
with at most « + 3 facets.
Let S" be the class of all convex bodies K eX" for which each nonempty
intersection of translates of ^C is a summand oi K. Ey Lemma 3.2.4, K eS"
if and only if K - {K - L) = L implies that L is a summand of K, thus if
and only if /C ~ L = A/ and /C -M = L implies that L + M = K. From a

1S4 Minkowski addition
slightly different viewpoint, this problem (for centrally symmetric K) was
also studied by Sallee A987). A result of Maehara A984) implies that
M-balls belong to cV".
A general study of relations between summands and intersections of
translates was presented by Wieacker A988a), from which Theorem 3.2.7 is
taken. He has other, more general, results in this spirit, also for unbounded
convex sets. Restricted to convex bodies, one of his results reads as follows.
Theorem. Let K^Lj^JC" for i€/ (an index set) and suppose that
flic/i-t 4= 0- If for each subset 7 of / with « - 1 or fewer elements there is a
subset 7' C / such that fl/cjuyi-/ is a summand of K, then flic/L, is a
summand of K.
This result can be considered as a higher-dimensional extension of
Theorem 3,2.3, to which it reduces if « = 2 and each Li is a translate of K.
Geivaerts A972, 1974a) has studied the class ^(K) of all convex bodies
homothetic to a summand of the body K € JC". For « = 2 he proved that
S{K) is closed if and only if ^C is a polygon, and that ^{K) is dense in X" if
and only if K is smooth. He also investigated the class S'{K)cX" of all
bodies L with the property that for each x ebdK there exist A;j > 0 and
f^ € E" such that x € X^L + t^d K.
Criteria for summands. Theorem 3.2.8 was proved by Shephard A963) in the
case where K is a polytope and by Weil A974b) in the general case, though
in a more indirect way using deeper results. The proof given here is taken
from Schneider A974b).
Weil A980b) proved the following theorem and explained that it can be
considered as a generalization of Theorem 3.2.8 to general convex bodies.
Theorem. Let K, L eX". Then L is a summand of K if and only if, for
each translate L' oi L not contained in K., the set of all points of L'
farthest from K {% & support set of L'.
If L is strictly convex, this simplifies as follows: L is a summand of K if
and only if each translate L' of L not contained in K has a unique point
farthest from K.
Wieacker A988c) found a way of generalizing Weil's theorem to
unbounded closed convex sets. Since for such sets the vector sum need not be
closed, and sets of farthest points may be empty, this requires a careful
reformulation.
Weil A980b) and Firey (oral communication, Oberwolfach 1980)
proposed to characterize summands L of convex bodies K by imposing
topological conditions on {L + t)r\hdK. Such a result was proved by Goodey
A982):
Theorem. Let K, L € JCq. If bd ^C n int L' is acyclic (in particular, if it is a
topological ball) for every translate L' of L, then L is a summand of K.
In fact, Goodey did not speak of summands but of freely sliding bodies,
but this is equivalent by Theorem 3.2.2.
Burton A979a) obtained the following sufficient criterion for a summand:
Let K, L e X" be convex bodies with LC K whose nearest-point maps
commute, that is, pi" Pk = Pl (= Pk ° Pl)- Then L is a summand of K.
He also proved: Suppose that L is smooth. Then Pl'> Pk = Pl if and only if
K is an outer parallel body of L.

Notes for Section 3.2 155
Criteria for summands involving mixed volumes or mixed area measures
were proved by Weil A974b) and McMullen A990); see also Matheron
A978b) (compare Section 4.2, Note 2, and Section 5.1, Note 9).
Rolling theorems. Extensions of Blaschke's rolling theorem, also to non-
compact convex sets, were treated by Koutroufiotis A972), Rauch A974),
Delgado A979), and very thoroughly and in great generality by Brooks &
Strantzen A989). Analogues for convex curves in spherical or hyperbolic
two-dimensional space are due to Karcher A968). In a question involving
geometric probabilities (see Section 4.5, Note 10), freely rolling convex
bodies were used, and some characterizations proved, by Firey A979) and
Weil A982a). In particular, these authors (Firey, §5, for the plane case;
Weil, Theorem 1, in E") give versions of Theorem 3.2.9 that hold for
general convex bodies, without differentiability assumptions. Brooks &
Strantzen A989, Theorem 4.3.2) have a very general result of this kind
where, however, the conclusion is not in terms of the summand relation but
in terms of the inclusion relation.
Apparently the first to consider indecomposable convex bodies was Gale
A954). Without proof he gave some examples of decomposable and
indecomposable convex bodies, and stated the assertion of Theorem 3.2.11. For
this theorem, an analytic proof was given by Silverman A973a) and a
geometric one by W. Meyer A969, 1972). Meyer proved a weaker form of
Theorem 3.2.3 and then used a characterization due to Rogers & Shephard
A957) of the simplex, which holds in n-dimensional space (see Section 7.3,
Note 2, for further results related to this theorem).
Simple examples of indecomposable convex bodies in E" (n = 3) that are
not polytopes can be constructed in the following way. A general frustum
over A and B in E" is, by definition, the convex hull of two convex bodies
A and B that lie in different parallel hyperplanes. Sallee A974) proved:
Suppose that K is a general frustum over A and B where A is
indecomposable and no homothet of /I is a summand of B. Then K is indecomposable.
Summands and decomposability of polytopes. The first results on summands
of polytopes and on indecomposability are due to Shephard A963) (see also
Griinbaum [15], Chapter 15). Shephard proved the polytope case of
Theorem 3.2.8 and deduced the following sufficient condition for
indecomposability.
Theorem. A polytope P € ^)" is indecomposable if it has an edge to which
each vertex is connected by some strong chain of indecomposable faces.
Among the consequences that Shephard deduced are Corollary 3.2.13 and
the fact (in a more general version) that every simple M-polytope, except the
simplex, is decomposable.
W. Meyer A969, 1970b, 1974) has some weaker sufficient
indecomposability conditions for 3-polytopes. He further studied the cone S{P) of all
polytopes homothetic to a summand of the polytope P and deduced a
necessary and sufficient condition for indecomposability in terms of the rank
of a certain system of linear equalities and inequalities determined by the
facets of the polytope. A clear picture of the situation and new proofs of
these results can be obtained if one uses a certain diagram technique
(polyhedral representations); see McMullen A973) and also McMullen
A979), Section 5. Pourneau A979) used McMullen's representations to
define a certain metric related to decomposability.
Kallay A982) studied the decomposability of a polytope in connection
with the decomposability of its (geometric) edge graph. Smilansky A986)

156 Minkowski addition
gave an example of an indecomposable 4-polytope all of whose facets are
decomposable.
Although some sufficient criteria for the indecomposability of a polytope
are of a combinatorial nature, such as Corollary 3.2.13, decomposability is
not a combinatorial property. W. Meyer A969, 1970b) and Kallay A982)
both gave examples of two combinatorially equivalent 3-polytopes, one
decomposable and the other not. Decomposability of poly topes is, however,
a projective property: If r is a projective transformation of E" that is
admissible for the polytope P, then the cones S{P) and S{TP) have the
same dimension. This was shown by Kallay A984); see also Smilansky
A987), pp. 40-1.
Smilansky A987) studied the decomposability of polytopes (and
polyhedral sets) by investigating the space of affine dependences of the vertices
of the polar polytope. He obtained new proofs for results of W. Meyer
A974) and McMullen A973) and several new results, in particular on
combinatorial decomposability, for instance: If a 3-polytope has more
vertices than facets, than it is decomposable. If a 3-polytope has at most
three triangular facets, then it is decomposable.
9. Summands and symmetry. Shephard A966b) called a convex body K with
centre o irreducible if it is not the difference body of some body that is not
homothetic to K\ otherwise K is called reducible (this notion of reducibility
is different from the one in Section 3.1, Note 13). He proved that K is
reducible if and only if it has a summand without a centre of symmetry, and
gave examples of reducible and of irreducible convex bodies. An application
of Shephard's investigation to Banach space geometry can be found in Pay^
& Yost A988). A thorough study of irreducible polytopes, including various
necessary and sufficient criteria, is due to Yost A991). One of his results
states that each w-dimensional convex polytope, for « S 3, with less than 4n
vertices is irreducible; the bound An is best possible.
For /C€3:S, Tennison A967) defined /(/C) := supF(L)/V(/C), where
the supremum is taken over the centrally symmetric summands L ({o}
admitted) of K. He showed that / has many of the properties usually
associated with a measure of symmetry, except possibly continuity.
However, it follows from Corollary 3.2.13 and the denseness of simplicial
polytopes that / is not continuous.
10. Asymmetry classes. Symmetric summands also play a role in the following
question. For Kx, K^^ DC" write KxaKz if there exist centrally symmetric
convex bodies 5i, 5: € X" such that Kx + Si= K2 + Si. Thus KiOcKi if and
only if the support functions of K\ and ^^2 have identical odd parts. The
relation a is an equivalence relation on JC", and the corresponding classes
are called asymmetry classes. Ewald & Shephard A966) posed the question
of whether each asymmetry class has a minimal member, where M i{K]
(the or-equivalence class of K) is called minimal if each other element
L€[K\ can be expressed in the form
L = M + S
with a centrally symmetric body S iX". Schneider A974e) proved that for
M = 2 the answer is in the affirmative, but not for « S 3. The latter follows
from the result that a polytope P is a minimal member of its asymmetry
class if and only if there does not exist a direction u for which F{P, u) and
F{P, -u) are parallel edges.
11. Relative indecomposability. As explained in Section 3.2, indecomposable
convex bodies correspond to extreme points of the compact convex set X".
More generally, if M is a nonempty, compact, convex subset of X", we say

3.3 Approximation and addition 1S7
that K € Jfl is extreme in M if /C = A - K)Kx + A/C2 with /C,, /C2 € Jtl and
Ac @,1) implies that K^ = K^. For a closed convex subset .//?C X" we also
say that K €,//? is indecomposable relative to JH if ^C = A - K)K'^ + A^C:
with ^Li, ^^2 € jn and Ac @,1) implies that Kx and ^^2 are homothetic to
K. The following cases have been considered. Kallay A974, 1975) gave a
complete characterization, in the plane, of all convex bodies that are
extreme in the class of bodies with a given width function. (This corrected
an erroneous statement by Silverman A973b).) In particular, a convex body
K &1C^ ol constant width unity is indecomposable relative to the class of
bodies of constant width if and only if its radius of curvature function (which
exists almost everywhere on 5') is almost everywhere either 0 or 1. In
higher dimensions, it seems unknown whether the relatively indecomposable
bodies of constant width have a similar characterization, and whether they
are dense in the set of all bodies of constant width.
McMulIen A984) determined (and used for an extremum problem) the
convex bodies that are indecomposable relative to the set of bodies of
revolution.
Grza^lewicz A984) determined the convex bodies that are extreme in
.1^(C) (the set of convex bodies contained in C) for a strictly convex set
C € X\. He showed that K is extreme in X{C) if and only if either A) ^C is
a line segment with endpoints on bd C or B) int K^Qi and each connected
component of bd ^CVbd C contains at most one extreme point of K.
12. Most of the preceding treatment of decompositions K = L + M was
restricted to convex bodies K, L, A/ in E". For unbounded convex sets and in
infinite-dimensional vector spaces, many new problems arise. A detailed
study was begun by Bair A975, 1976b, 1977a, b) and Bair & Jongmans
A977).
13. Decomposition of non-convex seti. In this note only, 'indecomposable' will
have a different meaning. The set Z C^" (not necessarily convex) is called
indecomposable if it cannot be represented as a vector sum Z = Z\ + Z2
with Zi, Z2 C E" where each Zj contains more than one point. Milka A973)
investigated the indecomposability of convex surfaces (i.e., boundaries of
convex sets with interior points). Milka showed that every (relative)
neighbourhood of the set of extreme points of a closed convex surface in E^ i.s
indecomposable and that a closed strictly convex hypersurface in E" is
indecomposable. Among further results, he proved: li K CE-^ is closed and
convex, with interior points and with recession cone rec K a ray, and if bd K
is decomposable in two different ways, then bd JC is a paraboloid.
14. Extremal convex functions are, in some sense, analogous to indecomposable
convex bodies. They were investigated by, e.g., Bronshtein A978b).
3.3. Approximation and addition
Our objective in this section is the treatment of approximation results
for convex bodies, with particular emphasis on the relation to
Minkowski addition. First we describe an approximation procedure by smooth
bodies that has particularly useful properties with regard to addition.
Then we study briefly the possibility of approximating a convex body by
Minkowski sums of bodies from a given class.
It is often necessary to approximate a convex body by sufficiently
smooth bodies. We can do this by applying to its support function a

1S8 Minkowski addition
suitable regularization process (the usual convolution, with a minor
modification to retain homogeneity).
Theorem3.3.1. Let £>0 and let q?: [0, »)^ [0, oo) be a function of
class C* with support in [e/2, e] and with
Im\^\)<
\dz = 1.
// /: E" ^ R is a support function, then the function f defined by
fi^) •= /eJ(^ + \x\z)(p(\z\)dz for X € E"
is a support function of class C^ on IE"\(o}. The map T:X" -* X"
defined by hjx - h^ has the following properties:
(a) T{K + L)=TK+TL and T{XK) = XTK for K,LeJi" and
(b) T{gK) = gTK for K e X" and each rigid motion gofE";
(c) d{K, TK) ^ Rs for all convex bodies K C RB" {where R > 0);
(d) d{TK, TL) ^ A + eNiK, L) for K, Le X".
Proof. Let (p be as in the theorem and let / be a support function. It
follows from the definition that / is positively homogeneous. Let z € E"
and put
g^{x) := f{x + \x\z) + f{x - \x\z) ior X e E".
For X, y € E" and a e [0,1] we have
8zix + y) = fix + y + \x + y\z) + fix + y - \x + y\z)
S fix + 0-1;^ + y\z) + fiy + A - a)\x + y\z)
+ fix - a\x + y\z) + fiy - A - a)\x + y\z).
Without loss of generality we may assume that x and y are linearly
independent. Put
.^ 1^1 r> .^ \x + y\ .^ 1^ + y|
"' \x\ + \y\' ^' \x\ ' ^- |y| '
then 1 - o-jS > 0 and 1 - A - a')y > 0. From
2ix + 0-1;^ + y\z) = A + aP)ix + \x\z) + A - cxP)ix - \x\z),
2ix - a\x + y\z) = A - a^Xx + \x\z) + A + afiXx - \x\z)
we infer that
fix + a\x + y\z) + fix - a\x + y\z) ^ fix + \x\z) + fix - \x\z)
= 8zix).
Similarly we obtain
fiy + A - a)\x + y\z) + fiy - A - a)\x + y\z) g g,iy).

3.3 Approximation and addition
159
Together with the inequality for gj.{x + y) as obtained above this yields
g,ix + y)^ g^{x) + g,iy).
Since
fix) - 'j j^M^M\zl)dz
and q>^0, the convexity of f follows. By Theorem 1.7.1, / is a support
function.
For M € 5""' we have
/(«) = j^Jiu + zM\z\)dz = j^JiyMly - «l)dy
and hence, for x e IE"\{o},
fix)=\x\j^Jiy)<f{\y--^\)<iy,
from which we see that f is of class C" on IE"\{o}.
The map T: J(" -^ X" defined by hjK = ^k clearly has property (a)
of the theorem. For t e E" and u e 5""' we have
^T(,){") = Jj„(" + z, t)q>{\z\)dz = {u, t) = /!{,,{«),
hence T{t} = {t}. Together with (a) this yields T{K + t) =
TK + T{t} = TK + t, thus T commutes with translations. Let p be a
rotation of E", proper or improper. Defining pf by {pf)ix) '■= f{p~^x)
for ;t € E", we obtain
iphix) = j^Jip-'x + \p-'x\zM\z\)dz
= j^Jip-Hx + \x\pz))(pi\pz\)dz
hence ipf) = pf. This shows that T{pK)^pTK for KeX", which
completes the proof of part (b).
For K eJC" with K C RB" and for u e 5""^ we obtain
\hTK{u) - M")l =
/j„M" + z)(fi\z\)dz - h^iu)
^ j^Jh^iu + z) - hK{u)\(pi\z\)dz
^j^R\z\cfi\z\)dz^Rs,

160 Minkowski addition
where Lemma 1.8.10 together with q>{\z\) = 0 for \z\ > e was used. Since
u € 5""^ was arbitrary, this proves (c). In a similar way assertion (d) is
obtained, again using Lemma 1.8.10. ■
Although the bodies TK obtained in Theorem 3.3.1 have support
functions of class C", they are not necessarily smooth since they can
have singular points. This can easily be remedied by defining T'K '■=
TK + sB" for KeX". Then T'K is of class C+, and the map T' has
properties that are easily derived from those of the map T.
If a convex body K is approximated by the use of Theorem 3.3.1,
then the approximating bodies inherit many properties from K. For
instance, TK has (at least) the same symmetries as K, and if K is of
constant width w, then TK - TK = T{K - K)= TiwB") is a ball by
(b); hence TK is of constant width.
The approximation procedure of Theorem 3.3.1 can be refined
further, as we now briefly describe. For this, we use spherical harmonics
(see the Appendix for the necessary explanations and definitions). By
n„ we denote the orthogonal projection from the space CE""') of the
real continuous functions on 5""' onto the subspace of spherical
harmonics of degree m. Let K e JC" be a convex body of class C+.
Since Hk^ (the restriction of the support function of K to 5""') is of
class C°°, we have
00
m=0
with convergence in the maximum norm. If, for some integer k^2, the
partial sum 2m=o'7^m^/c happens to be the restriction of a support
function, we denote the convex body which it determines by SkK. For
given k, the map 5^ is defined only on a subset of 5^", but if K and L
are in its domain, then it follows from the properties of spherical
harmonics that
Sk{K + L)= SkK + SkL,
SkigK) = gSkK
for each rigid motion of E". Now for K of class C+, the body S^K does,
in fact, exist for all sufficiently large k (see Appendix).
Now let K €JC" and e > 0 be given. We can choose a map T
according to Theorem 3.3.1 so that d{K,TK) < e. Then the body
TK + eB" is of class C+, hence k can be chosen so that
TkK ■.= SkiTK + sB") exists and satisfies d{TK + eB", TkK) < e. Thus
the body TkK satisfies d(K, T^K) < 3e and is of class C+ (for
sufficiently large k). Finally, the support function of TkK is, in fact, algebraic.

3.3 Approximation and addition 161
since TT„h (for any h e CE"~^)) is the restriction to 5""^ of a
homogeneous polynomial on E". Hence, h{TkK,x) = |;ic|P(VI^I)> where P{y) is
a polynomial in the cartesian coordinates of y. The map T^ commutes
with rigid motions, and if K and L are in its domain, then
Tk{K + L) = TkK + TkL for ;t > 0.
Our second topic in this section is the approximation of convex
bodies by Minkowski sums of bodies from a given class. This vaguely
formulated program gives rise to many interesting questions. Here we
treat only two special aspects.
li K e X" is & convex body, we say that K' h & rotation mean of K
if there are a number m eN and rotations pi, . .. , p„€ SO{n) such
that
K' = ^ip,K + ... + p„K).
Theorem3.3.2 (Hadwiger). For each convex body K e X" with
dim /C > 0 there is a sequence of rotation means of K converging to a
ball.
Proof For L e X" let d{L) := min (A^O|L C XB"). Clearly the
function d is continuous. A ball XB" containing K contains all rotation
means of K, hence the family ^A{K) of rotation means of K is bounded.
The function d attains a minimum dQ>0 on the compact set cl'^(/C),
say at L. Assume that L ^ doB". Then there is a vector u^e S"~^ with
h{L, Mo) < d(i, hence h{L, u) < do for all m in a suitable neighbourhood
U of Mo on 5""'. Since 5""' is compact, we can find finitely many
rotations pi, . . . , p„e SO{n) such that U" lAf^ = 5""'. Put
L := — (piL + .. . + p„L).
Let M € 5""'. There is a number i e {I,..., m} with u e p,(/, hence
p,~^M € U and thus h{L, p^^u) < do, which implies
hit, m) = — J^hipjL, m) = — J^hiL, pj^u) < do-
m j^i m y=i
By the continuity of h{L, •), this yields d{L) < do- There is a sequence
iKj)jeiH in '^(/C) converging to L, and we deduce
Ki :=-ip,Ki+ ... + p„Ki)^L
for /-»'», hence diKj) < do for large j. Since Kj e ^/?(/C), this
contradicts the minimality of do. Thus L is a ball, which proves the theorem.■

162 Minkowski addition
To put the foregoing result into a more general context, we say that a
convex body K is approximable by the class ci C DC", if K is the limit
of a sequence of bodies of the form
Ai/Ci + . . + XrKr with Ki € cyf, A; ^ 0 (f = 1, . . . , r), r € N.
Theorem 3.3.2 shows that a ball is always approximable by a class c-4 if
this class contains the congruent images of some convex body of positive
dimension. On the contrary, the following theorem shows that for many
convex bodies, namely the indecomposable ones, such an approximation
is only possible in an essentially trivial way.
Theorem 3.3.3. If the indecomposable body Ke'X" is approximable
by the class c4 C JC", then the closure of cyt contains a homothetic copy
ofK.
Before the proof of this theorem, a few remarks are in order. The
theorem was proved for the case of indecomposable polytopes by
Shephard A964a); this proof is also given in GrUnbaum [15] (Section
15.2). To formulate a consequence, call a subset c^CJC" a universal
approximating class if any convex body in X" is approximable by c^^.
For n-2, the set of all triangles and all line segments is such a
universal approximating class, since each convex polygon is a sum of
triangles and line segments. For n ^ 3, however, the set of
indecomposable polytopes is dense in JC", by Theorem 3.2.14. Hence, for n ^ 3,
the only closed, homothety invariant, universal approximating class in
X" is X" itself.
Shephard's proof seems to be restricted to polytopes. In its general
form, Theorem 3.3.3 is due to Berg A969a), who observed that it is an
immediate consequence of Milman's converse to the Krein-Milman
theorem (see, e.g., Phelps 1966, p. 9) applied to the set (ip{X^) (see
Section 3.2), which is a compact convex set in CE"~'). Since we have
not treated this theorem about convexity in topological vector spaces,
we present here an elementary version of the proof, adapted to our
particular situation.
Proof of Theorem 3.3.3. Let K e X" be an indecomposable body of
positive dimension that is approximable by a class c4. We can assume
that c-i is closed.
By assumption, there is a sequence (/C,),£n converging to K where
m,
Ki = E^y^y. f^a e <^, Ay S 0. C.3.1)
For L € X" with dim L > 0 we denote its normalized homothet by L,

3.3 Approximation and addition 163
that is, L = 6(L)-'[L-5(i,)]. Then Ki^ K implies Kj^ K for
f ^ oo, and we have
Ki = XkK^i with ^ = Y/^^'
so that
SA,y = 1.
/
Since indecomposability is invariant under homothety, we may,
therefore, assume from the beginning that cyf C J^", K e J(" and that C.3.1)
holds with
SA^y = 1.
i
We assert now that K e (tI. Suppose this were false. Then for each
A € (^ v/e have d{A, K) > 0, hence the set
^UiA) := {L € JC"\6iA, L) ^ ^6(A, K)}
has interior points. For ieN, let (ii{'7{{A)) be the sum of those
coefficients Ay for which Kfj e ^M{A). We assert that
limsup//i('7^(>l)) > 0
I—* 00
for at least one A e ci. Suppose this were false. The compact set c4- can
be covered by finitely many sets 'U{Ax)y .. ., ^{Ap). Then
p '"i
2//,(^(^r)) ^ SA// = 1,
r=l >=1
but on the other hand, //,(^(i4r)) ^ 0 for i^'», a contradiction.
Hence, there is some A^ecA. for which limsup//,('iY(i4o)) >0, and
(taking a subsequence and changing the notation) we may assume that
YimHi{ai{Ao))>0. C.3.2)
Now for I e N we define
'■ ■ S'A, ' '■ ■ S"A, '
where the sums X' extend over those / for which Ki; e QI{Aq) and the
sums 2" extend over the remaining / (with the convention that
^'t = {o} if S'Ai; = 0, and similarly for K"). Then we have
K, = t,a:; + A - t,)a:7
with
T/ = S'A/y = M^(^o))- C.3.3)
Since K'i, K"i belong to the compact set ^" U {{o}} and t, belongs
to the compact interval [0,1], we may assume, after selecting
subsequences and changing the notation, that K'l^K', K"^ K",

164 Minkowski addition
with K', K" ej:"U {{o}} and t, ^ t € [0,1], and thus
K = tK' + {I - x)K".
From C.3.2) and C.3.3) we have t>0, in particular K] + {o} for large
i. For these i, K\ is a convex combination of elements from ^l({A(t),
hence K] e ^1^{Aq) and thus K' e ^U{Aq), which implies K' ^ K. Since
K', K"eX", K' and K" are not homothetic to K, thus K is
decomposable. This contradiction shows that K e c-i, which proves the
theorem. ■
In the preceding proofs, some closed convex subsets of the space of
convex bodies have appeared. In studies related to Minkowski addition
and approximation, it is convenient to have a special notion for these
sets. By an M-dass of convex bodies in E" we understand a subset
M C X" that is closed in the Hausdorff metric and closed under
Minkowskian convex combinations, that is, K^, . . . , K„e Jfl implies
Ai/C, + ... + X^K„ € M for A, s o, X^. = 1-
Such an M-class will be called homothety invariant if together with a
convex body K it contains all homothetic images of K; similarly for
other types of transformations of E". Examples of similarity invariant
M-classes are the set of all one-pointed convex bodies, which we call the
trivial M-class, the set of all centrally symmetric convex bodies, the set
of all bodies of constant width, the set of all closed balls, etc. The set
X" of normalized convex bodies is a rotation invariant M-class. The set
of convex bodies K for which K - K \% equal to a fixed convex body S
is an example of a translation invariant M-class. The set of all
summands of a given convex body is an M-class that, in general, has no
invariance properties. In this terminology. Theorem 3.3.2 tells us that
each non-trivial rotation invariant M-class contains a ball.
If <^ C X" is any set of convex bodies and G is a set of
transformations of E", let us consider the smallest M-class (smallest G-invariant
M-class) containing ci; we call this the M-class (G-invariant M-class)
generated by cA. Theorem 3.3.3 implies that the M-class generated by a
homothety invariant subset ci C X" is equal to X" only if dl is dense in
X"(ng3).
In Section 3.5 we shall return to this topic; in particular we shall
consider the similarity invariant M-class generated by a segment.
Notes for Section 3.3
1. Regularization and approximation. A regularization process very similar to
that of Theorem 3.3.1 was first proposed by Radon A936), but it appears that

Notes for Section 3.3 165
his paper remained unnoticed for a long time. Radon considered (in a more
general form) the transformation defined by
/(^) ■= jjf^x - M^) + Mf(z)](p(\z\)dz
and showed that it yields a support function if applied to such a function. He
used this to show that a convex body can be approximated by convex bodies
with analytic support functions.
The map T of Theorem 3.3.1 appeared in essence in Schneider A974a),
where it was considered for a different reason, and where the convexity of /
was proved on the basis of a communication from W. Weil. As a special
approximation procedure for convex bodies, the method of Theorem 3.3.1
was used by Weil A975b). He investigated for which pairs of polytopes
P,Q s 'Vo with P C Q it is possible to find a convex body of class C+ with
P C K CQ. He proved that this is possible if and only if each point of
bd P n bd Q is a vertex of P and an internal point of some facet of Q.
Schneider A984) extended the method, in the way sketched above, by
using spherical harmonics. A regularization process close to that of Theorem
3.3.1 was independently proposed by Saint Pierre A985).
As mentioned, the approximation process of Theorem 3.3.1 automatically
yields bodies of constant width if applied to such a body. This does not hold
for other procedures, described in the literature, that involve approximation
by bodies with very smooth boundaries or support functions; see Minkowski
A903), §2, Bonnesen & Fenchel [8], p. 36, and Hammer A963) and Firey
A974b). Special approximations for plane ovals of constant width were
constructed by Tanno A976) and Wegner A977).
A thorough study of the approximation of centrally symmetric bodies
(mainly in the plane) by polynomial bodies was made by Faro Rivas A986).
2. Rotation means. Apparently the first proof of Theorem 3.3.2 appeared (for
M = 3) in Hadwiger [17], p. 27; see also Hadwiger [18], Section 4.5.3. The
basic idea of the proof seems to be older; for instance, it is very similar to an
idea in the usual construction of an invariant mean on a compact topological
group; compare Pontrjagin A957).
Interesting problems arise if one asks for quantitative improvements in
Theorem 3.3.2: How many rotations are required to reach a prescribed
degree of accuracy of the approximation? The following result was proved by
Bourgain, Lindenstrauss & Milman A989), Theorem 6.3: Lti K e^X" and let
0 < e < 1. There are a constant r - r{K) and rotations pi,..., p;v of E" with
N ^cnE'^\oge~^ (c an absolute constant) so that
A - e)rB" C ^ (piK + ... + p^K) C A + e)rB".
Bourgain, Lindenstrauss & Milman A988) have a number of similarly
interesting and deep results. We mention only one of these on random
Minkowski symmetrizations. For u s S"'\ let ff„ denote the orthogonal
reflection at the hyperplane through zero with normal vector u. For K e X",
the convex body jC^ + ^uK) is said to be obtained from ^ by a Minkowski
symmetrization (or Blaschke symmetrization; see Blaschke [5], §22, VII).
This Minkowski symmetrization is said to be random if u is chosen randomly
on 5"~' with rotation invariant distribution. Now Theorem 14 of Bourgain,
Lindenstrauss & Milman A988) says: Let ^ be a centrally symmetric convex
body in E" and let e >0. If « > noU) and if we perform N = cn\ogn + c{e)n
independent random Minkowski sjmimetrizations on K we obtain with
probability 1 - exp[-e(e)n] a body K that satisfies

166 Minkowski addition
A - £)rB" C K Cil + £)rB''
for suitable r = r{K).
3. Theorem 3.3.2 can be used, in a similar way to other symmetnzation
procedures, to prove extremal properties of the ball; see, for example,
Hadwiger A955b) and Schneider A967b).
4. M-classes generated by lower-dimensional bodies. Let k & {\, .. ., n) and let
Sk be the set of all simplices in E" of dimension at most k. The M-class
generated by S^ was investigated by Ricker A981). He found an integral
representation for its elements in terms of E"-valued measures, which
generalizes an earlier result for the case k = \ (for this, see Section 3.5).
Let ^ € B, ..., M - 1} and let JCu be the set of all convex bodies in E" of
dimension at most k. Hildenbrand & Neyman A982) proved a result that
implies that the M-class generated by JC^ is nowhere dense in JC". This
follows also from Theorem 3.3.3 and the denseness of the indecomposable
bodies for « = 3.
3.4. Additive maps
Since Minkowski addition is a basic structure on the set JC^ of convex
bodies, the maps defined on JC" and compatible with this addition
deserve special interest. Let q? be a map from X" into some abelian
group. (If the natural range is an abelian semigroup with cancellation
law, we embed it into an abelian group.) According to Section L7, q? is
called Minkowski additive if
qiK + L) = (p{K) + (dD for K, L e JC".
If cp takes its values in a real vector space or in JC", we say that cp is
Minkowski linear if it is Minkowski additive and satisfies
qiXK) = X(p{K) for K e X" and A s o.
Examples of Minkowski linear maps are the mean width, the Steiner
point, the map K •-» F{K, u) for fixed u e 5""' and the map K^-^hn
that associates with each convex body its support function.
In the first part of this section, we study Minkowski additive maps
with particular properties and prove characterization theorems. In the
second part, we relate Minkowski additivity to a more general notion of
additivity that will play a role in later chapters.
From a geometric point of view, the Minkowski additive maps
enjoying an invariance property are of particular interest. Under
additional continuity assumptions, one can obtain uniqueness results.
Theorem 3.4.1. Let n=2. If the map q):X" -^R is Minkowski additive,
invariant under proper rotations and continuous at the unit ball B" then
q) is a constant multiple of the mean width.
A map q) from X" into E" or into X" is called equivariant under a
group G of transformations of E" if qpigK) = gq>{K) for all ge G.

3.4 Additive maps 167
Theorem 3.4.2. Let n^2. If the map q):JC"^^" is Minkowski linear,
equivariant under proper rigid motions and continuous at the unit ball
B", then cp is the Steiner point map.
Remark 3.4.3. Let q? be a Minkowski additive map from JC" into R or
E". For K € a'" we have 2K = K + K, hence (pBK) = 2(p{K), and
induction gives (p{kK) = kq>{K) for k eN. For k, m eN one then
obtains k(p{K) = (p{kK) = (p{m{k/m)K) = mcpi{k/m)K), thus (p{qK) =
qcfiK) for rational ^ > 0. If now q? is assumed to be continuous on Ji.",
then we deduce c({^kK) = Xcp{K) for real A^O. Thus the assumption
'Minkowski linear and continuous at B"' in Theorem 3.4.2 is weaker
than the assumption 'Minkowski additive and continuous'.
Theorem 3.4.1 and its proof are essentially due to Hadwiger (e.g.,
[18], p. 213). We give a slightly modified version.
Proof of Theorem 3.4.1. For xe^" we can choose p€SO{n) with
px = -X, hence <)c<(o}) = c(i{x} + p{x}) = 2(fi{x)). This yields
(({{o)) = 0 and then (f{{x}) = 0 in general. Let K e JC" and dim K >0.
If
K' = XxPxK + ... + X„p„K
with positive rational numbers X-^,...,X„ and rotations pj,...,
Pm € SO{n), then <p(/C') = (Ai + ... + X„)(fiK) by the properties of cp
and Remark 3.4.3. Since the mean width has the same properties as (p,
we deduce that (fiK)/b{K) = (p{K')/b{K'). Now, it follows from
Theorem 3.3.2 that the integer m, the rotations p, and the rational
numbers A, can be chosen so that d(K', B") is smaller than a given
i>0. Since q) is continuous at B", we deduce that
<p{K)/b{K) = <KB")/2. ■
The following elegant proof of Theorem 3.4.2 is due to Positsel'skii
A973), except that he assumed equivariance under improper rigid
motions also.
Proof of Theorem 3.4.2. Since the rotation group is compact we can, for
every oO, decompose SO{n) into finitely many nonempty Borel sets
4i_£,. .. , i4„(f) f of diameter less than e. We choose Pk ^ e A,,^^ and
write v^ f := v{Ak^). Then the usual estimate shows that
for any continuous real function / on SO{n).

168 Minkowski addition
Let a: € JC". \i K = {x), then cfiK) = x = s{K) by the rigid motion
equivariance of (p. Assume, therefore, that dim K >0. We choose
vectors yeE", x e S"~^ and a number c> |y| and define for oO a
convex body K ^hy
m{e)
k = l
Applying the map (p to this body and using the properties of q? we get,
for an arbitrary vector y e S"~\
m(f)
{(piKe), y) = S [c + ("' pk.ex)]{c(iK), Pk,,y)vk,,
k=l
and hence
\xm{(fiK,),y) = f [c+ {v, px)]{(f{K), py)Av{p)
= lo(n)^^'P^)(^K),py)dvip),
since jso(n){^^)y P)')dv(p)> ^s a function of y, is odd and rotation
invariant, and hence zero.
On the other hand, the support function of K^ is given by
m(t)
hiKf, y) = ^[c + {v, Pk,,x)]h{K, pk,ey)Vk.,
k=l
and hence satisfies
lim;i(A:„ y) = f [c + {v, px)]h(K, py)A^ip).
Now
\ ch{K,py)Ay(p) = r
is a positive number (the integral is invariant under translations of K
and clearly positive if o € relint K) that depends only on K and v. The
integral
^(^'>'> •= l^,Av,px)h{K,py)Av{p)
JSO(n)
satisfies l{x,y) = l{y,x) if n^3, since l{xx,xy) = I{x, y) for each
rotation t e SO{n), and for n ^ 3 we can choose t such that tx = y and
ty = X. Writing
f p-'vhiK,px)dv{p) :=z
JSO{n)
for nS3, we thus have lix, y) = {z,y). For n = 2, Iix,y) is not
symmetric in x and y, but in this case we can write

3.4 Additive maps 169
1 f^'
lix, >') = ^ Jo <">' "(^ + c^))HK, u{r, + (x))Acx
with M(ar) := (cosa)ei + (sinar)e2. where (ei,e2) 's an orthonormal
basis of E^, and x = m(|), y = «(>?). An elementary computation now
yields
l{x, y) = (i4icos| + i42sin|)cos>? + (i4isin| - i42Cos|)sin>)
where the ^4, depend only on K and v; hence again we have I{x, y) -
{z, y) with some vector z depending only on K, v and x. Thus in both
cases we have arrived at
\\mh{K„ y) = r + <z, y) = h{B{z, r), y),
e—0
where B(z, r) is the ball with centre z and radius r. This holds for each
y € 5""', hence limj._o K^ - B{z, r) in the Hausdorff metric (observe
that pointwise convergence of support functions implies uniform
convergence on 5""^ by Theorem 1.8.12). From this we get
r'\K,- z)^ B" for e^Oand hence (p{r-\K,- z))^cpiB") by the
assumed continuity of q) at B". Since <p(B") = o by the rotation
equivariance of q>, we arrive at cp{Kc)^ z and thus
lim{(piK,),y) = {z,y)
e-*0
for y € 5""'. We have proved that
f {v,px){(piK),py)dvip)= {z,y),
JSO(n)
which depends only on K, v, x, y and not on q). Since the Steiner point
map has all the properties of q>, this yields
/.
,^, (v, px){(p{K) - siK), py)dvip) = 0.
SO{n)
The choices v := (p{K) - s{K) and x = y now yield (p{K) - s{K).
Remark3.4.4. Inspection of the proofs of Theorems 3.4.1 and 3.4.2
shows that in neither case is it necessary to assume that q> is defined on
all of J^". In fact, the methods work even if the domain of q? is a
similarity invariant M-class.
It is, however, not known whether similar characterization theorems
hold for maps that are only defined on polytopes. Here, of course, the
continuity at the ball must be replaced by some other assumption. To
formulate a definite problem, suppose that the map (p: ^" ^ R is
Minkowski additive, invariant under rigid motions and continuous. Is it
then true that (p{P) = cb{P) for all P e^", with some constant c? A
similar question can be posed for the Steiner point.

170 Minkowski addition
Although the continuity assumptions in Theorems 3.4.1 and 3.4.2 are
rather weak, they cannot be omitted. Examples showing this can be
constructed in different ways. Here we refer to the fact that in Chapter
4 (see especially Section 4.3) we shall associate with each convex body
KeJC" & certain finite positive Borel measure Si{K, •) on the unit
sphere 5""^, its area measure of order one. The dependence on K is
Minkowski linear, that is,
5,(A:+ L, •) = S,{K, •) + S,{L, •),
Sy{XK, ■) = XS,{K, •)
for K, LeX" and A so. If S\{K, •) denotes the singular part (with
respect to spherical Lebesgue measure) of Si{K, •), then it follows
from the uniqueness of the Lebesgue decomposition of a measure into
its singular and its absolutely continuous part that the map
K>-*S\{K,-) is also Minkowski linear. Furthermore, the map
K>-^ Si{K, •) is translation invariant and rotation equivariant, that is
S,iK + t, ■) = StiK, ■),
SiipK, pco) = 5i(A:, (o)
for any translation vector f € E", any rotation pe SO{n) and any Borel
set a)CS"~^. Again, these properties carry over to 5*. Now we can
define
b'{K) ■.= S\iK,S"-'),
s'{K) := l^^_^udS\iK, u) + siK)
for K e X". Then b' and s' are Minkowski linear functions, respectively
invariant and equivarient under rigid motions, but b' is not a multiple of
b and s' is not equal to s. This can be shown by constructing suitable
convex bodies (as will become clear in Chapter 4), and it implies that b'
and s' are not continuous.
Having found that real-valued and vector-valued Minkowski additive
maps satisfying certain natural invariance and continuity properties are
essentially unique, we may go a step further and consider body-valued
Minkowski additive maps. We call a map <p: JC" -^ JC" an endomorph-
ism of Ji" if it is Minkowski additive, rigid motion equivariant and
continuous. Thus such a map is compatible with the most basic
geometric structures on JC". Examples of endomorphisms are the maps
T occurring in Theorem 3.3. L They show that there is a great variety of
endomorphisms, and, hence, that simple ones can only be singled out by
imposing additional assumptions. We shall present, without proofs,
some of the known results in this direction.

3.4 Additive maps 171
The two-dimensional case differs considerably from the
higher-dimensional one, as a result of the commutativity of the rotation group in E^.
A particular example of an endomorphism q?: jO- —» JO- is given by
c^K) := |iA,p,[A: - 5(A:)]| + s{K)
for K € Ji}, where A],. .. , A„ so and pi,. .. , p„e 50B). One can
show that the general endomorphism is a limit of such rotation averages.
For a convenient formulation, let (ei, ej) be an orthonormal basis of E^
and write u{a) := (cosar)ei + (sina')e2 for a € [0,2n^). For K e JC^ v/e
then write h{K, a) instead of h{K, u{a)). The following was proved by
Schneider A974c).
Theorem 3.4.5. If cp is an endomorphism of jC^, then there exists a
positive measure fi on the Borel subsets of [0, In) such that
hicKK), a) = I h{K - s{K), a + p)MP) + <^(*^). «(^)) C-4.1)
for a € [0, 2;t) and all KeX'^.
Of course, any positive Borel measure (i on [0,2n^) defines an
endomorphism cp by means of C.4.1). If cp is given, the measure (i in
C.4.1) is unique up to the indefinite integral of (c, u{a)) with a
constant vector c. By subtracting a suitable measure of this kind from fi
we can obtain a signed measure Ji with
/in ,2ff
I cosard/i(ar) = 1, I %\naAp(a) = 0,
and then C.4.1) can be written in the more concise form
.21,
hicfiK), a) = \^ hiK,a+ p)dfi{P)-
This normalized measure ji is unique but not, in general, positive.
From Theorem 3.4.5 one can deduce (see also Schneider 1974a):
Corollary 3.4.6. The only surjective endomorphisms cp of y} are of the
form
(p{K) = Xp[K - siK)] + siK) for K e X^ C.4.2)
with fixed X>Oand pe 50B).
The endomorphisms described in this corollary can also be character-
zed in a different way. For endomorphisms <p, v we define

172 Minkowski addition
{<p + vKA:) ■■= (piK) + ^K) - s{K),
iXcp)iK) := XcpiK) + A - X)siK)
for /C € ;Z^ and A^O; then (p+ i/> and Aq? are endomorphisms. Now an
endomorphism cp may be called indecomposable if <p = <Pi + <P2 with
endomorphisms (pi, qsj implies that q?, = A,(p with A, SO (f = 1, 2). Then
Theorem 3.4.5 can be used to show that the indecomposable
endomorphisms of J^^ are precisely those of the form C.4.2).
In dimensions greater than two, no general description of
endomorphisms like the one given by Theorem 3.4.5 is known. It is,
however, possible to characterize special endomorphisms by means of
simple additional assumptions. The following results were obtained by
Schneider A974a). Their proof uses a combination of elementary facts
from harmonic analysis (for the rotation group SO{n) acting on the
sphere 5""') with convexity arguments. The first theorem restricts the
endomorphisms by imposing assumptions on the images.
Theorem 3.4.7.
(a) The only surjective endomorphisms (p of Ji." (n S 3) are given by
(p{K) = X[K - siK)] + siK) for K e Ji."
with A 4 0.
(b) Every endomorphism of 3L" is uniquely determined by the image of
one suitably chosen convex body, for example, a triangle with at
least one irrational angle.
(c) Let (p be an endomorphism of Ji.". If the image under (p of some
convex body of positive dimension is zero-dimensional, then
cpiK) = E(/C)} for K e JC". If the image under q> of some convex
body is a segment, then
<fiK) = k[K - s{K)\ ■¥ n{-K ■¥ s{K)] + s{K) for K e X"
with real numbers X, fi^O, X+ (i>0.
Another possible way of reducing the variety of endomorphisms
drastically is to replace the rigid motion equivariance by a stronger
assumption:
Theorem 3.4.8. Let q>:JCo^JC" be a Minkowski additive and
continuous map such that cp{aK) = aq>iK) for every non-singular affine
transformation a of E". Then
cpiK) = K + X[K - K] for K e X^,
where X^O is a real constant.

3.4 Additive maps 173
Observe that in this theorem the map cp need only be defined on the
set JC(, of convex bodies with interior points (otherwise the theorem
would be of little interest, cf. Schneider 1974a, pp. 76-7).
We turn now to a notion of additivity which is much less restrictive
than Minkowski additivity and which plays a role in later chapters, and
also in other contexts that are outside the scope of the present book.
A function (p defined on a family S of sets and with values in an
abelian group is called additive or a valuation if
<KA^ u L) + <ka: n L) = qiK) + qiL)
whenever K, L, K (J L, KULeS. If0€c$, we moreover assume
that <p@) = 0.
For functions on JC" this is, in fact, a weaker property than
Minkowski additivity:
Lemma3.4.9. Every Minkowski additive map on 7^" is additive.
Proof. Let K, L eJC" be convex bodies such that /C U L is convex.
Then C.1.4) states that
(a: u L) + (a: n L) = a: + L.
If now q? is a Minkowski additive map on Ji", then
<Ka: u L) + <ka: n L) = <ka:) + <KL),
which shows that cp is additive. ■
A stronger result (Theorem 3.4.13) will be proved below. Another
useful remark relating Minkowski addition and the valuation property is
the following lemma.
Lemma 3.4.10. Let cp be a valuation on X". If C eJ(" is a fixed convex
body and if
(pc{K) := cdK + C) for K e JC",
then (pc is a valuation on JC".
Proof Applying (p to C.1.1) and to C.1.3) and adding, we obtain the
assertion. ■
Additivity of a function on the space Ji" of convex bodies is an
important property in view of possible extensions of this function to
certain non-convex sets. Let q? be a valuation on a family S of sets. This
family is called intersectional if K,LeS implies Kr\LeS. US is
intersectional, we denote by U{S) the lattice consisting of all finite

174 Minkowski addition
unions of elements of S. It is a question of general interest whether a
valuation q> on an intersectional family cV can be extended, as a
valuation, to the lattice U{S). If this is the case (and if the extension is
denoted by the same symbol (p), then the inclusion-exclusion principle
(p(K,u...u Kj= i(-i)'-' [ S <KKi^ r^...r^ kA
/• = ! lii<-..<i, J
holds for Ki,. . , K„€ U{S). This follows by an obvious induction
argument, using the additivity property. To write this equality in a more
concise form, we let S{m) denote the set of nonempty subsets of
{1, . . . , m}, and for v e S{m) we write |y| = card v. U Ku . . . , K„ are
given, then we use the abbreviation
K„ ■= Ki^n .. n Ki^ for V = (/,, ...,/,}€ Sim).
The inclusion-exclusion principle now takes the form
(PiK^u ...u K„)= S (-i)|"|-Va:.). C.4.3)
veS(m)
In general, of course, this formula cannot be used to define an additive
extension of cp from <^ to Ui^), since the representation of an element
of U{S) in the form KiV . . . (J K„ with Kj e <S will not be unique. But
if such an extension exists, then C.4.3) gives its values, so that the
extension is unique.
Let us say that a function q) from an intersectional family S of
sets into an abelian group is fully additive if it satisfies the inclusion-
exclusion principle on S, that is, if C.4.3) holds whenever Ki,...,
K„ e S and K iV . . . V K„ € S. U <p is fully additive and it K e U{S)
has the representations K = K^yj . . .(J K„ and /C = L i U . . . U L;t
with Ki, Lj € S, then
S (-i)i"i-Va:.) = S (-!)'"'-'<?(U(^.nLy))
('eS(m) v€S(m) \ j=\ /
= S (-1)'"'-' S (-i)I"'I-Va:. nL„)
()e5(m) H'e5(*)
= S (-i)i"'i-vfU(^''^^-))
w€S(k) \i=l I
-=S(k)
= S (-i)W-^<K^J-
weS(k)
Hence, in this case cp can be unambiguously defined on the lattice U{S)
by means of C.4.3). This extension is additive on U{S), as can be
shown by some combinatorial argument (see Perles & Sallee 1970). We
do not prove this here, since no use will be made of it, but we formulate
it as a theorem.

3.4 Additive maps 175
Theorem 3.4.11. Let cp be a valuation on an intersectional family S of
sets. Then cp has an additive extension to the lattice U{S) if and only if cp
is fully additive. The extension is unique.
Of particular interest for us is the lattice U{X") generated by the
intersectional family X" U @}. Thus U{X") consists of all subsets of
E" that can be represented as finite unions of convex bodies. U{X")
was much studied by Hadwiger, who called it the 'Konvexring'. We shall
use the disputable translation convex ring. The following result is
fundamental in a study of extensions of valuations to the convex ring.
Theorem 3.4.12. There is a unique valuation x on the convex ring
U{IK"), which satisfies
X(K) = 1 for K eX". C.4.4)
That C.4.4) defines a valuation on X." is clear, since Kid Ki^ 0 ii
Ki, K2 € JC" and /Ci U /C2 ^ ^"- The uniqueness of an additive
extension to U{Ji.") follows from C.4.3). The existence is well known since
the Euler characteristic as defined, for example, in singular homology
theory, yields a valuation on U{X"). For this reason, the function % on
f/(Z") given by Theorem 3.4.12 is, of course, called the Euler
characteristic. We shall present Hadwiger's elementary existence proof
(Hadwiger 1955a; see also [18], p. 239).
Proof of Theorem 3.4.12. The proof proceeds by induction with respect
to the dimension. The existence for n = 0 being trivial, suppose that the
existence of x has been proved in dimension n - 1. Choose u e 5""'
and define
X(K) := 2[x(Kn //„,,) - limxiK U //„,^)] C.4.5)
For K € U{X"), where on the right-hand side x denotes the (unique)
Euler characteristic, which by the inductive assumption, exists in each
lyperplane //„,a (AeIR). Obviously, x(K) = 1 for KeJC". For
K = K^\J ...\J K„ with Ki e X" we have from C.4.3)
x(K n //„,,) = 2 (-i)^''^-'x(K„ n //„.,).
veS(m)
Jince X>~*x(Ku<^ Hu,>.) is the indicator function of a compact interval,
t is clear that in C.4.5> the limits exist and the sum is finite. It is also
:lear (using the inductive assumption) that x^ thus defined on U{X"), is
idditive. ■

176 Minkowski addition
In a similar way, for K e U{JC") and for x e E"\(o} we define,
following Mani A971),
h{K, X) := X^X(K n //,,,) - limxiK n H,,,)]. C.4.6)
For K € JC", the right-hand side evidently gives the value of the support
function ot K at ;«:, thus the definition is consistent. Since the additivity
of X carries over to h{-,x), we see that C.4.6) extends the map
K >-* h{K, •) as a valuation from JC" to UiJC"). Also for K e U{JC"),
we call h{K, •), as defined by C.4.6), the support function of K. It
should, however, be observed that it no longer determines K uniquely;
for instance, h{{o}, •) = 0 = /i@, •). Observe also that
h{K ■¥ t,x) = hiK,x) + x(K){t,x)
for teE".
We can now use formulae A.7.2) and A.7.3) to extend the mean
width and the Steiner point additively to U{JC"), that is, we define
biK) :=— \ h{K,u)dX"-\u),
siK) ••= — \ „.,hiK, u)udX"-\u)
for K € UiJC"). We still have 6(/C + t) = biK) for t e E", but
s{K + 0 = s{K) + xkK)t.
The additive extension of the functions b and s from JC" to U{JC")
was achieved here by means of explicit formulae. The mere existence of
such extensions would also follow from Theorem 3.4.11 and the
following result concerning arbitrary Minkowski additive functions.
Theorem 3.4.13. Every Minkowski additive function on JL" is fully
additive.
Proof By formula C.4.6), the support function was extended as a
valuation to U{JC"), hence it satisfies the inclusion-exclusion principle.
In particular, for K e JC" of the form K = K^V . . .U K„ with /C, € X"
(i = I, . . . , m) this implies that
hence
HK, ■)= S (-i)I''I-'Ma:.. •),
veS(m)
h{K, ■)-*- S h{K,, ■)= S h{K,, •)■
veS(m) veS(m)
|u|even |u|odd

Notes for Section 3.4 177
Since h{0, •) = 0 by C.4.6), it follows that
'^ + Zj ^v - Zj ^v<
veS(m) veS(m)
|u|even |«|odd
where the sums 2' extend only over those t; e S{m) for which /C„ 4 0-
If now q? is a Minkowski additive function on X", we apply it to both
sides of this equation. Defining q>{0) := 0, we obtain
cfiK)= 2 (-i)|''|-Va:.),
veS(m)
which shows that (p is fully additive. ■
Finally we remark that there is no point in studying Minkowski
additive maps on €" instead of JC". In fact, let q? be a Minkowski
additive function on 6". From Theorem 3.1.2 we know that
A + n conv A = conv A + n conv A
for i4 C E"; hence for A eS" and any Minkowski additive map q> on g"
we obtain
(p{A) + q>{nconvA) = q>{conv A) + q){n conv A)
and thus (f>iA) = q>{con\ A). Thus cp is completely known if we know its
restriction to JC".
Notes for Section 3.4
Characterizations of the Steiner point. For the early history of the Steiner
point we refer to the notes for Section 5.4. The problem of characterizing the
Steiner point by some of its properties was first posed by Griinbaum [371,
p. 239, who asked whether Minkowski additivity and similarity equivariance
are sufficient to characterize the Steiner point. Sallee A971) constructed an
example showing that one needs a continuity assumption. The example given
above after the proof of Theorem 3.4.2 is taken from Schneider A974a). By
the properties of Minkowski additivity, rigid motion equivariance and
continuity, the Steiner point was characterized by Shephard A968b) for n = 2
and by Schneider A971a) for mS3. Before that, W. Meyer A970a) had
proved a slightly weaker version of the characterization, by assuming uniform
continuity. Two earlier attempts (Schmitt 1968, Hadwiger 1969a) contained
errors. For dimensions two and three, interesting elementary proofe were
given by Hadwiger A971) and Berg A971).
Positsel'skii A973), whose proof we presented above in essence, assumed
equivariance under improper motions also; that proper motions suffice was
pointed out (for « S 3) by Saint Pierre A985).
Abstract Steiner points of polytopes. If P 6 'P" is a convex polytope, then its
Steiner point can be represented by
MP)
s{P) = 2 ft);'a5"-'(N(P, V,) n 5"-')i'„

178 Minkowski addition
where vi, ..., Of^^p) are the vertices of P. This is a special case of E.4.11).
Motivated by this, Berg A971) defined
MP)
s^P) ■■= S V<^(/'- V,) n 5"-')t;„
1=1
where i/« is a real-valued simple valuation on spherical polytopes (that is,
V(Gi '-' Q2) = HiQi) + HiQi) if Qi, Q2 are intersections of 5""' with convex
polyhedral cones having convex union and without common interior points).
It is not difficult to show that the map s^ is Minkowski linear, and that it is
translation equivariant if VK5""') = 1. Berg A971) calls a map ()p:'^"-»E" an
abstract Steiner point if it is Minkowski additive and equivariant under
(proper and improper) similarities. The definition of s,, above yields an
abstract Steiner point if V >s invariant under rotations and reflections and
satisfies ^E""') = 1- It is not known whether every abstract Steiner point is
of this form, but Berg A971) proved this for n = 2 and « = 3. As a
consequence, every abstract Steiner point on •^", for « = 2 or 3, that is
bounded on bounded sets of polytopes, is the usual Steiner point.
3. Additive maps preserving some functional. Further results in the style of
Theorems 3.4.7 and 3.4.8 can be obtained if one considers only endomorph-
isms preserving some functional such as volume or surface area. For example,
the following theorem was proved in Schneider A974a) (the quermassinte-
grals W/i are defined in Section 4.2 below).
Theorem. If (p is an endomorphism of X" satisfying Wk(q){K)) = W*(/l) for
some A: e {0,1, ...,«- 2} and all K e X", then
cp(K) = X[K - s(K)] + s(K) for K e X\
where Ae {1, -1}.
The following result of Schneider A974d) requires no continuity
assumption. Here V„ denotes the volume.
Theorem. Let «S2. If q>:X"^*X" is a Minkowski additive map satisfying
V„(q)(K)) = V„(K) for K e X", then there exists a volume-preserving affine
map *: E" -»E" such that (piK) is a translate of aK for each K e X".
4. Characterization of Hammer body maps. According to Theorem 3.4.8, every
Minkowski additive and continuous map (p: X" —» X" that commutes with
affine transformations is of the form cp{K) = K + XDK with a fixed A^o.
Thus <p associates with each K its outer Hammer body K{k + 1) (see Section
3.1, Note 13). The question arises of whether it is possible to characterize the
map that associates with each convex body K its inner Hammer body
K(X - A) (where Ae [0, l/«]) in a similar way. Such a characterization was
obtained by Valette A974). He proved (in a more general form): The family
of all maps (p: X" -* X" that are continuous, commute with affine
transformations and satisfy
<p{K + L)D <p(K) + (p(L) for K, L e X",
<p(P) C A - 2A)(P - p) + p,
where P is some parallelotope with centre p and where A e [0, l/n], has a
largest element (with respect to inclusion of images); this is precisely the map
given by q>iK) = K(l-k)= K~ XDK.
5. The Euler characteristic. The elementary proof given by Hadwiger A955a) for
the existence of the Euler characteristic on X" (Theorem 3.4.11) appears in
the literature in various forms: see Hadwiger A959, 1968a, 1969c) and
Hadwiger & Mani A972); see also Hadwiger A974) for an elementary

Notes for Section 3.4 179
treatment of the Euler characteristic for polygons in the plane. Applications
of the Euler characteristic on the convex ring to some questions of
combinatorial geometry were treated by Hadwiger A947, 1955a, 1968a) and Klee
A963b). Klee's paper put the Euler characteristic in a lattice-theoretic setting,
and the general treatment of the Euler characteristic in combinatorial and
algebraic terms was continued by Rota A971); see also Rota A964). Eckhoff
A980) solved a problem posed by Hadwiger & Mani A974), namely to
determine sharp lower and upper bounds for the Euler characteristic of a
union of k convex bodies in X".
The recursive definition C.4.5) of the Euler characteristic can also be used
for non-closed convex sets. In particular, let ^Ia denote the set of all relative
interiors of convex polytopes in E". Using C.4.5) for K e {/G?"o), one
obtains a valuation x on the lattice UifPI^) that evidently satisfies
^P) = (_l)dimp for Pe-p^V
This extended Euler characteristic was considered by Lenz A970) and
Groemer A972) and in special cases (and with a different sign for
odd-dimensional P) also by Hadwiger A969c, 1973). If one decomposes a compact
convex polytope P e fj)" into the relative interiors of its faces (Theorem
2.1.2) and applies the valuation x to this representation of P, one
immediately obtains the well-known Euler relation
t(-mr(p) = I,
where /,(P) is the number of r-faces of P.
6. Extensions of valuations. The first extension theorem of the type of Theorem
3.4.11, which is due to Perles & Sallee A970), was obtained by Volland
A957). He proved that every valuation on the class •^" of polytopes in E"
admits a unique additive extension to the class U('J)"). (Related but simpler
extension results can be found in Hadwiger [18], p. 81, and in B5hm &
Hertel A980), p. 47.) Volland's method can be extended to establish further
results. Let T)"^ be the set of all relatively open convex polytopes in E". A
function q) from S into some abelian group is called a weak valuation if either
S ="/)" and
<p(P) = <p(Pn H*) + <p(Pn H-) - <p(Pn H) for P e 'p"
or S = ■7?,"„ and
(p{P) = (fiPn H$) + (fiPn Ho) + (p{Pn H) for P e ^"„.
for any hyperplane //CE", where H*, H' and H^, Hij are respectively
the closed and open halfspaces bounded by H. Schneider A987c) proved:
Every weak valuation on either •''P" or ^)"„ can be uniquely extended to a
valuation on lliff)"^).
If q> is extended in this way, its value at a relatively open polytope is given
by
<p(relintP) = 2(-l)'''""'"''''"''<P(f)
F
for Pe^p", where the sum extends over all faces of P (including P). This
can be proved by induction with respect to dim P if one uses the Euler
relation for faces PC G of a polytope (see, e.g., Brondsted [11], p. 103). For
a given valuation (pon ''/?", Sallee A968) defined
g>*(P) := 2(-!)'"'"''<?<f) for P e ^i".
F
Hence, (p*(P) = (-l)<"'"''<)!<relint P). Using the additivity of <p on UifP';^) it is

180 Minkowski addition
easy to see that <p* is a valuation on '7^". Sallee proved this in a different
way.
The function <p on ^P" is said to satisfy an Euler-type relation if
<p*{P) = EqiriP) for all Pe^J)", where e, r;e{l,-l} are fixed. Shephard
A966a, 1968a) showed that the Steiner point and the mean width satisfy
Euler-type relations; see also Shephard A968c). A general result of this kind
is due to McMullen A975c, 1977), who proved that valuations satisfying
certain invariance and homogeneity properties automatically fulfil an Euler-
type relation. For further information, see Sallee A982) and §12 of McMullen
& Schneider [40].
Valuations and linear functionals. In studying extensions of valuations, the
following viewpoint can be useful. Let <$ be a nonempty class of subsets of
some nonempty set 5. For K eS, let K* be the characteristic function of
K*, that is.
K:*(r\ — fl foTx e K,
'^ ^-^^ •" lOforj*: e 5\A:.
Let V(S) denote the real vector space generated by the functions K*, K e S.
If cV is intersectional, then X* e V(S) if and only if X belongs to the ring
R(S) generated by S (this is the smallest subclass of 5 containing S that,
together with A and B, contains A U B and A\B). This result is due to
Groemer A987b).
If K,L,KU L,KnL€ S, then (A: U L)* + (K D L)* = K* + L*, thus
the map K >-* K* is a valuation. In particular, if S is intersectional, then it
follows from this or from Groemer's result that K* e V(S) for all K in the
lattice U(S),
Now suppose that ^ is a homomorphism from the additive group of V(<^)
into some abelian group. Defining q>(k) := ^K*) for those K C S tor which
a:* e V(S), we obtain
<p(K \j L) + (p(K n L) = ^(a: u L)*) + mK n z.)*)
= q>((K U L)* + (K n L)*) = ^K* + L*)
= q^K*) + ^L*) = (piK) + (piL),
provided that q> is defined in each case. Thus g> is a valuation. If S is
intersectional, then this procedure yields a valuation on U(S).
These general remarks can be used to extend the notion of an Euler
characteristic. First, Groemer A974) has proved an abstract theorem from
which results on Euler characteristics in special concrete situations can be
deduced. The class S of subsets of 5 is called separable if to any two disjoint
sets A,BeS there exists a pair X,YCS such that XDCeS and
YnCeS for every CeS, ACX, AnY = 0, BCY, BnX = 0,
JV U y = 5, and Z n A' + 0, Z n y =t= 0 for Z e <$ only if Z n ^ n y + 0.
Groemer proved the following:
Theorem. Let S be a separable intersectional class of subsets of S. There
exists exactly one linear functional x on the vector space V{S) such that
X(li*) = 1 for every nonempty set K of S.
Second, Hadwiger's recursive procedure for the introduction of the Euler
characteristic can also be modified to extend the Euler characteristic to a
linear functional on V(S), for suitable S. For instance, consider the class
rV = 'P"U {0} of (possibly empty) polytopes in E". We want to show the
existence of a linear functional x on Vi'p") such that x(P*) = 1 for P e ■J)".
For n = I, such a linear functional %i is evidently given by

Notes for Section 3.4 181
Uf) := S[/(A) - lim/(^)] for / e VG)')-
Let n ^ 2 and suppose that the existence of the required linear functional
Xn-\ in dimension n - \ has already been proved. Consider E""' as a linear
subspace of E" and let u e 5""'\E""'. For given / e V('p"), define
fix. A) := f(x + Am) for x e E""' and A e R.
One can proceed in either of two ways: (a) define the projection n\f of /
onto E"-' by
(T,/)(J^) := H/(^, •)) forJ.eE"-'
and then put ^„(/) := Xn-ii^xf)', or (b) define the projection uif of / on to R
^ (t2/)(A):=^„-,(/(-,A)) forAeR
and put Xn(f) '■= Xii^if)- In each case it is easy to see that Xn has the desired
properties.
A procedure equivalent to method (a) was employed by Hadwiger A960)
and later by Groemer A972), and a procedure equivalent to method (b) by
Lenz A970). Lenz further generalized this procedure and extended the Euler
characteristic, as a valuation, from X" to the system of all subsets of E".
Further investigations of an extended notion of the Euler characteristic in
certain situations are due to Groemer A973, 1975).
We turn now to more general valuations on S and their extension to linear
functional. Here we assume that q>is a function from S into some real vector
space ^. By a linear extension of q> to V(^) one understands a linear map q>
from V(S) into C(f for which qj{K*) = qj{K) for K e S. U q> admits such an
extension, it is clearly a valuation. The following was proved by Groemer
A978).
Theorem. Let q> be a function from the intersectional class S into a real
vector space so that q>{0) = 0. Then the following statements are equivalent.
(a) q) has an additive extension to U(S);
(b) q> has a linear extension to V(S)\
(c) aiKt+ ... + a^Kl = 0 with meN, K, e S, a, e R for / = 1,..., m
implies ai(p{Ki) + .. . + a„cp{K„) = 0.
Using this theorem, Groemer also proved an extension theorem for
valuations on JC". Let us say that the function (p from X" into a topological
(Hausdorff) vector space is a-continuous if for every decreasing sequence
{Ki)ieH in X" one has
\m<p{K,) = <p\r\K\.
Then Groemer A978) proved that every a-continuous valuation on X" (with
values in some topological vector space) has an additive extension to U^X").
It seems to be unknown whether every valuation on X" has an additive
extension to UiX").
A thorough study of the extension of the Euler characteristic, and of many
operations over convex sets to multilinear mappings on linear spaces by
replacing convex sets by their characteristic functions was made by Przeslaw-
ski A988).
Minkowski additivity and full additivity. Theorem 3.4.13 was first formulated
in McMullen & Schneider [40], Theorem E.20). It generalizes earlier results
of Sallee A966), Malii A971) and Spiegel A976b) referring to special cases.
For certain additive functions on 'J)" satisfying additional assumptions one
can deduce Minkowski additivity; see Spiegel A976a) and §10 of McMullen &
Schneider [40].

182 Minkowski addition
9. An identity for polytopes. An Euler-type relation (see Note 6) is satisfied, in
particular, by the support function. It P e^" and m e E", then
ti-iy S h(F,u) = -h(p,-u),
as proved by Shephard A968c). This can be written as an identity for
Minkowski sums, namely
2idimF 2>(limF
where the sums extend over all faces of P of respectively even and odd
dimensions (including 0 and n).
3.5. Zonoids and other classes of convex bodies
At the end of Section 3.3 we introduced the notion of an M-class of
convex bodies. In some sense the simplest non-trivial M-class is that
generated by the segments; it is also the smallest non-trivial affine-
invariant M-class. Its elements are called zonoids. Thus a zonoid in E"
is a convex body that can be approximated by finite sums of line
segments. From the viewpoint of Minkowski addition, the zonoids are
generated in a particularly simple way and deserve, therefore, special
attention. It turns out that zonoids appear in several different contexts.
Here we collect some basic remarks on zonoids, and we also consider a
more general concept.
A Minkowski sum of (finitely many) segments is called a zonotope.
Thus a zonotope in E" is a polytope of the form Z = Sy + .. . + Sk with
A: e f^ and segments Si, . . . , 5^. After applying suitable translations to
the 5, (which results in one translation applied to Z) we may assume
that each Sj has its centre at the origin; hence Z is centrally symmetric.
Let M e 5""'. By Theorem 1.7.5,
FiZ, u) = FiSi, u) + ... + FiSk, u),
hence each face of a zonotope is itself a zonotope and, in particular, is
centrally symmetric. This fact characterizes zonotopes:
Theorem 3.5.1, A convex polytope is a zonotope if and only if all its
two-dimensional faces are centrally symmetric.
Proof. Only the sufficiency has to be proved. Let P e ^P" be a polytope
of dimension at least two all of whose 2-faces are centrally symmetric.
Let S be an edge of P, and let u e S"~^ be orthogonal to 5. By /T we
denote the orthogonal projection from P on to a hyperplane orthogonal
to S. Let p be a vertex of FIFiP, u). In the polytope FIP we can find
an edge path (£i, . . . , E„) connecting the vertex 775 with p. The set
/T"^(£,) is a 2-face of P and hence centrally symmetric and has.

3.5 Zonoids and other classes of convex bodies 183
therefore, an edge 5' =t= 5 that is a translate of S. The projections of S
and S' under /T are the endpoints of f]. Applying the same argument
to £2. and so on, we deduce that i7~'(p) is a translate of S. Thus the
face F{P, u) contains a translate of S. Since u ± S was arbitrary, it
follows from Theorem 3.2.8 that 5 is a summand of P, hence
P = S+ P' with a polytope P'. Since F(P, u) = F(S, u) + F{P', u) for
Me5"~', it is clear that each 2-face of P' is centrally symmetric and
that P' has no edge parallel to S. Repeating the argument, we conclude
after finitely many steps that P is a sum of segments. ■
If P = S + P' with a segment 5, then each support set F(^P, u) with
u e 5""' orthogonal to S contains a translate of S. The union of all
these sets f (P, w) makes up a 'zone' in the boundary of P; this explains
the name 'zonotope'.
If Z = Si + . . . + Sk with Sj = conv{a,u,-, -a/U,} where u,- e 5""'
and a, > 0 for i = 1, .. . , A:, then the support function of the zonotope
Z is given by
KZ, -) = 2«i\<-''^i)l C-5.1)
1=1
Conversely, a convex body Z with such a support function is a zonotope
with centre at the origin.
By definition, a zonoid in E" is a convex body that can be
approximated, in the Hausdorff metric, by a sequence of zonotopes.
Thus each zonoid has a centre of symmetry. The representation C.5.1)
is generalized as follows. By a signed measure (a measure) on 5""' we
understand a real-valued a-additive (and non-negative) function on the
a-algebra 3E""') of Borel subsets of 5""'. A signed measure on 5""'
and a function on 5""' are called even if they are invariant under
reflection in the origin.
Theorem 3.5.2. A convex body K e X" is a zonoid with centre at o if
and only if its support function can be represented in the form
h{K, X) = \^J{x, y)| dp(i;) for x e E" C.5,2)
with some even measure p on S'
n-l
Proof Suppose that C.5.2) holds. For k eN we decompose S"'^ into
finitely many nonempty Borel sets 4*,. .. , Ai^^k) of diameter less than
l/k and choose vf e Af. Then
lim S \{x, oh\p(M) = ij{x, o)\dp(v),

184 Minkowski addition
uniformly for x e S"~^. Hence, the zonotopes Z/, defined by
m(k)
i=l
satisfy h{Zk, •)-* H^y ') and thus Z^-* K for A:-* oo. By definition,
K is a zonoid.
For the converse, observe that C.5.1) can also be written in the form
C.5.2), with an even measure p concentrated in finitely many points.
Hence, we may assume that
h{Z,, ■)- lj{-,o)\dp,iu) C.5.3)
with an even measure pk on 5""' (A: e N) and that Zk-* K for A:-* <».
Integrating C.5.3) over 5""' and using Fubini's theorem, we obtain
l„.MZk, u)d^"-\u) = c(n)p,E"-').
Since the left-hand side converges for k-^<x>, the sequence
{Pk{S"~^))kef<j is bounded. In the space of signed measures on 5""' with
the total variation norm, which can be identified with the dual space of
C{S"~^), bounded sets are relatively weak*-compact. Hence, some
subsequence (Pk)ieN is weak*-convergent to a signed measure p,
necessarily an even measure, and we get
hiK, x) = Mm h{Zk,, x) = lim \Ux, v}\dpk.{v)
= \^J{x,u}\<ip{u)
for each x e E".
If the support function of a zonoid K with centre at the origin is
represented in the form C.5.2) with an even measure p, we say that p is
the generating measure of K. The next theorem shows that in fact this
generating measure is unique. It also shows that a representation of the
form C.5.2) with a signed measure p is always possible if K is centrally
symmetric with respect to o and has a support function that is
sufficiently often differentiable.
Theorem3.5.3. If p is an even signed measure on S"~^ with
J^„_,Km, i;)|dp<i)) = 0 forueS"-' C.5.4)
then p = 0. If G is an even real function on 5""' of differentiability class
C*, where k^ n + 2 is even, there exists an even continuous function g
on 5""' such that

3.S Zonoids and other classes of convex bodies 18S
G(") = L,-,K"' v)\giv)d7r."-\u) C.5.5)
for Me5""'.
Proof. We use spherical harmonics and refer to the Appendix for some
of the details. In the following, all integrations without specified domain
are over 5""'. Let Y„ by a spherical harmonic of order m on S"~^. The
Funk-Hecke theorem yields
j\{u, o)\Y„io)d'?()"-\o) = A„y„(M) C.5.6)
with
(_'t\m (n-\)/2 f^ /d\'"
'" 2'"-'r(/w + (n - l)/2) J-i' ' [dtj ^ ^
= -^^-r ; ^ r- for even m C.5.7)
2'"-2r(/w/2)r((/w + n + l)/2) ^ '
and A„ = 0 for odd m. If now C.5.4) holds, we have
= k„JY„(^v)dp{v)
and hence
jy„(i;)dp(i;) = 0.
For even m this holds since X„ =t= 0, and for odd m it holds since Y„ is
odd and p is even. From the completeness of the system of spherical
harmonics it now follows that p = 0.
Now let G satisfy the assumptions of the theorem. To solve the
integral equation C.5.5), we develop G into a series of spherical
harmonics:
G~- 2y„ with Y„ := 7T„G.
m=0
If the series
8(.v) := S ^m'Y„iv) C,5.8)
m=0
m even
converges uniformly on S"~^, the continuous function g solves C.5.5),
by C.5.6). For the uniform convergence of C.5.8), it is sufficient to
prove that

186
Minkowski addition
||A;'y„|| = Oim-'). C.5.9)
Now from C.5-7) we get, using StirHng's formula, that
\X-J\ = 0(/w<"+2)/2) C.5.10)
for even m-^^. By Appendix equation (A.12), to which we refer for
the notation.
\Y I S
I ■• mil =
CO„
1/2
mll2'
From Appendix equation (A.l),
Oi„
1/2
= 0(//l("-2)/2).
C.5.11)
C.5.12)
It remains to estimate ||ym||2- Let {Ym,)^^"''"^ be an orthonormal basis
of S"" (m e f^o). and put a„j := (G, Y„j), so that
N(n,m)
IV l|2
I^mll2
N(n.m)
= (G, Y„) = 2 «m; =: «m-
/=1
Using ^^Y„ =-mim +n-2)Y„ and applying Green's formula
(/, Asg) = (As/, g) on the sphere k/2 times, we get
ai = (G, Y„) =
-1
m(m + n — 2)
k/2
(A|/'G, y„)
for m>0. Writing
b„i := (a|/'G, y„y),
N(n,m)
fei:= E bi,-=\\n„M^G\\i
we deduce, using the Cauchy-Schwarz inequality, that
flm = m
By the Parseval relation.
Sfei = ||As*^G||^<-,
m=0
hence |a„| = o(/«"*). This, together with C.5.10), C.5.11) and C.5.12)
yields C.5.9), which completes the proof. ■
Remark3.5.4. For n =2, the solution of the integral equation C.5.5)
can easily be given explicitly. It is convenient to write the argument of
G as an angle. If G: [0,2n-] -» R is a periodic function of class C^, then

3.5 Zonoids and other classes of convex bodies 187
Ci<p) = \ j/|cos(<p - v)| g|v - y) + G"(v - y)]dV,
as one may check by partial integration.
The existence part of Theorem 3.5.3 is a motivation for considering
also convex bodies K eJC" whose support function can be represented
in the form
h(^K,u) = j^J{u,o)\dpiv)
with an even signed measure p. Such a body K and each of its translates
is called a generalized zonoid. First, information on these bodies is
obtained from the following lemma, showing that each support set of a
generalized zonoid is itself a generalized zonoid.
For e e 5""' we write
Qe ■■= {v e S"-^\{e,v) > 0},
0),:= {v e S''-'\{e, o) =0},
and for K eJC" v/e abbreviate the support set F(K,e) by K^.
Lemma3.5.5. If the support function of the convex body K e DC" is
given by
hiK, u) = j^Jiu, o}\dp(v) for uel"
with an even signed measure p, then, for e e 5""',
hiK„ u) = f Km, v)\dp(v) + {ve, u) for u e E",
where
Ve = 2 J^udp(u).
Proof Theorem 1.7.2 tells us that
h{K,, u) = h'^ie; u)
= \imX-^[hiK, e + Xu) - h^K, e)].
The assertion of the lemma is true for u = ±e, because h(^Ke, ±e) =
±h{K, e); hence we may assume that u and e are linearly independent.
Put
A
B
C
= {ue 5"-'Ke, u) > 0, <e + Am, u) > 0},
= {u e 5"-'Ke, u)^0, <e + Am, u) > 0},
= {u e 5"-'Ke, v) > 0, <e + Am, v) ^ 0}.

188
Minkowski addition
Then Qe+Xu = A U B and Qg = A U C, and we obtain
hiKe, u) = limA-' [£„.J<e + Am, u)\Mo) - lj\(e, o)\Mo)
= 21imA-' |j^^^<e + Am, v) Mo) - j^^.^^e, v) df^v)
= 21imA-' U{e. u) dp{u) - j^{e, o) Mio)
where all limits refer to AjO. For u e fi we have |(ee,u)| ^cA with a
constant c independent of A. Writing
B' := {v e 5"-'Ke, u) < 0, <e + Am, v) :> 0},
we obtain
A-'j^<e, i))dp(i;) = |a-i j^,<e. t'>dp(o)
^ c|p|(fi').
where |p| denotes the total variation measure of ip. Since (in the
set-theoretic sense) lim fi' = 0, we have
lim|p|(fi') = 0,
hence
lim A"'I {e, v)dp(v) = 0.
J B
From lim C = 0 we similarly find
limA-^J (e, v) dp{v) = 0.
Further, lim A = Qg and
limfi = D := {v e S"-^\{e, v) = 0, <m, v>) > 0}.
This yields
h(Ke, M) = 2J^<M, U}dp{u) + 2J^<M, U}.dp(v)
= 2/m, j„i'dp(i))\ + £ Km, i))|<dp(i;),
which completes the proof of the lemma. ■
Corollary 3.5.6. Each support set of a generalized zzonoid is itself a
generalized zonoid and is, in particular, centrally symmetric. Each

3.5 Zonoids and other classes of convex bodies 189
support set of a zonoid is a summand of the zonoid. A polytope is a
generalized zonoid only if it is a zonotope. In the set of centrally
symmetric convex bodies in E", the class of generalized zonoids is dense
but not closed, and the class of zonoids is closed and nowhere dense.
Proof If
h(^K,u) = j^J{u,v)\<ip{o)
with an even signed measure p, then Lemma 3.5.5 shows that
h(Ke - Ve, U) = J^„.,Kw, v)\dpeiv),
where Pe is the restriction of p to 0)^; hence K^ is a generalized zonoid.
If p is a measure, then
hiKe - Ve, U) + j^Jiu, U}\dip - PeXv) = h(^K, u).
Since p- p^^O, the integral defines a support function, hence Kg is a
summand of K. If the polytope P is a generalized zonoid, then each of
its faces is centrally symmetric, hence P is a zonotope by Theorem
3.5.1. That the class of generalized zonoids is dense in the class of
centrally symmetric convex bodies follows from Theorem 3.5.3, since
the set of convex bodies with support function of class C°° is dense by
Theorem 3.3.1. A convex body with a non-symmetric support set cannot
be a generalized zonoid; hence the set of generalized zonoids is not
closed, and the set of zonoids, which is closed by definition, is nowhere
dense in the set of centrally symmetric convex bodies. ■
Generalized zonoids were obtained by extending the integral
representation C.5.2) from measures to signed measures. The fact that the
generalized zonoids are dense in the class of centrally symmetric convex
bodies but do not exhaust this class is a reason for a further extension,
namely to distributions. It was shown by Weil A976b) that for every
convex body K e JC" with centre at the origin there exists an even
distribution T^ on S"~^, whose domain can be extended to include the
functions Km, •)!, we 5""', such that Tk(\{u, ■){) = h(K,u). The
properties of the correspondence K<-^Tn and some applications were
studied further by Weil A976b, 1979a). We shall not pursue this matter
here.
We conclude this* section with a brief look at a different type of
A/-class. An M-class that has been thoroughly studied is the class of
bodies of constant width. These bodies are, in a certain sense, a

190 Minkowski addition
counterpart to the centrally symmetric convex bodies: A convex body is
centrally symmetric if and only if its support function is essentially (that
is. up to a linear function) an even function, and a body is of constant
width if and only if its support function is essentially (meaning, here, up
to the support function of a ball) an odd function. For the various
aspects from which the bodies of constant width have been studied, we
refer the reader to the comprehensive survey article of Chakerian &
Groemer [31].
There is a special way of looking at bodies of constant width that
readily suggests a generalization. A body of constant width can be freely
turned inside a suitable cube while always touching the facets of the
cube. More generally, let PCE" be a polyhedral set, that is, an
n-dimensional intersection of finitely many closed halfspaces. A convex
body K e X"\ is said to be a rotor of P if for each rotation p of E" there
exists a translation vector f e E" such that pK + t is contained in P and
touches each facet of P. Clearly the set of all rotors of a given
polyhedral set is a motion invariant M-class. If it is not empty, then it
contains a ball, by Theorem 3.3.2. Thus P can have a rotor only if it has
an inscribed ball. If the exterior normal vectors of the facets of P are
linearly independent, then obviously every convex body is a rotor of P.
Leaving this trivial case aside, we ask for the polyhedral sets admitting
non-spherical rotors. A complete classification is possible. Of course, if
/l is a rotor of the polyhedral set
F= [\H-{P,uX
where the u, for i e / (a finite index set) are the exterior normal vectors
of the facets of P, then K is also a rotor of
P' := f]H-(P,ud
whenever /' is a subset of /. Such a polyhedral set P' is said to be
derived from P. The following theorem, which we quote without proof,
lists all the non-trivial pairs of polyhedral sets and corresponding rotors.
In the formulation of this theorem, support functions are restricted to
cn-l
Theorem3.5.7. Let F C E" be a polyhedral set whose facets have linearly
dependent normal vectors. Suppose that K is a non-spherical rotor of P.
Then one of the following assertions holds.
(a) P is derived from a parallelepiped with equal distances between
parallel facets; K is a body of constant width.

Notes for Section 3.5 191
(b) n = 2; P is derived from a regular k-gon with A: e {3, 4, . . .}; the
Fourier coefficients a,, 6, of the support function of K are zero
except for i = 0 and / = ± 1 (mod k).
(c) n = 3; P is a regular tetrahedron; the support function of K is a
sum of spherical harmonics of orders 0, 1, 2, 5.
(d) n = 3; P is derived from a regular octahedron, but is not a
tetrahedron; the support function of K is a sum of spherical
harmonics of orders 0, 1, 5.
(e) n = 3; P is congruent to the cone
{xe E3Kx,e,)^0 for i = 1,2,3,4},
Wi
here
e\
ex =
-^ (V6, 0, 1), ^2 = ^ (-V6, 0, 1),
^ @, V6, 1), e4 = T^@,-V6, 1)
with respect to some orthonormal basis; the support function of K is
a sum of spherical harmonics of orders 0, 1,4.
(f) n^A\Pisa regular simplex; the support function of K is a sum of
spherical harmonics of orders 0, 1, 2.
Vice versa, each of the cases listed here does really occur, so that no
further reduction is possible. We wish to point out that cases (c) to (f),
unlike (a) and (b), describe M-classes that depend only on finitely many
real parameters.
References are given in Note 8 below.
Notes for Section 3.5
I. Zonotopes and generalizations. The characterization of zonotopes contained
in Theorem 3.5.1 must have been folklore for a long time, at least for « = 3;
compare Blaschke A923), p. 250, and Coxeter A963), Section 2.8. Proofs for
^-dimensional space were given by Bolker A969) and Schneider A970a).
Some results have been proved for polytopes with centrally symmetric
faces of dimension greater than two. Aleksandrov A933) (see also Burckhardt
1940) proved that a polytope of dimension at least three all of whose facets
have a centre of symmetry is itself centrally symmetric. As a generalization of
the zonotopes, one may consider the class ^Pl of «-polytopes all of whose
^-faces are centrally symmetric, where A: e {2, ..., n). Aleksandrov's
theorem implies that "PI c 7>t+i c ... c 7);; for k^2\ see also Shephard
A967) for further results and McMullen A976b). McMullen A970) showed
that 'PS-2 + 'Vl-x for « g 4 and proved that 'T^J for 2 S A: g « - 2 is equal to
the class of ^-dimensional zonotopes.

192 Minkowski addition
Various combinatorial and geometric results on zonotopes have been
obtained by McMullen A971) and Shephard A974a); for a survey and further
references see McMullen A979), §7.
Zonotopes are also related to space-filling polytopes, that is, polytopes that
tile space by translations. Every space-filling polytope in E' is a zonotope,
with the additional property that every zone contains four or six facets.
Coxeter A962) used this fact for a simple classification of the space-filling
polytopes in E^. In higher dimensions, the space-filling zonotopes were
investigated by Shephard A974b) and McMullen A975b).
Hadwiger A952) proved for « = 3 and Miimer A975) proved for general n
that every space-filling «-polytope is T-equidecomposable (in the sense of
scissors congruence) to a cube, where T denotes the group of translations. In
three-space, this property leads to a characterization of zonotopes, and, more
generally: An «-polytope is T-equidecomposable to a cube if and only if it
belongs to the class ^l-i defined above. This was conjectured by Hadwiger
at\d proved by Miimer A977).
For lattice zonotopes in E" (i.e., zonotopes whose vertices belong to the
integer lattice 2"), Betke & Gritzmann A986) have proved two interesting
inequalities of discrete geometry, which for general polytopes are respectively
not true and unknown.
Several aspects of the geometry of zonotopes are discussed in Martini
A987). Some extremum problems for zonotopes were treated by Linhart
A986, 1987) and Filliman A988).
2. The sum of a zonotope and an ellipsoid. For a convex body /l C E", consider
the following property: (A) For each u e 5""', there exists e(u) > 0 such that
the section [H(K,u)- au]r) K is centrally symmetric for 0<ar<f(M).
Generalizing a classical characterization of the ellipsoid. Burton A976)
proved the following remarkable result: An M-dimensional convex body K
(n ^ 3) has property (A) if and only if K is the Minkowski sum of a (not
necessarily ^-dimensional) zonotope and an (^-dimensional) ellipsoid.
3. The integral equation for zonoids. Proofs of Theorem 3.5.2 may be found in
Bolker A969), p. 336, Schneider A970a), Lindquist A975a) and Matheron
A975), p. 94.
Theorem 3.5.3 goes back (at least) to Blaschke [5]. For the uniqueness
part, two proofs for « = 3 and special measures appear on pp. 152 and 154-5
of that book. The general uniqueness theorem was first proved by Aleksan-
drov A937b), §8, and rediscovered by Petty (l%la), Rickert (l%7a,b) and
Matheron A974a) (reproduced in Matheron 1975, Section 4.5). Aleksandrov,
Petty and Rickert use spherical harmonics for the uniqueness proof, while
Matheron's proof is motivated by probability theory. Clearly the uniqueness
assertion of Theorem 3.5.3 is equivalent to the fact that the real vector space
spanned by the functions M>-»|(M,t;)|, veS"~^, is dense in the space
^■,E""') of even continuous real functions on 5""' with the maximum norm.
A simple elegant proof of this fact was given by Choquet A969b, p. 53;
1969c, p. 171).
Matheron A975), p. 73, had conjectured a generalization of the uniqueness
assertion of Theorem 3.5.3, with the sphere replaced by Grassmannians. This
was disproved by Goodey & Howard A990).
The fact that the integral equation
<^(") = L„-J<"' o)\g(o)d^.'-'(o), C.5.13)
where G is a given even function on S"~', can be solved by means of
spherical harmonics was mentioned by Blaschke [5], p. 154 (for « = 3 and

Notes for Section 3.5 193
without mentioning the necessary smoothness assumptions). The general
argument given above was carried out in Schneider A967c). Geometric
applications of a solution of C.5.13) appear in Chakerian A967b), Firey
A967a), Petty A967) and Schneider A967c). Schneider A975c) proved a
(weak) stability result for the solutions of C.5.13), namely ||g||S||G||2t, where
||-|| denotes the maximum norm and ||-||2t is a certain norm involving
derivatives up to order 2k. This was used to prove the existence of zonoids,
other than ellipsoids, whose polar bodies (with respect to the centre) are also
zonoids.
For special functions G, an explicit solution of the integral equation
C.5.13) may be found. In this way (and using the uniqueness result) one can
obtain explicit examples of centrally symmetric convex bodies of class C+ that
are not zonoids, for instance the body K e X^ with support function given by
h(K, m) = 1 + aPzde, u)) for u e 5^
where e e S^ is fixed, ^2@ = Cf^ - l)/2 denotes the second Legendre
polynomial and *e [-2/5,-1/4] is a real constant (see Schneider 1970a,
p. 69).
Approximation problems for zonoids. By definition, a zonoid can be
approximated arbitrarily closely by finite sums of segments. Interesting problems
arise if one asks for the minimal number of segments necessary to reach a
given degree of approximation. For a zonoid Z in E" (with centre at the
origin) and for f > 0 (sufficiently small, in dependence on the cases
considered below) let N(Z,e) be the smallest number N such that there is a
sum P^f of N segments satisfying
Z c P,v C A + e)Z.
Improving a slightly weaker result of Figiel, Lindenstrauss & Milman A977),
Gordon A985) showed for the ball B" that
N(B\ e) g ce-^n,
where c is an absolute constant. The remarkable fact here is the linear
dependence of the upper bound on n. For general zonoids Z, Bourgain,
Lindenstrauss & Milman A989, announced 1986) obtained results that are
only slightly weaker. For r > 0 they showed that
N(Z, e) S c(r)f-B+')(log«)'«
with a constant c(r) depending only on r, and
N(Z, e) S c(r, 6)f-<2+'>«
if the zonoid Z is the unit ball of a uniformly convex norm, 6 being the
degree of uniform convexity. For given n, the above bounds are not optimal
in e, but here for the ball sharper estimates are known. There are constants
Ci(n), depending only on n, such that
ci(«)f-«"-')/<"*^' S N(B\ e) S C2(«)(f-^|logf|)<"-'W"*2>.
The left-hand inequality is due to Bourgain, Lindenstrauss & Milman A989),
Theorem 6.5, and the right-hand one to Bourgain & Lindenstrauss A988c).
The latter authors also have similar, but in general weaker, results for
arbitrary zonoids. Weaker inequalities of the above type were obtained
earlier by Betke & McMullen A983) and Linhart A989).
Characterizations of zonoids. Theorem 3.5.1 characterizes zonotopes in a very
simple and intuitive way. An equally simple characterization of zonoids
apparently does not'exist. In particular, it is not possible to characterize
zonoids by a strictly local criterion. The question of such a characterization
was posed repeatedly; see Blaschke A923), p. 250, Blaschke & Reidemeister

194 Minkowski addition
A922), pp. 81-2 and Bolker A971). However, the following was shown by
Weil A977). There exists a convex body K e X" (« S 3), arbitrarily smooth,
that is not a zonoid but has the following property: For each u e 5""' there
exists a zonoid Z with centre at the origin and a neighbourhood [/ of m in
5""' such that the boundaries of K and Z coincide at all points where the
exterior unit normal vectors belong to U. Thus, no characterization of
zonoids is possible that involves only arbitrarily small neighbourhoods of
boundary points. Instead, Weil A977) proposed the following conjecture: Let
K e X" be a convex body such that to any great sphere aC 5""' there exists
a zonoid Z and a neighbourhood U ot a in 5""' such that the boundaries of
K and Z coincide at all points where the exterior unit vector belongs to U;
then K is a zonoid. Affirmative answers for even dimensions were given
independently by Panina A988, 1989a,b) and Goodey & Weil A990+).
It is also not possible to characterize zonoids by means of their projections:
Weil A982c) constructed a convex body K e X" (n S 3) that is not a zonoid
but has the property that all its projections onto hyperplanes are zonoids.
An approach to characterizations of zonoids by means of a different
integral representation was proposed by Ambartzumian A987).
Non-trivial characterizations of zonoids are possible if systems of
inequalities satisfied by support functions are taken into account. Some definitions
are needed. A function /: E" —»IR is called positive definite if
2/(x,-x>,*,gO
holds for all A: e N, Xj,.. . , jt^ e E". orj,..., or^ e R, and it is called
conditionally positive definite if these inequalities are only assumed for X*/ = 0-
The function / is of negative type if for every t>0 the function e"'^ is
positive definite. A real normed vector space (V, || • ||) is called hypermetric if
k
for all ^ e N, jTi,..., jT/i e V and all integers a^,..., au satisfying 5)*, = 1-
With these definitions, the following properties of an ^-dimensional convex
body Z e 7^" with centre at the origin are equivalent:
(a) Z is a zonoid;
(b) /i(Z, •) is of negative type;
(c) e"'"''^'* is positive definite;
(d) E" with norm || • ||z = /i( Z, •) is hypermetric;
(e) —h(Z, ■) is conditionally positive definite.
Here the equivalence of (a) and (b) is a classical theorem due to Levy A937),
§63; see also Choquet A969c), p. 173. The equivalence of (b) and (c) is
trivial; for the equivalence of (c), (d) and (e) see Witsenhausen A973).
Witsenhausen A978) and Assouad A980) also obtained characterizations of
zonotopes among the polytopes by means of norm inequalities.
6. Relations of zonoids to other fields. 2Lonoids appear, and are useful, in
various contexts. We mention briefly some of their occurrences; for more
information we refer to the survey articles of Bolker A969) and Schneider &
Weil [44]; we have already quoted from the latter above.
A well-known theorem of Liapounoff A940) says that the range of a
non-atomic E"-valued measure is compact and convex; shorter proofs are due
to Halmos A948), Blackwell A951), Lindenstrauss A966) and others. It is not
difficult to show that the convex bodies occurring as such ranges are precisely
the zonoids; see Rickert A967b) and Bolker A969).
In Banach space theory, zonoids occur as follows. Let Z C E" be a convex
body with o as centre and interior point. Then ||-||2 = ;i(Z, •) defines a

Notes for Section 3.5 195
norm on E", for which the polar body Z* is the unit ball. The body Z is a
zonoid if and only if the space E" with norm || • \\z is isometric to a subspace
of Z-i = Z,i([0,1]) (a proof and further discussion can be found in Bolker
1969). Thus, questions about the isometric embedding of finite-dimensional
Banach spaces in Z-i reduce to problems on zonoids. For instance, Bolker
A969) pointed out that the space Ip (which is R" endowed with the norm
defined by ||(|i,.. -, f„)|| = B,"=il§/|'')'/'') embeds in Lj isometrically if
1 = p = 2, and he conjectured that this is not the case if n = 3 and p>2.
This was proved for p > 2.7 by Witsenhausen A973) and for all p > 2 by Dor
A976). Bolker A969, 1971) also formulated the following question: If E is an
n-dimensional Banach space (n = 3) such that E and its dual space both
embed isometrically in Z-i, is E isometric to /J'? The result of Schneider
A975c) mentioned in Note 3 provides counterexamples.
In stochastic geometry, zonoids have proved useful as auxiliary bodies. If p
is an even measure on 5""', which can be, e.g., the orientation distribution of
a random hyperplane or a measure associated in some way with a stochastic
process of geometric objects, one defines an associated zonoid K by means of
equation C.5.2). Some parameters describing the stochastic situation can then
be expressed in terms of functional of /l, so that, for instance, inequalities
for convex bodies can be used to solve extremal problems of stochastic
geometry. This method was initiated by Matheron A974b, 1975), who called
the associated zonoid a 'Steiner compact' (for reasons that are not entirely
convincing). Later applications were given by Schneider A982a,b, 1987a),
Thomas A984), Wieacker A984, 1986, 1989) and Weil A988).
Alexander A988) made an interesting contribution to Hilbert's fourth
problem, using zonoids and generalized zonoids. In particular, the
equivalence of (a) and (d) in Note 5 plays an essential role in his argument.
An application of zonoids to mathematical economics may be found in
Hildenbrand A981).
Generalized zonoids. Lemma 3.5.5 is taken from Schneider A970a).
The existence of generalized zonoids that are not zonoids motivates two
questions: to characterize the even signed measure p on 5""' for which
h(K, ■):= j^J{-,v)\dpiv) C.5.14)
yields a support function, and to characterize the convex bodies K obtainable
in this way. Aiming to answer these questions, Weil A976a) developed
formulae for mixed area measures of generalized zonoids (Section 5.3). With
their aid and using results from Weil A974a) he obtained the following result.
For M],..., Uj e 5""' let £>,(mi, ..., My) denote the y-dimensional volume of
the parallelepiped spanned by ui, ...,Uj. Further, define a partial map
T: E""')""' ^5"-' by the normalized vector product. Then C.5.14) defines
a support function if and only if
il)^"-
d(p' X &>"-/-•)
go
for /=1,...,«-1 (where (o denotes the normalized rotation invariant
measure on 5""'). A different characterization of the signed measures p that
generate convex bodies by means of C.5.14) can be obtained by considering
projections. For a signed measure p on 5""' and for a y-dimensional great
subsphere a, of 5""S Weil A982c) defined a projection of p onto ay such
that, for any generalized zonoid K with generating measure p and any
(j + l)-dimensional linear subspace £y+i of E", the orthogonal projection of
K onto £y+i has the projection of p on £y+, n 5""' as its generating measure.

196 Minkowski addition
He deduced, for instance, that p generates a convex body if and only if all its
projections onto great circles are non-negative. Another application
generalizes a convexity criterion of Lindquist A975b).
Several characterizations of zonoids, generalized zonoids and centrally
symmetric convex bodies, expressed in terms of inequalities involving mixed
volumes, were investigated by Weil A976b, 1979a) and Goodey A977,
1984a,b) (some of the assertions in Goodey A984a,b) have to be modified;
see the conections in Goodey & Weil A991)). See also Section 5.1, Note 9.
Rotors. Special rotors in regular polygons were considered, often in
connection with kinematic considerations, by Reuleaux A875), Meissner A909),
Wunderlich A939), Goldberg A948, 1957), Schaal A963) and others. All the
rotors of the regular polygons were determined by Meissner A909). Case (b)
of Theorem 3.5.7 is due to Fujiwara A915); simpler proofs were given by
Kameneckii A947) and Schaal A962).
Meissner A918) determined the rotors of the three-dimensional regular
polyhedra. Models of such rotors are shown in the survey article by Goldberg
A960), who also discussed the problem of the existence of rotors in higher
dimensions and for non-regular polytopes. The complete classification given
in Theorem 3.5.7 is due to Schneider A971b).
Since the class of rotors of a regular ^-gon depends on infinitely many
parameters, it is sufficiently interesting to be the subject of extremal
problems. In particular, one may ask which rotors of a given regular ^-gon have
the least area. The case ^ = 4, where the Reuleaux triangle is the solution,
was solved independently by Lebesgue A914) and Blaschke A915). The
survey [31] (p. 67) lists several later proofs, of which the one by Chakerian
A966) is especially elegant. The case ^ = 3 was settled by Fujiwara & Kakeya
A917); alternative proofs were given by Weissbach A972, 1977a), see also
Jaglom & Boltjanski [20]. For arbitrary k^'i, the problem was treated by
Focke A969a) under special symmetry assumptions and solved completely by
Klotzler A975).
Some other extremal problems for rotors of regular polygons were
investigated by Fujiwara & Kakeya A917), Fujiwara A919), Schaal A963),
Focke A969b), Focke & Gensel A971) and Weissbach A977b).
Other M-classes in the plane, more general than rotors, are made up by
the so-called i7ji-curves, for which all circumscribed equiangular ^-gons have
the same perimeter, or by the Pyt-curves, for which all circumscribed
equiangular fc-gons are regular. These classes have been studied by Meissner
A909), Jaglom & Boltjanski [20], Goldberg A966) and Weissbach A977a).

Curvature measures and
quermassintegrals
4.1. Local parallel sets
For a convex body K e X" and a number p>0, the Minkowski sum
K + pB" is called the outer parallel body of K at distance p. According
to the well-known Steiner formula (to be proved in Section 4.2), its
volume can be expressed as a polynomial of degree at most n in the
parameter p:
V„(K + pB") = ip'|"Jw,.(A:). D.1.1)
The coefficients defined by this expansion turn out to be important
quantities, carrying basic geometric information on the body K.
Formula D.1.1) is only a very special case of a similar expansion valid for a
general Minkowski sum A,/l, -f ... + X„K„; this leads to the notion of
mixed volumes and will be extensively studied in later chapters. The
particular case of adding a ball deserves a separate study, as its strong
dependence on the Euclidean metric results in a theory of different
character and leads to a number of notions and results related to the
Euclidean geometry of convex bodies.
The Minkowski sum K + pB" is at the same time the set of all points
of E" having distance d(K, •) from K at most p. This point of view
leads naturally to the more general notion of a local parallel set: Given
a subset /3C /C, we may consider the set of all points x e E" for which
d{K,x) = p and for which the nearest point p^K, x) belongs to /3.
Alternatively, we may prescribe a set /3c 5""' of unit vectors and then
consider the set of all x eU for which d(K, x) = p and for which the
unit vector u(K, x) pointing from p(^K, x) to x (see Section 1.2) belongs
to /3. Restricting /3 to Borel sets, we find that the Lebesgue measure of
the local parallel set thus defined is again a polynomial in the parameter
197

198 Curvature measures and quermassintegrals
p. The coefficients now depend on K and /3, and as functions of the
latter argument they are measures. These are the curvature or area
measures studied in the present chapter. Their basic properties will be
derived from corresponding properties of the local parallel sets as
established in this section. Instead of treating the two types of local
parallel sets separately, we consider a common generalization.
We need some more notation. The product space E" x 5""' is
denoted by 1. \i K & JC", & pair (x, u) e .T is called a support element of
K if JT ebd/C and u is an outer unit normal vector oi K at x. In
particular, if x^^"\K, then the pair {p{K,x), u{K,x)) is a support
element of K. The set of all support elements of K, also called the
generalized normal bundle of K, will be denoted by Nor K. For a
topological space X, the a-algebra of Borel subsets of X is denoted by
3(/l). 'Measurable' without further comment means Borel measurable.
Now let a convex body K eJC" and a number p> 0 be given. The
outer parallel body K + pB" is abbreviated by Kp. The map
/p: Kp\K-*S
x>^ipiK,x),u(K,x))
is continuous and hence measurable. We may, therefore, consider the
image measure Hp{K, ■) under /p of the Lebesgue measure. Thus
(XpiK, •) is a finite measure on ^B), and for a Borel set r;e^B),
fXp(K, rj) is the Lebesgue measure of the local parallel set
MpiK, ri) := /;'(;?)
= {x e E"|0 < d{K, x)^p and (p(K, x), u{K, x)) e rj}.
We establish some basic properties of the map ftp: JC" x -2B)-* R.
For the measure-theoretic notions and results to be used we refer, e.g.,
to Ash A972) and Bauer A974). By A we denote weak convergence of
Borel measures.
Theorem4.1.1. Let (^Kj)jef^ be a sequence in X" with Kj^ K for j-* <».
Then Hp{Kj, ■) A Hp{K, ■) for j -* oo.
Proof. Let r;C2 be open, and let xeMp(K,Tj) be a point with
d(K,x)<p. From Lemma 1.8.9 it follows that d{Kj,x)^ d{K,x)
and {p{Kj,x), u{Kj,x))^(p{K,x), u(^K,x)) for ;-*<». Hence, for
almost all ; we have d{Kj,x)<p and {p{Kj,x),u{Kj,x)) ^r], thus
X e Mp{Kj, ri). This shows that
MJK, Ti)\bdKp C liminf Mp(A:y, rj)
/■-.oo
and hence that

4.1 Local parallel sets 199
g a5"(liminf Mp(/^y, rj))
,-«
gUminf5i5"(Mp(/:y, rj))
= lim'mffip(Kj, rj).
This holds for all open sets rjCl. Further, we have iXp(K,F) =
X"{Kp\K) = X"{Kp) - 7Cj"{K). The continuity of the volume on X"
(Theorem 1.8.16) together with the continuity of Minkowski addition
shows that
Hpi^Kj, 2) ^ fipiK, 2) for / ^ oo.
The assertion follows (see, e.g., Ash 1972, p. 196). ■
Theorem4.1.2. For each r; e 3B), the function fip(- ,t}):X"-^U is
measurable.
Proof In the preceding proof it was shown that Kj -* K implies
liminffipi^K,, rj) ^ fip(K, t})
for open t], hence for such r; the function Hp( • ,i]) is lower semicon-
tinuous and thus measurable. Let ai be the family of all sets r) ^"li{I)
for which Hp{ •, r;) is measurable. We show that d is a Dynkin system.
For r)x, t}^ e d with t}2 C r)i we have Mp(K, TJ2) C Mp(K, t]i) and
Mp(K, TJi\T]2) = MpiK, m)\MpiK, 7J),
hence
IXp(,K, r){\rJ) = Hp(,K, r;,) - Hp{K, 7/2)
for K ^JC", thus r]i\rJ e (d. For a sequence {,r)j)j^f^ of pairwise disjoint
elements of c4. we have
\ /=i / y=i
for K ^JC", since jUp(/L, •) is a measure. It follows that U/LiJ?, ec^.
Thus c^ is a Dynkin system that contains the open sets and, therefore,
the a-algebra generated by them (Bauer 1974, p. 20). Hence ^B) C d,
which was the assertion. ■
Theorem4.1.3. For each rj e 3B), the function fip{ • ,tj) is additive, that
is,
HpiKi U Kj, V) + Mp(/^i n K2, T,) = Hp(Ki, rj) + Hp{K2, rj)
ifKi, K2, KiUK2eX".

200 Curvature measures and quermassintegrals
Proof. Let Ku K2e X" be convex bodies such that /iCi U /iCj e X". By
IpiK, T], ■) v/e denote the characteristic function of the set Mp(K, t]).
Let JT e E" be given and suppose that y '■= p^Ky U K2, x) e Ki, without
loss of generality. Then clearly
p(Ki U K2,x) = p(Ki,x).
Let p(/l2, x) ='■ z. Since KiU K2 is convex, there is a point a e[z, y]
for which a e Kif) K2- Since y - p{Ky U K2, x), we have |>' - xj S
|z - jr| and hence |a-jr|^|z-jr|. Here strict inequality is impossible;
hence a = z and thus z e /C1 n /l2. It follows that
p(/^,n K2, x) ^ p{K2, X).
We deduce that
d{Ky U K2, x) = d{K^, x), d{Ki D K2, x) = diK2, x)
u{Ky U K2, x) = u{K„ x), u{Ky n K2, X) = u{K2, X),
from which we get
/^(A", U K2,r],x)^ If,iKi,V,x),
IpiKi n K2, T), x) = lp{K2, r?, Jr).
As JT was arbitrary, we conclude that
lp{K,\J K2,r}, •) + Ip(Kin K2,r], •)
= Ip(.Ki,rt, •) + lpiK2,V, •)■
Integration of this equality with respect to X" yields the assertion. ■
4.2. Curvature measures and area measures
We shall now show that the measure Hp(K, rf) of the local parallel set
Mp{K, rj) can be expressed as a polynomial in the parameter p. First let
P e X" be a polytope. For x e E"\P, the nearest point p(P, x) belongs
to the relative interior of a unique face of P. For a given face F of P,
the Lebesgue measure of the set
Mp(P,r,)n p(P, O'^CrelintF),
where r; e 7JB) and p > 0, can be computed by means of Fubini's
theorem. For this, we observe that
(A-^XA") n p(P, •)-HrelintF)
is, up to a set of measure zero, equal to the direct sum
F © iN{P, F) n pB"),
where NiP, F) is the normal cone of P at F. If F is of dimension
m e {0, . .. , n - 1}, we obtain

4.2 Curvature measures and area measures 201
a5"(Mp(P, T}) n p(P, ^"'(relintF))
= jp^"-'"[z e 7V(P, F)|0 < |z| ^ p, ^y, ^J e r?|da5'«(>')
= /^^^""'"{'^"lO < A ^ p, M e 7V(P, F) n S"-\ (y, u) e r;}da5'"(>')
where
r;, := (u e 5"-'|(>', u) e r;}.
Thus we arrive at
= "i P"-" —^— E f 5t^"-'-'"(N(P, F) n r?,)d5t^'"(>').
„=0 n - m Fe7„{py'^
D.2.1)
We define
( " " ^H„(P, r?) := S L^"-'-'"(^(P, F) n r?,)d^'"(>')
D.2.2)
for m = 0, . . ., n - 1, so that
HpiP, V) = -l P"-'"i " ]&miP, V)- D.2.3)
« m=0 \ '"/
This representation can be extended to arbitrary convex bodies:
Theorem 4.2.1. For every convex body K e DC" there exist finite positive
measures Q^^K, •),..., 6)„_|(/l, •) on '2B) such that, for every
T} e ^B) and every p > 0, the measure Hp{K, rj) of the local parallel set
Mp{K, rj) is given by
fipiK, r?) = -^ S p"-'"i " ]e„(K, r,). D.2.4)
« m=0 \ '"/
The mapping K >-> Q„{K, •) {from JC" into the space of Borel measures
on I) is weakly continuous and additive, that is,
Kj -* K implies Q„(,Kj, •) A Q„{^K, ■),
and Ki, K2, K^U K2& JC implies
QJ^Ky U ATz, •)^QJ,K, nK2, •) = 0„(A:„ •) + ©JATj, ■).
For each rj e '3B), the function 0„{ •, rf) {from ZK." to R) is
measurable.

202 Curvature measures and quermassintegrals
Proof. First let P e :X" be a poly tope. Let t] e1i(U) and p > 0 be given.
Writing down equation D.2.3) successively for p = 1, ..., n, we obtain
a system of linear equations that can be solved for 0o(P, t]), .. .,
0„_i(P, r;). We get representations of the form
n
with certain constants a„k' With these constants we define
n
0m(A:. •) := E«m*M*(A:. •) D.2.5)
for arbitrary convex bodies K e JC". Then 0„(K, •) is a finite measure
on '2B), for the moment possibly a signed one. From Theorem 4.1.1 it
follows that Kj-* K (for K,, K e DC") implies Q„{Kj, •)^ Q„{K, •).
Applying this to a sequence of polytopes converging to /C, we see that
0m(K, •) is a positive measure and that D.2.3) can be extended to give
D.2.4). The final assertions of the theorem follow from D.2.5) and
Theorems 4.1.3, 4.1.2. ■
The measure Q„{K, •) is called the mth generalized curvature
measure of /C, for a reason to be explained later. Although we have
defined the measure ©^(/l, •) on the whole space S, it is clearly
concentrated on the set Nor/C of support elements of K: If
X eMp{K,ri), then (p(/l,x), m(/l,x)) e r;n Nor/C, hence iXp{K,rj) =
Hf,(K,r)n NotK) and, therefore, by D.2.5)
0„(A:, r,) = 0„(A:, r, n Nor K) D.2.6)
for T] e 3B) and m = 0, ..., n - 1.
The existence of a polynomial expansion of the type D.2.4) carries
over to the generalized curvature measures. For p > 0 we define a map
tp-. 2-* 2 by tp(x, u) ■= (x + pu, u). Then we have:
Theorem4.2.2. Let K eX", t) e (B^IJ), p> 0 and m e @,.. ., n - 1}.
Then
Qm^Kp, tpT,) = f,p'i"']0„.i(K, ri). D.2.7)
Proof. Let x^^"\Kp. There is a unique point y ^[x, p{K,x)\
Dbd/Cp. Assume that p{Kp,x)^y. Then \x-p(^Kp,x)\ <|jr->'|.
Let p{K,p{Kp,x))=- z- From p(Kp,x)&bAKp and yebAKp it
follows that
\p(,Kp,x) - z\ = p^\y ~ p{K,x)\
and hence

4.2 Curvature measures and area measures 203
\x - z\^\x - p{Kp, x)\ + |p(/Lp, x) - z\
<k- >'l + !>' - PiK,x)\
= \x- piK,x)\,
a contradiction. Hence p(Kp, x) = y. This imphes u(Kp, x) = u(K,x),
d{Kp,x) = d(K,x)-p and p(Kp,x) = p(K,x) + pu(K,x). Now we
obtain the disjoint decomposition
Mp^x(K, ri) = Mp{K, r,) U M.^Kp, tpt,)
for A > 0 and hence the equality
fip+xiK, TJ) = fipi^K, TJ) + Hx{Kp, tpTj).
Here we insert D.2.4) and compare the coefficients of equal powers of A
to obtain the assertion D.2.7). ■
From the measures 0m{K, •), which are defined on sets of support
elements, we derive two series of measures defined either on sets of
boundary points or on sets of normal vectors. For m = 0, ..., n - 1 we
put
C„(A:, /3) := 0UK, ft X 5"-') for /3 e ^(E"),
S„iK, CO) := 0„iK, E" x co) for o) e ^E""').
The measures Co{K, •),... ,C„-i(K, •) are called the curvature
measures of K, and So(K, •),..., S„-i(K, •) are called the area
measures of K. Observe that the curvature measures are Borel measures
on E", while the area measures are Borel measures on the unit sphere
S""'. For the reader's convenience, we repeat the definition of these
measures by specializing formula D.2.4) to these cases. Writing
Api^K, p) := {x e E"|0 < d(,K, x) ^ p and p^K, x) e /3}
for /3 e '2(E") and p > 0, we have
X"iApiK, /3)) = -"i p"-'"i " ]c„iK, 13), D.2.8)
and with
Bp(K, CO) ;= {x e E"|0 < d^K, x) ^ p and u^K, x) e co}
for 0) e ^E""') we obtain
^^"(Bpi^K, CO)) = -2 p"-'"{ " ]s„iK, CO). D.2.9)
It is evident that the need to introduce two distinct series of measures
is a result of the fact'that a convex body can have singular boundary
points and singular normal vectors. The following lemma expresses one
aspect under which both types of measures are related to each other.

204 Curvature measures and quermassintegrals
Lemma4.2.3. Let m € {0,. . . , n - 1}, let a)C5""' and /3CE" be
closed. Then
C„iK, x{K, oi) n regA:)^ 5„(A:, m) ^ C„{K, t{K, a)),
S„iK, a{K, /3) n regnA:)^ C„{K, /3) ^ S„{K, a{K, /3)).
Proof. Let {x,u)&lioxK be a support element of K for which
X € t(/l, 0)) n reg/C; then u e N{K,x). Since jreT(/L,a)), there is a
normal vector of K at x belonging to co, and this is equal to u since
X eiegK. Thus w e o). This shows that
([t(/:, CO) n reg/C] x 5""') D Nor/C C (E" x o)) D Nor/C.
The sets occurring here are Borel sets, since t{K,co) and Nor/C are
closed and bd K\ieg K is an f ^-set. We deduce that
Q„{K, {{t{K, <a) n reg/C] x 5""*) D Nor/C)
^ ©„(/:, (E" X a))n Nor/C)
and hence, using D.2.6), that
C„{K,T{K,oi)r\xtgK)
= 0„(/^, ([t(/^, <a) n reg/^] x 5""*) D Nor/^)
^ 0„(/^, (E" X a))nNor/^)
= 5„(A:, 0)).
Let {x, u) e Nor /C be a support element of K for which u e o); then
xe T(/i:,a)). Thus
(E" X 0)) n Nor/^ C [t(/^, 0)) x 5""'] D Nor/^
and hence
5„(/^, 0)) = Q„{K, (E" X 0)) n Nor/^)
^ 0„(/^, [t(/^, 0)) X 5"-'] n Nor/^)
= C„(A:, t{K, 0))).
This proves the first two inequalities of the lemma. The proof of the last
two is completely analogous. ■
If /l e JC" is smooth, then the spherical image map aj^ is defined on
all of bd K, and t{K, <a) = a'f^^oi) for <a C 5""'. If o) is a Borel set, this
shows that t{K, co) is a Borel set, since o^ is continuous. From Lemma
4.2.3 we obtain, first for closed sets and then for general co e ^E""'),
C„{K, ol\co)) = C„iK, T{K, CO)) = S„iK, co).
An analogous result holds for strictly convex bodies; hence we can state:

4.2 Curvature measures and area measures 205
Theorem 4.2.4. If K e JC" is smooth, then S„{K, •) is the image
measure of C„{K, •) under the spherical image map o^. If K is strictly
convex, then C„{K, •) is the image measure of S„{K, •) under the
reverse spherical image map Tj< (m = 0, . .. , n - 1).
Thus for convex bodies K that are both smooth and strictly convex
there is in fact no essential difference between both types of measures,
since
C„(A:, /3) = S„(^K, OKiP n bd K)) for /3 e ^(E").
In the literature, curvature measures and area measures also appear
with a different normalization. One defines <Pm(K, •) and Vmi^K, •) by
(:)
nK„.„<P„iK, ■)={ \C„iK,-), D.2.10)
-(:)
nK„^„W„iK, ')=\\S„(^K, ■) D.2.11)
for m = 0,..., n - 1 (recall that Kj is the /-dimensional volume of the
/-dimensional unit ball). It is reasonable to supplement the first
definition by
4>„(K, p) := ^^"{K n /3) for /3 e ^(E").
Neither of the two nomenclatures deserves unique preference over the
other. In studies related to differential geometry the €„, S„ appear
more natural, while in integral geometry the 4>„, •?„ yield slightly less
clumsy formulae. One advantage of the second normalization is that
<PmiK, /3) depends only on K and /3 and not on the dimension of the
surrounding space in which it is computed. This follows from D.2.17)
below.
It is clear that the properties established for the generalized curvature
measures carry over to curvature measures and area measures. The
functions K >-^ C„{K, •) and K >-* Sm{K, •) are weakly continuous and
additive; the function C„(-,/3) is measurable for each /3e'2(E"), and
5„( • ,co) is measurable for each co e 3E""'). The following invariance
and homogeneity properties are easily obtained from D.2.8), D.2.9): If
g is a rigid motion of E" and go denotes the corresponding rotation,
then
C„igK,gP)=C„{K,p), D.2.12)
SMK, goco) = S„iK, CO). D.2.13)
If A > 0, then
CMK,Xp) = X'"C„iK,p), D.2.14)
S„iXK, CO) = X'"S„{K, CO). D.2.15)

206 Curvature measures and quermassintegrals
The measure C„{K, •) is concentrated on the boundary of K, since
Ap(/:,/3) = 0 if /3nbd/: = 0. It is defined locally, in the following
sense: If /3C E" is open and if Ki, K2 e J^" are such that /C, D /3 =
K2 n /3, then C„(/:,, /3') = C„(K2, /3') for every Borel set /3' C /3. This
follows from the observation that p(Ki, x) e /3 (i = 1 or 2) for an open
set p, together with /C, D/3 =/l2 n/3, implies p(Ki,x)= p(K2,x),
hence ApiKi, /3') = Ap(K2, P') for /3' C /3. The assertion then follows
from D.2.8). Also, the measure S„{K, •) is defined locally, which is
now meant in the following sense. If we 7i(S"~^) (not necessarily open
in this case) and if K^,K2^X" are such that t(/li, o)) = t(/l2, o)),
then S„(Ki,co)= S„{K2,co). In fact, the local parallel set Bp(Ki,co)
depends only on the reverse spherical image t(/l,, co). Finally we remark
that, for area measures, formula D.2.7) takes the simple form
=tAl)
S„iKp, 0)) = 2.P' . \Srn-iiK, 0)) D.2.16)
y=o \7 /
for we'2E"'') and p>0.
We turn to the consideration of special cases where the curvature and
area measures can be given a more direct and intuitive intepretation.
First let P e X" be a polytope. From D.2.2), D.2.10) and the definition
of the external angle y{F, P) (see Section 2.4) we obtain
^m{P,P)= E y(f, P) 5K'"(f n/3) D.2.17)
for /3 e 3(E") and m = Q,. . . ,n. Since y{F, P) depends only on P and
F, as remarked in Section 2.4, we see that <Pm(P, /3) does not depend
on the dimension of the surrounding space, that is, it can be computed
in any affine subspace containing P. Similarly we get
^.(P,„,= 2 ^-I^ZMf^QHii!) ^.(„ D.2.18)
for CO e ^E""') and m = 0, . . . , n - 1.
If K is of class C+, then a comparison of B.5.31) with D.2.8) and
D.2.9) shows that
C„(A:, /3) = J^^^^^//„_,_„d5r.'-', D.2.19)
S„(K,a})= I s„<iX''-' D.2.20)
JO)
for /3 € ':?(£"), CO e ^E""'), m = 0,.. ., n - 1 (first for open sets /3, co,
but then for Borel sets, since both sides of D.2.19) and D.2.20) are
measures in respectively /3 and co).
Formula D.2.19) is, of course, the reason for the name 'curvature
measure'. These measures replace, for general convex bodies, the

4.2 Curvature measures and area measures 207
elementary symmetric functions of the principal curvatures that can be
defined in the C^ case. Similarly, the measures S„(K, •) replace the
elementary symmetric functions of the principal radii of curvature (as
functions on the spherical image). The name 'area measures' for this
series of measures comes from the fact that 5„_,(/l , co) is the area of
the reverse spherical image of K at co (see D.2.24) below for the
general case) and that S„(K, co) can be obtained from this by means of
the Steiner-type formula
5„_,(A:p, •)= "ip"~'""i"~ %m(.K, •),
m=0 \ '" /
which is a special case of D.2.16). S„{K, •) is also called the area
measure of order m.
We return to general convex bodies and establish the intuitive
meaning of the measures C„, S„ in the extreme cases m = 0 and n - 1,
whereby we shall show that they are, in fact, suitable Hausdorff
measures.
Theorem4.2.S. Let K e X", p e S(E") and we ^E""'). Then
CoiK, 13) = X"-\aiK, /3)), D.2.21)
5o(/^,a))= ^"-'@)). D.2.22)
If K is n-dimensional, then
C„.iiK, p) = ^"-'(/3 n bd K), D.2.23)
S„.i(K, CD) = ?Cj"-\t(^K, co)). D.2.24)
Proof. First we remark that the formulae are valid if /C is a polytope, as
follows from D.2.17) and D.2.18).
For K eX" andpe ^(E""') we define
KiK, p) := X"-\a(^K, p)).
By Lemma 2.2.10, o{K, P) is a Lebesgue measurable subset of S"~^. In
the proof of that lemma it was shown that /3i n /32 = 0 implies that
X"'\oiK, pi) n oiK, p2)) = 0.
Since 70"''^, restricted to the Lebesgue measurable subsets of 5""', is
a-additive, it follows that k{K, •) is a measure on 2(E").
Let {Kj)j^fi be a sequence in X" converging to a convex body K. We
take an open set /3C E" and assume that u e o{K, P) D regn K. There
is a unique point xeF{K,u); it belongs to p. For /e(^ choose
Xj e F(Kj, u). Any accumulation point y of the sequence {Xj)i^fq belongs
to K and lies in H(K, u), hence y = x. Thus Xj^x for ;-♦<» and,
therefore, Xj e P and u e o(Kj, p) for almost all /. This shows that

208 Curvature measures and querraassintegrals
o(K, /3) n regnK C \\min{o{Kj, /3),
and in view of Theorem 2.2.9 we deduce
KiK, /3) = ?Cj"-\a(K, /3) D regn A")
^5!5"-'(liminf(T(/:y, /3))
^liminfa5"-'(a(/^y,/3))
= liminfjc(/:y,/3).
y-oo
Since this holds for all open sets /3cE" and since k(Kj,^") =
5i5"~'E"~*) = k(/l, E"), we have proved the weak convergence
KiKj, ■)^k{K, •). Now the equality ic(K, •) = Cq^K, •) follows
immediately if K is approximated by a sequence of polytopes, since it is
true for polytopes and both sides are weakly continuous. This proves
D.2.21).
Equality D.2.22) holds since it is true for polytopes and So(K, •)
depends weakly continuously on K.
For the proof of D.2.23) we put
T}(K,p) := X"-\pnbdK)
for K e JCq and /3e'2(E"). Then r](K, •) is a measure, and as above
it suffices to prove that Kj-* K (for Kj,KeXS) implies
We may suppose that o € int /C. Define
(p: E"\{o} ^ 5"-' by cpix) := -^.
\x\
Let p(K, •) be the radial function of K (see Section 1.7) so that, for
y e 5""', the point p(K,y)y is in the boundary of K. By n(K,y) we
denote an arbitrary outer unit normal vector of K at this point.
Theorem 2.2.4 tells us that n(K,y) is unique for 35"~'-almost all
>' e 5""'. We can now write r]{K,p) as the integral
r){K,^)^\ ,^^'/J" .X d^"-^^). D.2.25)
"• ' ^' Ji>{pr\b<SK) {y, n{K, y)) ^^' ^
This can be proved with the aid of Theorem 3.2.3 in Federer A969). If
now the sequence {Kj)j^^ of convex bodies converges to K, we may
assume that oeint/Cy for all /. We have p^Kj, •)-*p{K, •),
n{Kj, ■)-*n{K, ■) almost everywhere on 5""', and the functions
( ■ ,n{Kj, ■)) are bounded from below by a positive constant. Using
Fatou's lemma and the bounded convergence theorem, it is now easy to
see that r){Kj, •)Ar;(A:, •) for /-*<». This completes the proof of
D.2.23).

4.2 Curvature measures and area measures 209
For the proof of D.2.24) we put
FiK, 0)) := ^"-\t(^K, (!)))
for KeJCo and we^E""'). Again we may assume that oeint/C.
From Lemma 2.2.11 we know that q>{T{K,co)) is Lebesgue measurable
on S"~\ If 0), n ft>2 = 0, then
^"-\t(K, 0),) n t(K, 0J)) = 0,
as a consequence of Theorem 2.2.4. We deduce that F(K, •) is a
measure on '2E""'), and from D.2.25) we have
FiK, CO) = f P(A:,y)"~' d5K''-'(>').
*■ ■' )>p{T(K,<o)) {y, niK,y)) ^^'
As above, this can be used to show that Kj-* K (with K e JCq) implies
F{Kj, •)-^F(K, ■) for /-*<». Using this, D.2.24) is extended from
polytopes to general convex bodies. ■
If the body K eJC" is of dimension less than n, it is easy to see
(e.g., from D.2.8), D.2.9)) how the equations D.2.23), D.2.24) have to
be modified: If dim K = n - I, then
S„-i(K, 0)) = 2X''-\K), 7C>"-\K), or 0,
according to whether both, one, or none of the unit normal vectors of
the affine hull of K belongs to ot. If dimK<n-\, then
C„_,(/l, /3) = 0 = 5„_,(/l, oi) (so that D.2.23), D.2.24) are true again).
Next we consider the total curvature measures
Q„{K, I) = C„{K, E") = S„{K, 5"-'),
for which special notation is common. These functional were in use
long before the curvature measures. The traditional notation is
Wi{K) := - Qn-i{K, 2)
n
for I = 1,.. ., n, and Wo(/l) •= V„{K), the volume of K. The
functions W(i,. . ., W„ are called the Minkowski junctionals or quermass-
integrals. The latter term comes from the German 'QuermaB', which can
be the measure of either a cross-section or a projection. The reason for
this terminology will be clear when certain integral-geometric
interpretations of the functions W, have been obtained in Section 4.5.
We denote the total values of the renormalized curvature measures
by
V„(^K) := *„(A:, E") = W„iK, 5"-'),
so that

210 Curvature measures and quermassintegrals
=(:)-
Kr,-„V„iK) = I ^ \W„.„(K) D.2.26)
for m = 0, ... ,n. McMullen A975a) proposed that V„(K) should be
called the intrinsic m-volume of K since, if K is m-dimensional, then
V„(K) is the ordinary m-dimensional volume of K.
In order to illustrate the meaning of quermassintegrals, we recall
some of the earlier formulae for this special case. The representation
v„{K + pB") - Sp'" w,(a:) = 2p'-'ic„_,v,(a:), D.2.27)
1=1 \' / 1=1
a very special case of D.2.4), is the classical Steiner formula, by which
quermassintegrals and intrinsic volumes can be defined. If K is of class
C\, then
^iK) = -^ J^^^H,-,d5K"-' D.2.28)
W,.(A:)=-^/^„.,5,-,d5K"-' D.2.29)
for / = 1, ... , n. If P is a polytope, then
Vnr{P)= E y{F,POC>'"{F) D.2.30)
for m = 0, ...,n. For general convex bodies, we list the following
special cases:
Wo = V„, the volume;
riWx = 2K„_,, the surface area;
— W„_, = -^^ K,, the mean width (formula E.3.12));
«■« ft>„
— W„ = Ko = 1> the Euler characteristic.
Because of D.2.28), riW2{K) is sometimes called the integral of the
mean curvature (even if K is not of class C+).
As a function on JC", the quermassintegral W, is rigid motion
invariant, additive and continuous. This follows from the corresponding
properties of the curvature measures; the continuity is clear from
D.2.27) and from the continuity of both the volume functional and
Minkowski addition. The following celebrated result shows that these
properties are sufficient to characterize the linear combinations of
q uermassintegrals.
Theorem 4.2.6 (Hadwiger). // (p-.JC-^U. is a function that is invariant
under rigid motions, additive and continuous, then

Notes for Section 4.2 211
(=0
for K e JC", where cq, . • • , c„ are real constants.
There is a counterpart to Theorem 4.2.6 with continuity replaced by
monotoneity. The quermassintegrals are monotone (or increasing), that
is, K C L implies W,(/l) ^ W,(L). This will become clear later, either
by means of the integral-geometric interpretation (Section 4.5) of the W,
or from their interpretation as mixed volumes (Chapter 5).
Theorem4.2.7 (Hadwiger). // q>:X"^U is a function that is invariant
under rigid motions, additive and monotone, then
n
cpiK) = Ec,W,(A:)
(=0
for K e X", where cq, . . . , c„ are non-negative real constants.
Both theorems characterize the series of quermassintegrals by simple
and natural properties and thus illustrate the importance and basic
character of these functionals.
The only known proof of these two theorems is the original one given
by Hadwiger. It requires some results from the dissection theory of
polytopes and would not fit well into our present context. We refer the
reader to the book of Hadwiger [18].
A result of McMullen A977; see also [40], Theorem A1.5)) permits
us to deduce Theorem 4.2.7 from Theorem 4.2.6.
Notes for Section 4.2
The Steiner formula. The Steiner formula goes back to Steiner A840b), who
derived it in E^ and E', for polytopes and surfaces of class C^. For general
convex bodies in E", this formula is just a special case of more general
expansion formulae for mixed volumes (see Chapter 5). For the volume of
the 'tube' around a smooth submanifold of Euclidean or spherical space,
Weyl A939) derived a similar expansion, now called 'Weyl's tube formula'.
The essential point here is that the coefficients can be expressed in terms of
the Riemann curvature tensor of the submanifold and thus depend only on its
intrinsic metric. Counterparts to the original Steiner formula in spaces of
constant curvature were treated by Herglotz A943), Vidal Abascal A947),
Allendoerfer A948) and Santal6 A950). The extension of Weyl's tube
formula to general Riemannian spaces was studied by Gray & Vanhecke
A981); see also Gray A990). Hadwiger A945, 1946a,b,c) used integral-
geometric methods to obtain Steiner-type formulae for certain non-convex
domains also. Other analogues, partly in the form of inequalities, are due to
Hadwiger A946d), Ohmann A955b), Sz.-Nagy A959), Makai A959) and P.
Meyer A977).

212 Curvature measures and quermassintegrals
2. Steiner formulae for Minkowski subtraction. A special case of D.2.16) is the
system of Steiner formulae for the quermassintegrals, that is,
W,(K + pB") = iVi" T ^]w,.i(K)
.=0 \ ' /
for K e X", p^O and p = 0,...,«. If B" is a summand of K, then it is not
difficult to see that
W,(K ~ pB") = 2i-p)i" T ^)wp,,(^) D.2.31)
for 0 = p= 1 and for p = 0,..., n. This was pointed out (in a more general
form, with B" replaced by an arbitrary convex body and with the
quermassintegrals replaced by corresponding mixed volumes) by Matheron A978b). He
showed, conversely, that the validity of D.2.31) for p e @,1) and
p = 0,..., n implies that B" is a summand of K. He conjectured that this is
already true if D.2.31) holds for p e @,1) and p = 0 (the case of the
volume); he proved this for « = 2.
3. Sets of positive reach. A natural common generalization of convex sets and of
smooth submanifolds, for which local parallel sets can be defined in such a
way that their measure has a polynomial expansion, yielding a Steiner
formula, are the sets of positive reach. A compact set A CE" is said to be of
positive reach if there exists a number p> 0 such that, for each point * e E"
whose distance from A is less than p, there is a unique point in A that is
nearest to x. For sets A of positive reach, Federer A959) introduced the
curvature measures 'P„,(A, •). He made an extensive study and proved many
of the relevant results which, in this book, are considered only for convex
bodies.
For extensions of the curvature measures to other classes of sets, see
Section 4.4 and the notes therein.
4. The simpler introduction of the curvature measures for the case of convex
bodies and the derivation of their properties, as presented here, is taken from
Schneider A978a). For the area measures (which are related to the theory of
mixed volumes, see Chapter 5), one should see the beautiful paper by
Fenchel & lessen A938). The generalized measures on sets of support
elements appear in Schneider A979a).
5. Differentiation of curvature measures. Let K e X" be a convex body of class
C+. If jic e bd/l and (/3,),gi^ is a sequence of Borel subsets of bdK with
positive measure and shrinking to x, then it follows from D.2.19) that
hm = H„-i-„(x).
-« 5f5"-'(/3,)
One may ask whether this relation holds true for a general convex body K if
Jic is a normal point (see Section 2.5), so that H„-i-„(x) is defined. Under
certain (not too restrictive) assumptions on the sequence (A')i6N this can, in
fact, be proved; see Aleksandrov A939c) for m = 0 and Schneider A979a) for
m = 0, ...,«- 1.
6. Integral and current representation of Federer's curvature measures. In a paper
under this title, ZShle A986a) defined (for sets of positive reach) principal
curvatures as functions on the unit normal bundle and expressed the
curvature measures as ;^;''"'-integrals of elementary symmetric functions of
such curvatures over the unit normal bundle. She also interpreted these
integrals as values of associated rectifiable currents on specially chosen
differential forms. This work was continued in Zahle A987a). (See also the
notes for Section 4.4.)
7. Arguments leading to special cases of Theorem 4.2.5 can be found at several

Notes for Section 4.2 213
places in the work of Aleksandrov A937a, §1, 1939b, 1939c, §6, [1], Chapter
V, §2).
8. Valuations on X". The important theorems 4.2.6 and 4.2.7 of Hadwiger
characterize the linear combinations of quermassintegrals as the only
valuations on X" that are rigid motion invariant and either continuous or
monotone. These theorems were one of the starting points for a series of
investigations on general valuations on the space of convex bodies or
subspaces thereof, aiming at establishing general properties as well as
characterization and classification results. We refer the reader to the survey
given by McMullen & Schneider [40], in particular to Chapter IV. More
recent contributions are due to Spiegel A982), Goodey & Weil A984), Betke
& Goodey A984) (this paper contains a mistake, see Goodey & Weil A991)),
Betke & Kneser A985) and McMullen A990).
9. Axiomatic characterizations of curvature measures and area measures. Linear
combinations of curvature measures or of area measures can be characterized
by some of their properties in a manner analogous to Hadwiger's theorem
4.2.6.
Theorem. Let q) be a map from X" into the set of finite Borel measures on
E" satisfying the following conditions (where (p{K)(p) is written as qj{K, p)).
(a) q) is rigid motion invariant, that is, q>(gK, gp) = q>{K, p) for K e X",
P e '3(E") and every rigid motion ^ of E".
(b) q> is additive.
(c) q> is weakly continuous, that is, Kj^*K in X" implies (p{Kj, •)
(d) q> is defined locally, which means: If pCE" is open and K H P =
Lnp, then <p{K, p') = <p(L, P') for every Borel set P' C p.
Under these assumptions, there are non-negative real constants cy,..., c„
such that „
(p(K, p) = 'Zc><P.(K, p)
ioT K eX" andperBil").
This theorem was proved by Schneider A978a). It gives a partial answer to
a question of Federer A959, Remark 5.17), who had asked for a
characterization, similar to Hadwiger's characterization theorem for quermassintegrals, of
his curvature measures on sets of positive reach. A similar result for area
measures was obtained by Schneider A975e):
Theorem. Let i/* be a map from X" into the set of finite signed Borel
measures on 5""' satisfying the following conditions (where t{;{K)((o) is
written as if'iK, (o)).
(a) V is rigid motion invariant, that is, y{gli, goto) = y{li, (o) for K e X",
we ^E""') and every rigid motion g of E", where ^o denotes the
rotation corresponding to g.
(b) V is additive.
(c) V is weakly continuous.
(d) V is defined locally, which means: If we7jE""') and r(K,(o) =
r(L, (o), then V</l , (o) = \f{L, w).
Under these assumptions, there are real constants Cq, .. ., c„_i such that
n-l
V<A:, w) = 'Zcirl>,{K,(o)
1=0
for K eX" and we •5?E"-').
A far-reaching generalization of these theorems was proved by Zahle
A990).

214 Curvature measures and quermassintegrals
4.3. The area measure of order one
The area measure Sm(K, •) of order m, for a convex body K, is a
special case of the mixed area measure to be treated in Section 5.1 and
is, therefore, closely related to the theory of mixed volumes. The area
measure of order one, however, exhibits some special features; this
suggests a separate treatment that is independent of mixed volumes. The
essential special property of 5i is its Minkowski linearity, that is, the
equality
Si{kK + nL, •) = kSi(K, •) + nSi{L, •) D.3.1)
ioT K, L € X" and A, n^O. For convex bodies of class C\ this follows
from D.2.20) and the linearity of Si, which is evident from B.5.19); the
general case is then obtained by approximation, using the weak
continuity of the area measures. In view of D.3.1), the area measure Si is
closer to Chapter 3 than to Chapters 5 and 6 and will, therefore, be
considered briefly in the present section.
First we show that Sii^K, •) determines K uniquely up to a
translation. It appears that the shortest proof of this is the one using
spherical harmonics.
Theorem4.3.1. If K, L e X" are convex bodies with 5i(/l, •) =
5i(L, •), then K and L are translates of each other.
Proof. Let ¥„ be a spherical harmonic of degree m. First let K be of
class C\. Then it follows from D.2.20) and B.5.23) that
/^„.7„(u)d5,(A:, u) = l„.Jn.{^h^ + -^ AshK^do,
where o denotes spherical Lebesgue measure on 5""'. Using Green's
formula (¥„, Ash^) = (AsY„, h^) and AsY„ =-m(m + n-2)Y„
(Appendix equation (A.2)), we obtain
l„_Jrn(.u)dSi(K, u) = k„_„j^^_^h^Y„da D.3.2)
with a real number X„„ satisfying X„„ ^0 tor m^l. By
approximation, D.3.2) extends to arbitrary convex bodies. The assumption of the
theorem now yields
K.n.j^U^'K - hL)Y„do = 0.
Since m e {0,1, . . .} was arbitrary, it follows from the completeness of
the system of spherical harmonics that h^ - hi is a spherical harmonic
of degree 1, hence hf^ - hi = {• ,t) with f € E". This is equivalent to
K = L + t. ■

4.3 The area measure of order one 215
In the plane, the uniqueness result holds even locally. This is not
surprising, in view of equation B.5.22) which is valid in the C+ case. To
formulate the general case, we understand by an open convex arc a
nonempty, relatively open and connected proper subset C of the
boundary of a convex body K e ^o; we also assume that C is not
contained in a line. The spherical image a{K,C) of K at C depends
only on C; it is called the spherical image of C and denoted by a^. The
definition of the measure Sy can be extended to open convex arcs by
putting
Si(,C, 0)) .= X\t{K, 0)) n C)
for Borel sets <aC Oq, in agreement with D.2.24). We call 5i(C, •) the
length measure of C.
Theorem 4.3.2. Assume that C, C C E^ are open convex arcs such that
Oc = Oc, this set is contained in a closed semicircle of 5', and
•^i(C, • ) = 5i(C', •). Then C and C are translates of each other.
Proof. The open convex arc C has two well-defined endpoints (not
belonging to it), say x and y. Let C be the image of C under reflection
in the point (x + y)/2. Since Oc is contained in a semicircle of 5', it is
easy to see that CD CU {x,y} is the boundary of a convex body, say
K. Let K' be constructed from C in the same way. Then
Si{K, co) = Si{K', co) if 0) is a Borel subset of Oc or of its reflection Oc
in the origin, and Si(K, co) = 0 = Si{K',co) if o) is a Borel subset of
5^((TcUac). Thus Si(K, ') = Si(K', •), from which we deduce that
K' is a translate of K. It is then clear that C must be a translate of C.
Here the following has to be observed. If, say, x is the endpoint of a
segment S contained in C, with outer unit normal vector u, then
u € Oc, u is an endpoint of Oc and 5i(C, {«}) is the length of S. It
follows that C contains a parallel segment of the same length. ■
Clearly, Theorem 4.3.2 can also be applied to open convex arcs
having spherical images not contained in a semicircle, by covering a
given arc with smaller arcs and applying the theorem to each of these.
In the plane, one also has an existence theorem for convex bodies
with prescribed measure Sf.
Theorem 4.3.3. Let (pbe a measure on S' satisfying
£,"<1<P(") = o.
Then there exists a convex body K e X^ for which Si(K, ) = q>.

216 Curvature measures and quermassintegrals
This is just the two-dimensional case of Minkowski's existence
theorem (see Section 7.1). Therefore, we do not give a proof here but
simply mention that the polygonal case (the two-dimensional case of
Theorem 7.1.1) is almost trivial (compare the construction after C.2.3)).
The n-dimensional existence problem for Si consists of finding
necessary and sufficient conditions for a Borel measure q> on 5""' in
order that there exist a convex body K e X" for which Si(K, •) = •?.
In the smooth version, one asks which conditions a real function / on
S"'^ must satisfy in order that there exist a (sufficiently smooth) convex
body K such that / is equal to si(K, •), the mean radius of curvature
of /l as a function of the outer unit normal vector. The
three-dimensional case was treated by Christoffel A865); the general case is now
called Christoffel's problem. In the smooth case, this problem comes
down to solving the elliptic linear differential equation
-^-^ Ash + h = f D.3.3)
on S"~^ and expressing the condition that the solution h has to be the
restriction of a support function in terms of conditions on the given
function / (the latter was not done by Christoffel). The first complete
solution was given by Firey A967b). Instead of using D.3.3), he based
his treatment on the differential equation
A|(u) = grad/(u), D.3.4)
where |(u) is defined as in Section 2.5 and / is continuously differ-
entiable and is extended to E"\(o} as a positively homogeneous function
of degree -1. Solutions for the general Christoffel problem were given,
independently, by Firey A968) and Berg A969b). We formulate Firey's
result without proof. He introduces the fundamental singularity gi,
defined for u,u' eE", u^ u', by
gi(u', u) := /„E(u', u)),
where
Us) ■■= - , \. s'-\ ifn>3,
(n - i)(t)„-i
s(u', u) ■= arc COS-, ,.. ■,
and the Green's function G associated with the Poisson equation D.3.4)
by
G{u', u) ■■= y(s(u', u)),

4.3 The area measure of order one 217
where
y(s) ■■= ^cosec""^f sin""^ Tdrldf.
a)„ Jn/2 \}n I
Theorem 4.3.4 (Firey). A Borel measure q) on S" ^ is the first area
measure S^i^K, •) of a convex body K eJC" if and only if it satisfies
Jj„.,gi("'. u)d(p{u)
< 00
for all u' eS"~\ and
\„_,A(u', v',u)d(p{u)^0
JS"
for all u', v' e E"\{o}, where
A{,u', v', u) := r{u', u) + r(u', u) - r{u' + v', u),
riu', M):=(n -2)<u',u)G(u',u)- <u', gradG(u', u)).
Berg A969b) developed a theory of potentials and subharmonic
functions on the sphere S"~^ and within this theory finally formulated a
necessary and sufficient condition for a measure q? to be the first area
measure of a convex body. In the course of his investigation, he
constructed real functions g„, for n = 2, 3, .. ., on [-1,1] such that
hi^K, u) = /^„.,g„«u, v))dSi(^K, 0) + {s(K), u) D.3.5)
for K eX" and ue 5""'. From this result and the fact that
j^Jgniiu, U})\d7r^"-\U) < 00, D.3.6)
also shown by Berg, a stability version of the uniqueness theorem 4.3.1
can be derived:
Theorem4.3.5. If K, K' eJC" are convex bodies with coinciding Steiner
points and satisfying
\SiiK, (!)) - Sii^K', (!))\ S £^"-'@)) D.3.7)
for all 0) e li^S"'^) nnd a given number e > 0, then
d(K, K') ^ c„e
with some constant c„ depending only on the dimension n.

218 Curvature measures and quermassintegrals
Proof. From D.3.5) and the assumption s{K) = s(K') we obtain
\h(K, u) - h{K', u)\ ^ \^Jg„{{u, vmv{v),
where v is the variation of the signed measure Si{K, •) - Si{K', •).
The assumption D.3.7) yields v(aj)^ e^""'(a)). The existence of the
constant c„ now follows from D.3.6). ■
Notes for Section 4.3
1. Christoffel's problem. It is clear from the Introduction in Christoffel A865)
that the surfaces that he studied are assumed to be convex ('allenthalben
gewolbt'), but in the existence part of his investigation no such property is
discussed (an erroneous statement to the contrary was made in Bonnesen &
Fenchel [8], p. 123). Favard A933b) and Suss A932b, 1933) claimed to prove
that the condition JMd(p(M) = o is sufficient for the existence of a convex body
K with 5i(/l, •)= <P (under smoothness assumptions). However, Aleksan-
drov A937c, 1938a) gave examples of positive measures on 3E""'), some
even with analytic densities, which satisfy that necessary condition and are
not the first area measure S\(K, •) of any convex body K. A sufficient, but
not necessary, condition was found by Pogorelov A953).
The proof of the uniqueness result. Theorem 4.3.1, by means of spherical
harmonics goes back essentially to Hurwitz A902). His proof for the three-
dimensional case is reproduced in Blaschke A924), §95, and Pogorelov [27],
and was extended to higher dimensions by Kubota A925b).
A brief review of the various treatments of Christoffel's problem was given
by Firey A981). An application of Firey's A967b) existence result to the
study of surfaces of constant width appears in Fillmore A969).
In the centrally symmetric case, Goodey & Weil A990+) found a
connection between Christoffel's problem and Radon transforms.
2. Christoffel's problem for polytopes. The necessary and sufficient conditions
for first-order area measures found by Firey and Berg are not easy to handle
in concrete cases. For instance, it seems difficult to derive from these criteria
a description of the first-order area measures of polytopes. However,
Christoffel's problem for polytopes can be given an independent treatment by
direct elementary methods. This has been done by Schneider A977a).
3. The length measure. In the plane, the map K*-^Si{K, •) establishes an
isomorphism between the convex cone of convex bodies with Steiner point o
and the cone of measures cp on 3E""') with \uA(p{u) = o. This fact is
sometimes useful for the treatment of decompositions of plane convex bodies.
Schneider A974e) used the length measure in the investigation of the
asymmetry classes of convex bodies. Kallay A975) characterized the extreme
convex sets K in the set of bodies in JC^ with a given width function in terms
of a property of the Radon-Nikodym derivative of S\{K, •).
A detailed study of the length measure was also made by Letac A983).
4. Stability results. Theorem 4.3.5, for the special case of three-dimensional
convex bodies with twice continuously differentiable support functions, is due
to Pogorelov [27], p. 502. The general case is in Schneider [43], Theorem
(9.8).

4.4 Additive extension 219
Sen'kin A966a, b) proved the estimate
\w(K, u) - w(K', u)\ g - sup \Si(K, (o) ~ Si(K', w)\
for the width k'( •, m) in any direction u e 5^, for convex bodies K, K' e X^.
He used an elementary argument for strongly isomorphic polytopes and then
Aleksandrov's approximation theorem 2.4.14.
4.4. Additive extension
The generalized curvature measure Q„{K, •) is an additive function of
K, by Theorem 4.2.1. In Section 3.4 we introduced the convex ring
U{JC"), consisting of all finite unions of convex bodies in E", and
collected some results about the additive extension of valuations from
X" to U{DC"). In the present section we shall treat the additive
extension of the generalized curvature measures &„ to the convex ring.
While the existence of such an extension could be deduced from general
results on valuations (see Section 3.4, Note 7), we prefer to give an
explicit construction that provides additional insight and at the same
time extends the Steiner formula to local 'parallel sets with multiplicity'.
The availability of curvature measures for sets more general than convex
bodies is useful for applications, and the additivity of the extension
permits the almost automatic generalization of some integral-geometric
formulae to be proved in the next section.
First let K €JC" be a convex body, let p>0 and r)e7i{S). In
Section 4.1 we defined the local parallel set Mp{K,rf), and equation
D.2.4) states that
^"(Mp(A:, r,)) = -2 P"-""! " ]0miK, V).
« m=0 \ '" /
In order to obtain a Steiner formula such as the above for non-convex
sets K € U(X") also, we consider, instead of the parallel set Mp{K, vj),
its characteristic function Cp{K,r), •) and extend this additively to
U{1C"). The value Cp{K,r), •) is then interpreted as counting the
multiplicity with which the point x belongs to the parallel set of K with
respect to r].
Recall that the Euler characteristic x 'S the (unique) valuation on the
convex ring U{X") satisfying j^K) = \ for K&X"; see Theorem
3.4.12. Now, for K € UiX") and points q,x e^" we define the index
of K at q with respect to x by
i{K, q, X) : =
1 - lim limx(K n B(^x, \x - q\ - e) n B(q, 6)) if q e K
^ \i q i K.

220 Curvature measures and querma&sintegrals
If K is convex, then clearly
^(^'^'^> = (o otherwise. ('•'•^>
To prove the existence of the limits, choose a representation K =
U'=i^^, with Ki € JL". Without loss of generality, we may assume that
q & Ki for /=!,...,/« and q i Kj for / > m, where 1 ^ m ^ r. We
can choose 60 > 0 such that Ki H B{q, 6) = 0 for / e (m + 1,. . . , r}
and 0 < 6 < 60. If 6 < 60 is fixed, then for all sufficiently small e > 0 the
inequality
K^ n B{x, \x - q\- e)r\ B^g, <5) + 0
holds for all v e S(m) with i(K^, q,x) = 0 (where we use the notation
introduced in Section 3.4). In fact, for v e S(m) we have q e K„, and if
j(K^,q,x) = 0, then D.4.1) implies that K^ and hence K^nB(q,d)
contains a point whose distance from x is smaller than that of q. For
sufficiently small e > 0 we thus have
JiK„ q,x) = l- xif^v n Bix, \x - q\- e)r\ B{q, S)).
Using the additivity of x, we deduce that
\- XiKn B(x, |jr - ^1 - e) n B^q, 6))
= S i-\)'''^-'[l - X(K, n Bix,\x - q\ - e) n B(q, 6))]
veS{m)
= 1 i-lt^-'KK,,q,x)
veS{m)
= 1 i-lt^-'KK,,q,x).
The right-hand side does not depend on e or 6, which shows the
existence of the limits.
The index function / thus defined is additive in its first argument, that
is, for fixed q,x eE" and for K, L e U^X") we have
KK U L, q, x) + /(A" n L, q, x) = /(A", q, x) + JiL, q, x).
This follows from the definition; in fact, for q i K U L both sides are
zero and for ^ e A n L it follows from the additivity of X- If q e K\L
(and similarly for q e L\K) one has to observe that (K U L) n
B{q,d)= K n B{q, 6) for all sufficiently small 6 > 0.
Now for K € ulx"), r) € ^B), p > 0 and x e E" we define
Cp{K,r),x):-= E JiK n B^x, p), q, x)

4.4 Additive extension 221
where x - q ■= (x - q)/\x - q\. Actually, the sum is finite, since
j(K,q,x)j=0 for K = U,-=i/l,- with KieJC" implies, by the additivity
of ;(' ,q,x), that there is some v e S{r) for which ;(/l „, q,x) ^0 and
hence q = p(K^,x).
If K is convex, then D.4.1) obviously implies that
1 ifO < d(K, x) ^ p and (p(K, x), u{K, x)) e rj.
.^ ,_|l ifO<d(A
Cp(A:,r?,Ar)-|p otherwise.
Thus Cp(K,i], •) is the characteristic function of the local parallel set
Mp{K, T}). From the additivity of j(- ,q,x) it follows that Cp(- ,tj,x) is
additive on U^X"). Hence, for K = [JUiKi with /C, e ^T" we have
CpiK,r,, ■)= 2 (-l)l"l-'cp(A:„ r?, •)•
Since the right-hand side is a finite sum of integrable functions, we can
define
HpiK,r,) ■■= j^CpiK,r,,x)d^,"(x)
for K e U(X") and r)e^{E). The notation is consistent with that of
Section 4.1, and Hp{ •, r;) is an additive function on U{JC"). Hence, for
K = \}U\I^i with Ki eJC" we have
np(K, r?) = 2 (-i)|''|-Vp(a:., n)
yeSfr) « m = I V'" /
Since the left-hand side depends only on the point set K and not on its
chosen representation as a finite union of convex bodies, the same is
true for the coefficients of the polynomial in p on the right-hand side.
Hence, we are now in a position to define
&n,iK,T]) := 2 (-1)"'I-'0™(A:., r?)
veSU)
for m = 0,. . . , n - 1. Thus 0m(^K, •) is a finite signed measure on
C>SB). From this representation (or from the additivity of Hp( -, rf)) we
deduce that ©„( •, r;) is additive on U{JC"). In this way, we obtain the
(unique) additive extension of the generalized curvature measures to the
convex ring, and for the generalized parallel volume fip(K, rj) we arrive

222 Curvature measures and quermassintegrals
at the Steiner formula
H,iK, r,) =: ^2 P"-^ "]e„iK, r,).
As in Section 4.2, we specialize the generalized curvature measures
by putting
C^iK, p) = —— 0„(A:, /3) = 0„(A:, /3 X 5"-'),
(:)
5^(a:, oj) = ^^^ •?„(/<:, oj) = 0„(a:, e" x w)
(:)
for Ke t/(X"), Borelsets/3e7J(E"), we'SCS""') and for m = 0,. . . ,
n - 1; further
<I>„(A:, /?) := X"{K n /3).
Of these additive extensions of the curvature measures and area
measures to the convex ring, it will mainly be the signed measures C„
(or <P„) that will be used in the sequel. We shall first extend the
interpretation of C„_i given by D.2.23).
Theorem 4.4.1. If K e U{X") is the closure of its interior, then
C„-i(A',/3) = 5ry"-i(/3n bdA")
/or/3e3(E").
Proof. Let K = Uf=i^, with /C, ^JC". If among the convex bodies K^,
. . . , Kr there were one of dimension less than n not covered by the
union of the others, then K would not be the closure of its interior.
Hence, lower-dimensional bodies in the representation may be omitted,
and we can assume from the beginning that dim /C, = n for i = 1, . . . , r.
For LeJC", let (f{L, •) denote the characteristic function of the
boundary of L. Let x ebAK. If, say, x e Kj precisely for f = 1,... , m,
then X ebdKv for v € 5(m) and hence
2 {-\t\''<fiK,,x) = 2 (-1)''''"' X 1 = 1-
veS(r) oeSim)
Let j3e^(E") and v e S(r). Theorem 4.2.5 together with the remark
after its proof tells us that C„-i(/l„, jS) = ^"-'(/3 n bd/C,,),
provided that dim /C„ 4= « ~ 1- If dim A:„ = n - 1, then relint K^, C int /C,
hence C„_,(/:„,/3n bd/C) = 0 = :^"-'(i3n bd/C n bd/:„). Further,
C„_i(/l, •) is concentrated on the boundary of K, since

4.4 Additive extension 223
Cp(/:,/3x5"-', •) = Cp(K,{pnbd K) x S"-\ •) by the definition of
Cp. For /3 e ^(E") we conclude that
C„_,(A:,/3) = C„-i(K,pnbdK)
= 2 (-i)i''i-'c„_i(A:„,/3nbdA:)
V€S(r)
= S (-i)'"'''^""'(/3n bdA" nbdA-^)
= 5J5"-H/3nbdA'). ■
For convex bodies K e JC", formula D.2.21) interprets the curvature
measure Cq as the content of the spherical image, namely
Co{K, /3) = X"-'{o{K, 13)).
If we want to extend this representation to the elements of the convex
ring, we have to count the points of the spherical image with a suitable
multiplicity. For this, we use a similar approach to the above and
introduce another index function. For K e U{JC"), a point ^ e E" and a
vector M e 5""' we define
i(K, q, u) := ■
1 - limlim;t(/: n B{g + F + e)u, 6)) iig e K,
0 iig i K.
If K is convex, then clearly
., „ ^ [l if (9, u) e Nor/iC, /^ ^ o-,
'(^' ^' "> = @ otherwise. ("•"•^^
As for the index function j{K,q, •) one sees that i{K,q,u) is well
defined and that f( • ,q,u) is an additive function on the convex ring
U{X"). For K e U(JC"), /3e ^(E") and u e 5""' we define
c{K,l3,u) := 2i{K,q,u).
Here we have to allow infinite values, but these can be neglected: Let
K = [JUiKi with Ki eX". If u e HUiKgn Ki and v e S{r), there is
at most one point q for which {q, u) e Nor K„ and thus /(/C„, q, u) =}= 0.
It follows from Theorem 2.2.9 that c(K,P, ■) is finite 5i5""'-almost
everywhere on 5""'. If K is convex and u e regn K, then D.4.2) implies
that c(K,P, ■) is the characteristic function of the spherical image

224 Curvature measures and quermassintegrals
a{K,p), hence D.2.21) can be written as
Co(A:, /3) = /^„.,c(A:, /3, u)AX"~\u). D.4.3)
Since both Co( •, /3) and c( •, /3, w) are additive on UQC"), the equality
extends immediately to sets K of the convex ring. This is the desired
interpretation of the curvature measure Cq{K, •) as the 3^""'-integral
of a multiplicity function for the spherical image.
Finally we remark that
— Co(A:, E") = V<,{K) = xiK) D.4.4)
<a„
for K e U(y."). This is true for convex bodies, and the general case
follows by additivity. In the same way, a corresponding result for the
index functions is obtained. This is in analogy to well-known index sum
formulae for smooth submanifolds. For K e U{X") and x e E"\/l we
have
2 KK, q, X) = xiK), D.4.5)
and if M e 5""* is regular for K, which means that u e n'=iregn Kj for
some representation K = \J'=iKj with Kie X", we have
2 i{K, q, u) = xiK). D.4.6)
Notes for Section 4.4
1. Index functions. The additive extension of the generalized curvature measures
to the convex ring by means of an index function was carried out by
Schneider A980a). This had previously been done for the curvature measure
Co in Schneider A977d), extending some work of Hadwiger A969c) and
Banchoff A967) for polyhedra and cell complexes. The index UK, q, u) is
also used in Zahle A987b).
As mentioned at the end of Section 4.4, the index functions that were
introduced are analogues of index functions known in differential geometry.
Calling the point q a critical point oi K & U{X") with respect to the height
function (•,«) if i(K,q,u)^Q, one may consider equality D.4.6) as an
analogue of the critical point theorem for smooth submanifolds. Similarly,
D.4.4) is an analogue of the Gauss-Bonnet theorem.
Suppose that K e UiX") is the point set of a polyhedral cell complex of
which 4* is the set of ^-dimensional cells. Then the index i(K, q, u) satisfies
n
i(K, q, u) = S(-l)* 2 'B- 9. -")• D.4.7)
This was proved by Shephard A968c) for the special case of the boundary
complex of a convex polytope, and it can be extended to the general case by
means of an argument due to Perles & Sallee A970). A definition equivalent

Notes for Section 4.4 225
to D.4.7) was used by Banchoff A967, 1970) in his investigation of critical
point theory, curvature and the Gauss-Bonnet theorem for polyhedra.
Absolute curvature measures on the convex ring. If the additivity assumption
is dropped, other extensions of the curvature measures to the convex ring are
possible. Of particular interest are non-negative, or absolute, curvature
measures. A non-negative extension of Federer's curvature measures to the
convex ring was proposed by Matheron A975), pp. 119ff. His construction
was extended in Schneider A980a), as follows. For given K e U(X") and
X e E", a point q eE" is called a projection of x in K if q e K and there
exists a neighbourhood N of q such that |jic - yl > |;ic - q\ for att y e K D N,
y "^ q. The set n(K, x) of all projections of jic in /l is finite. For 7} e '3(X)
and p > 0 let
Cp(K, T], x) := card(g e n(K, x)\0 < \x - q\S p and (q,x - q) e t])
and
{K,r,):=\^c,iK,n,x)dX''(x).
One can show that iip(K, t]) satisfies a Steiner formula, that is, a polynomial
representation
=0
and that this defines (positive) measures ©(,(K, •),..., &„_i(K, •) on
•3(r), which for K e X" coincide with respectively &o(K, •),...,
&„-i(K, •). The specialization
C„(K, fi) := e„(K, |8 X 5"-'), ^ e 7i(E"),
yields the measures introduced by Matheron.
The measure Co(K, -) can also be interpreted as follows. For K e U(X")
and X ebd K, a unit vector u e 5""' is called a normal vector of K at x if
there exists a neighbourhood N of x such that (x,u)^ {y,u) for all
y e K n N. For fi e ^(E"), let c(K, fi, u) be the number (possibly infinite)
of points X e K r\ fifoT which m is a normal vector. Then
Co(K, p) = j^„.S(K, p, M)d5K"->(w).
Thus Co(K, •) can be interpreted as the measure of the 'spherical image with
multiplicity'. The normalized total measure Co(K,E")f(o„ was called the
convexity number of K by Matheron A975). It is equal to one for convex
bodies, but not only for these.
Other generalizations of curvature measures. The treatment of curvature
measures given in this section was restricted to convex bodies and the sets of
the convex ring. Its extension to more general classes of sets is possible, but
requires in general deeper methods from geometric measure theory. We give
only some brief hints to the literature. Federer's A959) curvature measures
for sets of positive reach were further extended and investigated by Zahle
A984a,b, 1986a, 1987a,b). The extension is to finite unions of sets of positive
reach and later to so-called second-order rectifiable sets. Applications to cell
complexes and mosaics appear in Zahle A987c, 1988), Weiss A986) and
Weiss & Zahle A988). Fu A989) relates curvature measures to generalized
Morse theory.
The approach to curvature measures via so-called normal cycles, as begun
by Wintgen A982) and Zahle A986a, 1987a, b), is further extended in work
by Fu A990+a,b). General versions of absolute curvature measures were
introduced by Baddeley A980) and Zahle A989).

226 Curvature measures and quermassintegrals
Further different approaches have been followed by Stach6 A979), who
constructed generalizations of Federer's curvature measures to arbitrary sets
(but with weaker properties), and by Kuiper A971), who used singular
homology to define a sequence of curvature measures, which includes a
generalization of Cq.
Curvature measures for (non-convex) polyhedra were considered by
Flaherty A973). A thorough study of curvatures for piecewise linear spaces
was made by Cheeger, Miiller & Schrader A984, 1986). The latter paper is
also related to some (more general) work of Zahle quoted above. Part of the
work of Cheeger, Miiller & Schrader can be simplified; see Budach A989)
and also the remark at the end of Section 2 in Schneider A990c).
Approximation results, of different kinds, for curvature measures appear
in Brehm & Kuhnel A982), Cheeger, Muller & Schrader A984), Lafontaine
A987) and Zahle A990).
4. Curvature measures in spaces of constant curvature. Kohlmann A988)
introduced curvature measures for sets of positive reach in Riemannian space
forms. Among other results, he obtained integral representations, Minkowski-
type integral formulae (see Section 5.3, Note 3) and characterizations of balls
(see Section 7.2, Note 4).
5. Groemer's extension of the quermassintegrals. The following extension of the
quermassintegrals, which apparently does not generalize to curvature
measures, was proposed by Groemer A972). He introduced a vector space
A" of real functions on E" with a pseudonorm such that A" contains VifP")
(see Section 3.4, Note 7) as a proper dense subspace. The elements of A" are
called 'approximable' functions. The system Sa of subsets of E" whose
characteristic functions are approximable contains the convex ring UQL")
and, for example, the relative interiors of convex bodies. Groemer showed
that the quermassintegrals can be extended from JL" to continuous linear
functional on A". In particular, this yields an additive extension of the
quermassintegral Wi to the class Sa-
4.5. Integral'geometric formulae
Some integral-geometric formulae can be proved for curvature measures
and area measures. Generally speaking, integral geometry is concerned
with the computation and application of mean values for geometrically
defined functions with respect to invariant measures. In this section we
shall treat local versions of some intersection and projection formulae,
which in their global versions, namely for the Minkowski functional,
are central results of classical integral geometry. We shall also consider
mean value formulae for Minkowski sums, and counterparts to the
intersection formulae in the form of kinematic formulae involving the
distances between non-intersecting bodies, or between bodies and flats.
This yields, among other results, an integral-geometric interpretation of
generalized curvature measures.
We assume that the reader is familiar with the topological groups
SO{n) of proper rotations of E" and G„ of rigid motions of E", and
with their Haar measures. We normalize the Haar measure v on SO{n)

4.5 Integral-geometric formulae 227
by imposing that v{SO{n)) = 1. Every rigid motion g e G„ determines
uniquely a rotation p e SO{n) and a translation vector f e E" so that
gx = px + t for X eE", and we write g = g,f,. The map
y: E" X SO{n)-* G„
{t,p)^ g,,p
is a homeomorphism (products are always equipped with the product
topology). We can define a Haar measure fi on G„ by the image
measure
H := y(A„ (8) v),
where A„ denotes the restriction of 5J5" to the Borel sets of E"; this
implies a definite normalization of fi. The measure fi, defined on the
a-algebra ^(G„) of Borel subsets of G„, is left invariant and right
invariant, thus the motion group is unimodular.
On A{n,k), the set of /c-dimensional affine subspaces of E", where
/c e {0, . . . , n), the motion group G„ operates in the natural way,
making A{n, k) into a homogenous space. To facilitate the handling of
its invariant measure, we choose a /c-dimensional linear subspace Ek of
E" and denote its orthogonal complement by E^. The map
y^: Ei X SO{n)-* A{n, k)
(t,p)^ p(Ek + t)
is surjective. The usual topology of A{n, k) is the finest topology for
which Yk is continuous. With this topology, A{n, k) is a locally compact,
second countable Hausdorff space, the natural operation of G„ on
A{n, k) is continuous and A{n, k), with this operation, is a
homogeneous G„-space. The image measure
Hk := Yki^n-k ® v)
of the product measure A„_^ ® v, where A„_^ is the restriction of :^""*
to the Borel sets of Ej;, is a G„-invariant measure on A{n, k) with a
suitable normalization. It does not depend on the special choice of Ek-
In a similar, even simpler, way we could obtain the 50(n)-invariant
measure v^ on the compact homogeneous space G(n, k) of /c-dimen-
sional linear subspaces of E", but we shall not need it in what follows.
Below, we shall sometimes need to know that certain sets of
rotations, rigid motions or flats are of measure zero. Some of these facts
follow from the results of Section 2.3; a more elementary one is
contained in the following lemma.
Two linear subspaces £, F of E" are said to be in special position if
lin(£: U F) + E" and E f) F^ {o}.

228 Curvature measures and quermassintegrals
Lemma4.5.1. Let E eG(n,p), F e G(n, q), 0^ p, q ^ n - 1. The set
of all rotations p € SO{n) for which E and pF are in special position is
of v-measure zero.
To avoid computations requiring an explicit description of the Haar
measure v, we give a proof that uses only its invariance and finiteness.
Proof of Lemma 4.5.1. For given E, F let X be the set of all p e SO{n)
for which E and pF are in special position. We prove the assertion
v{X) = 0 by induction on p. The case p = 0 is trivial; we assume that
p ^ 1 and the assertion has been proved for dim E < p. Let dim E = p.
We choose a linear subspace U e G(n,n - p + I) and denote its
orthogonal complement by U-^. Further, we choose a number /c e N
and k vectors Wj,. .. , w^t € f/ of which any n-p + l are linearly
independent. Put £, := lin{U^ U {«,}); then dim£, = p. Let A", be the
set of all pe SO{n) for which E, and pF are in special position. As
E; =PiE for suitable p, e SO{n), we have X; = PiX and hence
v{Xi) = v{X). Let Y be the set of all peSO{n) for which U^ and pF
are in special position. Since dim f/-"- = p - 1, we have v(y) = 0 by the
inductive hypothesis. We assert that each pe SO(n)\Y lies in at most
n - p of the sets ^,.. ., Xk. Let pe SO{n)\Y and, say, pe A", for
1 = 1, . . ., r. Thus Ei and pF are in special position for i = 1, . . . , r.
In particular, lin(£, U pF) 4 E" and hence projupF^U, since
U^ <Z Ei. U^ and pF are not in special position since p$Y; however,
Ei and pF are in special position. This is only possible if w, e proji; pF.
Since the vectors wi,. .., M„_p+i are linearly independent and dim
projy pfS n - p, we deduce that r"^ n- p. Let ^, be the
characteristic function of A", on 50(n). We have proved that Sf=iS.(P) = n - p
for pe 50(n)\y. Since v(y) = 0, integration yields
kviX) = SH^.) = (,„, ,ES,(P)dv(p) ^ n - p.
1=1 •'iO(n)i=l
Letting /c -»<», we conclude that v(X) = 0. ■
Our first major goal in this section is the proof of the following
theorem.
Theorem4.5.2. Let K, K' € UQC") be sets of the convex ring, let /3,
/3' € 3(E") and] € {0, . . . , n}. Then
\ ^j(K n gK', p n gp')M8) = I.o'nik'PdK, /3H„+^.,(a:', /3')
D.5.1)

with
Oinjk =
4.5 Integral-geometric formulae 229
/*\ Jk + \\Jn + j- k + \\
.-J- i'¥]W]
[k-
The special case /3 = /3' = E" reduces to what Hadwiger [18] (with
different notation) called the 'complete system of kinematic formulae',
namely
I V,iK n gK')Mg) = S a„i,V,(K)V„^j-,(K'). D.5.2)
The further specialization / = 0 can be written in the form
\ XiKngK') dn(g) = —1 -T-n— V,{K)V„^,{K'). D.5.3)
U;
This result is known as the 'principal kinematic formula' (for convex
bodies). It expresses the total measure of the set of all rigid motions g
for which the moving body gK' meets K in terms of the intrinsic
volumes or in terms of the Minkowski functional of K and K'.
The proof of Theorem 4.5.2 is divided into several steps. As a
preliminary, we remark that it suffices to prove the theorem for convex
bodies K, K'. In fact, if D.5.1) is true in this case, then one can utilize
the fact that both sides of D.5.1), as functions of K, are additive on
U(X"). By additivity, D.5.1) remains true if K is replaced by a finite
union of convex bodies. In a second step, K' can similarly be replaced
by a finite union of convex bodies. Hence, from now on we assume that
K, K'eX".
First we have to show that the integrand in D.5.1) is, in fact, a
measurable function of g. For this, let K, K' e OC" and /3, /3' e 7i(E")
be given. Let X{K, K') be the set of all rigid motions g e G„ for which
K and gK' cannot be separated by a hyperplane. For given e>0, we
define a function h: G„-^Rby
Kg) := U,{K n gK', p n gp'),
where f/,(L, a-) := ^,(L, a-x 5""') = :^"(/l,(L, a)) for LeX",
are^(E") (jUj was defined in Section 4.1), and we define a function
f:X{K,K')xE"-^U by
fig, X) := tp{xKpig-'x)UAK n gK', E"),
where ^a 'S the characteristic function of a on E". The function
{g,x)i-> g'^x is continuous on G„ x E" and the function
g^U^iKn gK',E") is continuous on X(K,K') by Theorem 1.8.8
and by the continuity of [/e(-,E") (which follows from Theorem

230 Curvature measures and quermassintegrals
1.8.16). Since t/s. tp' are measurable on E", the function / is
measurable. Let
P{g, a) := UAK n gK', a)/U,{K n gK', E")
for g € X(K, K') and a e ^(E"). Since the function g>-^ KOgK' is
continuous on X(K,K') and Uc(-,a) is measurable on X" by
Theorem 4.1.2, it follows that P( •, or) is measurable on X{K, K'). For
fixed g, P{g, •) is a probability measure on ^(E"). Since, for
geX(K,K'),
Kg) = j^Spn,pix)U,(K n gK', dx)
L
J(g,x)P(g,dx),
it follows (e.g., Neveu 1969, p. 95) that h is measurable on X(K, K').
Now /i(g) = 0 if Kr\gK' ^Z, and the (closed) set M of all
g € G„\X(,K, K') for which /C n g/C' + 0 satisfies
JSO(n)J T(p)
where r(p) is the set of all t e E" for which pK' and /C meet but can
be separated by a hyperplane. It is easy to see that T{p) =
b(i{K - pK') and hence fJt{M) = 0. Thus h is measurable. Since e > 0
was arbitrary, we deduce from D.2.5) that the function g>->
4>j{K n gK', /3 n g/3') is measurable if / e {0,. .. , n - 1}; for j= n this
is easily seen directly.
Now we enter into the essential part of the proof of Theorem 4.5.2,
by writing the integral of D.5.1) in the form
IG
<Pi{KngK',i3ngi3')dfi{g)
= Lo(„)/p*^'(^ ^ ^P^' + '), /3 n (p/3' + f))df dv(p). D.5.5)
This is possible by the description of the invariant measure n given
initially and by Fubini's theorem (this argument was already used in
D.5.4)). We first consider the inner integral for the case of polytopes,
where an explicit computation is possible.
Some preparatory definitions are needed. First, for ACE" and
JT e E" we often abbreviate A + x by Aj,- For linear subspaces L, L' of
E" we define a number [L, L'] as follows. We choose an orthonormal
basis of L n L' and extend it to an orthonormal basis of L as well as to
an orthonormal basis of L'. The resulting vectors span a parallelepiped

4.5 Integral-geometric formulae 231
of dimension dimlin(L U L'); let [L, L'] be its n-dimensional volume.
This definition does not depend on the choice of the bases. Next, let P,
P' e ^P" be convex polytopes, let F be a face of P and F' a face of P'.
By L{F) := aff F- aff F we denote the linear subspace parallel to aff F.
Then we define [F, F'] := [L(F), L(F')]. We also define an external
angle
y{F, F', P, P') := y{F n F',, P n P;),
where x e E" is chosen such that relint F n relint Fi =(= 0; this definition
does not depend on the choice of x. Finally, we write
Af (/3) := X'^^'^iF n /3) for /3 e 3(E"),
so that Ayr is a measure with support F.
The faces F, F' are said to be in special position if the linear
subspaces L(F), L (F') are in special position.
As an intermediate result, we state the following 'principal translative
formula' for polytopes.
Theorem 4.5.3. Let P,P'e'p" be polytopes; let /3,/3'e (SCE") and
j e {0, . .. , n}. Then
j^^4>i(P np',,pn p',)dx = <Pi(P, p)<Pn(P', P') + <PniP, P)^i{P\ n
+ "i E E y(F, F', P, P')[F, F']AK/3)Ap(/3').
Proof. By D.2.17) we have
/ :=/^„«i>,(pnp;,/3n/3;)dAr
= L E y(F, pnp;)AK/3n/3;)dx.
In computing this integral, we may neglect those translation vectors x
for which a /c-face F of P meets an m-face F^ of Pi with /c -i- /w < n or
with F, F' in special position, since
{x e E"|Fn F':, ^0} = F - F'
is of measure zero in each case. As the vectors x with F n Fi =|= 0 but
relint F n relint Fi = 0 also form a set of measure zero, we may assume
that each /-face of P Pi P; is the intersection of a /c-face of P and an
{n + J - /c)-face of Pi, for some /c e {/,..., n}. It follows that

232 Curvature measures and quermassintegrals
1=11 1 IjiF r>F',,pn p;)AfnF;(/3 n /3;)dx
n
= 11 1 y(F, F\ P, P')J(F, F\ 13, /3')
with
7(F, F', /3, /3') := j^MnrSP n /3;)d;t.
We fix k e {],... ,n}, F e <7^kiP), F' e G^„+j-k{P') and write
J{F, F',P,P') = J. For the computation of 7 we may assume that
F and F' are not in special position (otherwise 7 = 0) and that o e
affFDaffF'. We put L{F) n L{F') := Li, Lj" n L(F) := L2,
Lf r\ L{F') '= Li and write x e E" uniquely in the form
JT = jTi + jr2 + ^3 with Xj e L, (i = 1, 2, 3). Writing Fr\ P= a,
f n 13' = a' we obtain, using the definition of [F, F'],
J = [F, 'P'l/JJ^^^'C* n a',)dXi{xi)d^j'-i(x2)dX"-'(xi).
Since (or D ori) — XjC L^, we obtain
f 555'(ana;)d^;'(x,)
= f X>{{a - X2) n (a-' + Xi + xi) n Li)dX>{xi)
= 555'((a- - X2) n Li)^'((a-' + xj) n Li),
by Fubini's theorem. Also by Fubini's theorem,
f X'(a n (Li + X2))d:^*-'(X2) = a5*(a-),
f a^'(a' n (L, - X3))d5J5"-*U3) = 5iJ"+'-*(a'),
thus
7 = [F, F'lAK/3)Af.(/3').
If F' e '7^„(P'). we have y(F, F', P, P') = y(F, P), [F, F'] = 1 and
A,.. = <J>„(P', O.thus
1 1 y(F, F\ P, P'O(F, F\ p, n = ^i^P. P)<PniP', /3')-
This holds trivially if dim P'<n, since both sides are then zero. A
similar remark concerns the case F e^„(P). This completes the proof
of Theorem 4.5.3. ■
To perform the outer integration in D.5.5) for the case of polytopes,
we need the following mean value formula.

4.5 Integral-geometric formulae 233
Lemma 4.5.4. Let P, P' e 'p" be polytopes, j e {0,. . . , n -2}, k e
{/ + 1,..., n - 1}, F e <7k{P), F' e r7„^,_,(P'), /3, /3' e ^(E"). T/ien
f y(F, pF', P, pP')[F, pF')dv(p) = <x„i,y{F, P)y{F', P')
with a real constant a„jk depending only on n, j, k (its explicit value is
given by D.5.13)).
Proof. Since a direct computation of the integral appears difficult, we
proceed in an indirect way.
Denoting by o'" the normalized spherical Lebesgue measure on
m-dimensional great subspheres of 5"~\ we have y(F,P) =
o"-'''\N(P, F) n S"'^) by the definition of outer angles. By Lemma
4.5.1 we may neglect those rotations p for which F and pF' are in
special position. Hence, we may assume that
y(F, pF', P, pP') = a"-i-\[N{P, F) + pN^P', F')] n S"-'),
where Theorem 2.2.1(b) was used. Now we generalize the integral to be
determined. For (t)CS"~\ let C{(t)) := {hc\x e (o, X^O) be the cone
spanned by co. For arbitrary Borel sets (oC L(F)^ r\ S"~^ and
CO' C L(F')-^ n S"'^ we write
7@), CO') := [ a"-'-'([C(a)) + pC(co')] n S"-'}[F, pF']dv(p)
JSO(n)
(the measurability of the integrand is proved in a standard way). First
we fix CO' and consider the function 7( •, co'). For v-abnost all p, C(co)
and pC{co') lie in complementary subspaces. For such p.
[#-)
+ pCico')
n 5"-' = \JC{co„) + pC(co')] n 5'
■n-l
for any sequence (co„)„^fq of Borel sets in L(F)"'" n 5""'. If the sets of
this sequence are pairwise disjoint, the union on the right is disjoint up
to a set of a""'"^-measure zero. From the a-additivity of a""'~' and
the monotone convergence theorem it follows that 7( •, o)') is a finite
Borel measure on the sphere L{F)^ n S"'K A rotation t of 5""^ that
carries L(F)-^ into itself and keeps L{F) pointwise fixed satisfies
C(to)) + pC(co') = t[C(o)) + T-'pC(ft)')] and [F, pF'] = [F, T~^pF'];
hence the invariance of a""'~' and v implies that J{rco, co') = 7(a), co').
Thus J{- ,ca') must be a constant multiple of the invariant measure on
the sphere L(F)^ CYS"'^. Interchanging the roles of co and o)', we
arrive at
7@), CO') = a„iko"-''-\co)o''-'-\co')

234 Curvature measures and quermassintegrals
with a constant a„jk that evidently depends only on n, /, k. The
particular choice o) = N{P, F) n S"'\ co' = N(P', F') n 5""' now
yields the assertion of the lemma. ■
If we now put equality D.5.5), Theorem 4.5.3 and Lemma 4.5.4
together, we arrive at formula D.5.1) for the case where K, K' are
polytopes. The coefficients are those of Lemma 4.5.4.
For general convex bodies. Theorem 4.5.2 will now be proved by
approximation. For given g e G„, we define a map Tf E" -* E" x E" by
Tg{x) ■= {x,g~^x). For convex bodies K,K'eX", we denote by
(pj{g, K, K', •) the image measure of <Pj(K CtgK', •) under the
(continuous) map Tg-, then
cp^ig, K, K', /3 X /3') = <PiiK n gK', p n gp')
for /3, /3'e3(E"). The set cd of all a-e^(E"xE") for which
q>j{ •, K, K', a) is measurable is a Dynkin system. Since c4 contains the
product sets /3 x/3' (/3,/3'e 3(E")), it contains the Dynkin system
generated by these products, which is equal to the generated a-algebra
:8(E" X E"). Thus (pj{- ,K,K',a) is measurable for each ae
7i(E" X E"), and we can define a measure on 3(E" x E") by
<Pi(,K,K', ■):=\^<p;{g,K,K', OdMg).
This measure is finite since
(Pi{g, K,K', ■)^ VjiK n gK'). D.5.6)
In JC" we consider convergent sequences Ki-^K, K'^-^K'. For
jU-almost all g, either K O gK' = 0 ot K and gK' cannot be separated
by a hyperplane, which by Theorem 1.8.8 implies that KiOgK'i-*
K n gK for /-* 00. For these g, it follows from the weak continuity of
the curvature measures that
liminf<l>/A:, n gKU T'g'ia)) s <p(K n gK', r-'(a))
if or C E" X E" is open and hence also Tg^(a) is open. Integration with
respect to g and application of Fatou's lemma yields
Wminf (pj{Ki, K'i, a) ^ (pj{K, K', a).
Similarly, we obtain
Mm (pj(K:, K\, E" x E") = cpji^K, K', E" x E")
if we use the bounded convergence theorem, which can be applied
because of D.5.6), and the fact that VjiKiD gK'i)-^Vi{K n gK') for

4.5 Integral-geometric formulae 235
jU-abnost all g. Thus we have proved that
(PjiKi, K'i, •) ^ (pj{K, K', ■) for / -^ oo.
This can be used to show that
q>i(.K, K', ■)= ta„j,4>,(K, •) (g> <P„^i-,(K', •) D.5.7)
k=j
if K, K' eJC". In fact, both sides are finite measures on '2(E" x E"). If
K, K' are polytopes, both sides are equal on product sets /3 x /3'
(/3, /3' e 3(E")), since D.5.1) holds for polytopes, and thus are equal on
an intersection stable generating system of -^(E" x E"); hence they are
equal on 3(E" x E"). Since both sides are weakly continuous functions
of {K, K'), approximation by polytopes shows that D.5.7) holds for
arbitrary convex bodies K, K'. Formula D.5.1) is the special case
obtained by applying both sides to /3 x /3'. This completes the proof of
Theorem 4.5.2, except that we still have to determine the explicit values
of the constants a„jk.
Before we compute the numbers *„;<., it is convenient to deduce from
Theorem 4.5.2 another integral-geometric formula for the case where
the moving convex body gK' is replaced by a moving flat.
Theorem4.5.5. Let K e U{X") be a set of the convex ring, let
/3e 0?(E"), A: e {0, .... n} and/e {0, ...,k). Then
f ^ 0y(A: n E, /3 n E)dM*(E) = *„,,<!>„+,-,(/<:, /3)
(where a„jk is the same number as in Theorem 4.5.2).
The special case /3 = E" gives
,^^ Vi(^K n E)dfikiE) = a„i,V„^j_,{K). D.5.8)
These equalities are called Crofton's intersection formulae. In particular,
one has
I XiK n E)dn,(E) = a„okV„-,(K), D.5.9)
JA{n.k)
and if K is convex, this gives
a„okV„-k(K) = n^aE e A(n, k)\K n £ =(= 0}). D.5.10)
We note also the special case
a„o(n-i)<Pi(K, p) ^ \^^^^_,4>o{K n E,l3n £)dM„-y(£) D.5.11)
giving a mean value interpretation of the curvature measure <Pj in terms
of <Po, which has a simple intuitive meaning.
L

236 Curvature measures and quermassintegrals
Proof of Theorem 4.5.5. We choose a A:«dimensional linear subspace Ek
of E" and recall that the invariant measure n^ can be defined as the
image measure of A„_4 ® v under the map
Yk- Et X SOi^n) ~* A{n, k)
it, p) -^ p{Ek + t),
where k„-k is the restriction of 35"~* to 3(Et)- Similarly, we denote
by Xk the restriction of ^* to ^{Ek).
We choose a bounded Borel set aC Ek with A^(ar) = 1 and consider
the integral
J := j^ 4>>(£* ngK,an gl3)dn(g).
We can choose a convex body K' C Ek such that a C relint ^' and that
every g e G„ with g/C D a- + 0 satisfies Ek^ gK = K' r\ gK.
Therefore, the integral J can be computed by means of Theorem 4.5.2, and
we obtain
n
Because orCrelint/L' we have ^,{K',a) = Xk{a) = \ if / = A: and 0
otherwise; hence
J = a^jk'Pn^i-kiK, 13). D.5.12)
In the following, we write f e E" in the form t = ti + t2 with tie Ek
and 12^ Ek- We find
J = J50(„Jp*^>(^* "^ (^^ + f), a n (p/3 + f))dfdv(p)
= L(„)L Je.*^'(^* "^ (^^ ^ '1 ^ '^)' * -^ (p^+'1 + '2))
X dAi(f2)dA„_4(fi)dv(p)
= Lix50(„)k*^'<^* "^ (^^ ^ '' ^ '^)' * -^ (P^ + '. + '2))
xdAfc(f2)d(A„_i ® v)(fi,p).
For the computation of the inner integral, put
<Pj(Ekn{pK + fi), ■)=■■ <p,
p/3 + fi =: /3';
then
4>y(£:i n (pK + ti + t2), or n (p/3 + fi + 12))
= 4>,(£fc n (pA: + r,), (a - fj) n /3')
= (p((ar - t2) n /3').
Denoting the characteristic function of ACE" by ^^, we obtain

4.5 Integral-geometric formulae 237
|^^4>y(Efc n {pK + f, + t2), an{pp+ ti + f2))dAfc(f2)
= /£^<p((*- f2)n/3')dA,(f2)
= IeJv^"^'' ^ '2)t^'(Ar)d(p(Ar)dA,(f2)
= /j£^t»-.(t2)t^'(Ar) dA,(f2) d(p(Ar)
= /,,t^'Wd<KAr) = <K/3')
= <!>,(£, n (pA: + fi), P/3 + t,)
= <Piip~\E, - f.) n a:, /3)
= «!>,(a: ny,(-f„p-i),/3).
This yields
Together with D.5.12) this proves the assertion of Theorem 4.5.5. ■
We are now in a position to compute the constants a„ik appearing in
Theorems 4.5.2 and 4.5.5. For the ball rB" of radius r > 0 we have by
D.2.27), for e > 0,
te"-'K„.,Vj{rB") = VMr + e)B") = i('!)r'£"-'^„,
hence
V.(rB") = l"]-^r>.
By Theorem 4.5.5 with /C = B" and /3 = E" we have

We deduce that
The identity
238 Curvature measures and quermassintegrals
Since K O p(Ek + f) is a A:-dimensional ball of radius A - |fp)^^, we
obtain
\^.Vj{K n p(£, + 0)dA„_,@ = (^] -^ L(l - M^)'/^dA„_,@
JEt \) I Kk-j J El,
( .VkKn+j-k
(Xnjk = ~i T • D.5.13)
w-ih"'
'if-) '^ ' ' ^^''
can be used to bring D.5.13) into the form given in Theorem 4.5.2. The
proof of Theorem 4.5.2 is now complete.
We come now to integral-geometric formulae of a different type. An
essential feature of the kinematic formulae of Theorem 4.5.2 and the
Crofton formulae of Theorem 4.5.5 is that they refer to mean values of
functions defined on the intersection of a fixed set and a moving set. We
will now replace intersection by other geometric operations. These are
the 'linear' operations of Minkowski addition and projection, which in
the context of the present book are of particular interest. For the rest of
this section, the investigations are restricted to convex bodies. First we
consider rotational mean values for area measures of Minkowski sums.
Theorem4.5.6. Let K, K' e X" be convex bodies, o), o)' e 3E""') and
;e {0, ..., n - 1}. Then
£^^^ 5,(A: + pK', CO n pa)')dv(p) = ~- ij^J5,(A:, co)S^-k{K', co').
D.5.14)
Here we prefer to work with the area measures 5, and not with their
renormalized versions f,, since in this way we can avoid complicated
coefficients.
Proof of Theorem 4.5.6. The proof of the measurability of the integrand
is similar to that for the integrand in D.5.1). For given e > 0, we define
V,{L, a) := n,{L, E" x a) = 3K"(S,(L, a))

4.5 Integral-geometric formulae 239
for L^X" and a-e ^E"-'),
Kp) ■= V,(K + pK', oi n po)')
for pe SO{n) and
/(p, u) := U")ta,(p-'")v^.(A: + p/i:'. 5"-')
for pe 50(n) and u e 5""^ where ^^ denotes the characteristic
function of A on 5""'. Then / is measurable. If we define
P(p, a) := K,(A: + pK\ a)IV,{K + pAT', S"'^)
for p e 50(n) and a e CE"~^), then P( •, a) is measurable and P(p, •)
is a probability measure, and from
Kp) = /^„.,ta,np.'(")V^.(A: + pA:', du)
we deduce that h is measurable. From D.2.5) we now infer that the
function p*-^ Sj{K + pK', wD pot') is measurable.
For the proof of D.5.14), we first consider the special case where
P, P' e 'V" are n-dimensional polytopes and j = n -\. Polytopes P, P'
are said to be in general relative position if no pair of faces F of P and
F' of P' is in special position.
Let p e SO(n) be a rotation for which P and pP' are in general
relative position. By Lemma 4.5.1, we thus exclude only a set of
rotations p of v-measure zero. Let G be a facet of the polytope
P + pP'; let u be the outward unit normal vector of G. Then
G= F + pF', where F = F{P, u) and pF' = F(pP', w). Let dim F = /
and dim F' = A:; then i + k = n - I since P and pP' are in general
relative position. By Theorem 2.2.1(a) we have
u e N{P, F) n A^(pP', pF').
Vice versa, if F is an /-face of P and F' is an (n - 1 - f)-face of P' with
N(P, F) n A^(pP', pF') 4= 0, then F -H pF' is a /-face of P -h pP' with
/ = n - 1. Again excluding a set of rotations of measure zero, we may
assume that ; = n - 1. Hence, for almost all p,
5„_i(P + pP', 0) n po)') = 2 ^""HG)
/v(P+pf",G)ntunpa>'+0
• =1 S S :^^"-'(F + pF')
,=0 Firfi(P) F'€r7„.,.,(f)
/i(p, F, F') + 0

240 Curvature measures and quermassintegrals
with
A(p, F, F') := [N{P, F) n 0)] n p[N{P', F') D co'].
Observing that, for the rotations and faces considered, the set
A(p, F, F') is either one-pointed or empty, we arrive at
f 5„_i(P + pP',a)npa)')dv(p)
JSO{n)
n-l .
= E E E i^, ,^"-\F + pf')cardyl(p, F, F')dv(p).
We fix le @,.. .,n-l}, FeG,iP) and F' eG„-i_i(P'). For
almost all p we have dim(F + pF') = n - 1 and hence
:^"-U^ + pF') = [F, pF']^'(F):^"-i-'(F').
It remains, therefore, to compute
I cardyl(p, f, F')[F,pF']dv(p).
JSO{n)
We argue in the same way as in the proof of Lemma 4.5.4, defining a
more general integral by
7(/3, /3') := f card(/3 n p/3')[F, pF'ldv(p)
JSO{n)
for Borel sets/3CL(F)^ D 5"-^ p' C L{F')^ nS"-K If re 50(n) is a
rotation that maps L(F)^ into itself and keeps L(F) pointwise fixed,
then card(T/3 D p/3') = card(/3 D t~V/3') and [F, pF'] = [F, t" V^'] and
hence 7(t/3, /3') = 7(/3, /3') by the invariance of v. Since 7( •, /3') is
obviously non-negative, finite and a-additive, it follows from the
uniqueness of spherical Lebesgue measure that 7(/3, /3') = ca"~'~'(/3) where
the constant c depends on /3'. In combination with the same argument
with the roles of /3 and /3' interchanged, this shows that
7(/3, /3') = a„,a''-i-'(/3)a'(/3'),
where the constant a„j evidently depends only on n and i. The special
case /3 = N{P, F) Hco, P' = N{P', F') D co' now yields
f ""'
5„_,(P + pP', 0) n pa)')dv(p) = E VnACP, a)M„_,_i(P', ft>')
-'*"(") it=o
with constants y„k, where D.2.18) has been used. By an approximation
argument strictly analogous to that used in the proof of Theorem 4.5.2
we extend this formula to general convex bodies K, K' instead of
polytopes P, P'. The explicit values of the constants y„k are found by
choosing balls of radii 1 and r for K and K' respectively, and
ft)= ft)' = S"-i, and then comparing equal powers of r. In the resulting
formula we now replace K by K + eB" with e > 0, develop both sides

4.5 Integral-geometric formulae 241
according to formula D.2.16) and compare equal powers of e. This
yields formula D.5.14) and thus completes the proof of Theorem 4.5.6.
Formula D.5.14) can serve as the source for several other integral-
geometric results. These include extensions to generalized curvature
measures, projection formulae and kinematic formulae for non-
intersecting convex bodies. We consider results of the latter type first.
For convex bodies K, K' e X" we denote by
d{K, K') := mm{\x - x'\\x e K,x' e K'}
their Euclidean distance. The principal kinematic formula provides an
explicit expression, in terms of Minkowski functional of K and K', for
the invariant measure of the set {g e G„\d{K, gK') = 0}. From this one
can deduce a similar expression for the measure of the set
{g e G„\0<d{K, gK')^ e} for given e>0, by applying the principal
kinematic formula to K + eB„ and to K, and subtracting. More
generally, we are interested in the measure of the set of all rigid
motions g for which 0<d{K,gK')^e and for which the points x e K,
x' e gK' defining the distance belong to certain preassigned sets. We
can impose a further condition on the set of rigid motions to be
measured. If x e K, x' e K' are such that
\x - x'\ = d{K, K') > 0,
we define a unit vector by
x' — X
u(K, K') := -rr^-
\x - x\
The pair (x,x') is not necessarily unique but one readily verifies that
this vector, pointing from K to K' 'in the most direct way', is unique.
Clearly we have {x,u{K, K')) e Nor K, the set of support elements of
K, and {x',-u{K, K')) e Nor K'. In addition, the vectors u{K,gK')
may be restricted to lie in preassigned sets. More generally, the
following theorem treats the case where shortest distances between a
fixed convex body and a disjoint moving convex body are realized within
given sets of support elements.
Theorem4.5.7. Let K, K' e X" be convex bodies and let rj, rj' e ^B).
For e > 0, let Nf{K, K',r],r)') be the set of all rigid motions g e G„ for
which there exist points x e K and x' e gK' such that
0<\x - x'\ = d{K,gK')^e,
(x, u(K, gK')) e Tj,
{X-, -u{K, gK')) e gr,'.

242 Curvature measures and quermassintegrals
Then
H{N,(K, K', T}, T}'))
= ^- i £'(")S(" " ']©k(K, ,,H„_,_,(/C', 7?'). D.5.15)
Before the proof, some preparations are necessary. In the following,
measurability for functions on SO{n) and G„ and for subsets of these
spaces refers to the completions of the measure spaces (SO{n),
7iE0(«)), v) and (G„,7i(G„), ju), respectively, The completed
measures are denoted by the same symbols.
If {x, u) e Z=l" X S"~^ and g = g,^p is the rigid motion defined by
gx = px + t with p 6 SO(n) and r e E", we define g(x, u) := (gx, pu);
also -(x,u)i-={-x,-u). Let K,K'eJC" be convex bodies. If
(a:, m) 6 Nor/<: and (a:', w) 6 Nor/C', then (a: + a:', w) e Nor (it +/<:')•
Vice versa, let (y, w) e Nor(/<: + K'). The point y e it + it' is of the
form y - X -^ x' with a: 6 it, x' e it', and necessarily {x, u) e Nor it
and (x', u) e Nor it'. In general, x and x' are not uniquely determined.
If y = z + z' with B,M)eNorit and (?', w) e Nor it' is a different
representation, then z - x = x' - z', hence it and it' contain parallel
segments lying in supporting hyperplanes with the same outer normal
vector M. In this case we say that it and it' are in singular relative
position.
We define an operation for sets of support elements which is adapted
to Minkowski addition. For r), r)' dS let
T]* T]' := {{x + X', U) 6 I\{X, U) 6 7J, {x', u) 6 J)').
In particular, for j8, j8' C E" and (o, w' C S"~^ v/e have
(j8 X w) * (P' X 0)') = (j8 + j8') X (w n (I)').
If TjcNorit and tj' C Nor it', then 7j*7j' C Nor (it + it').
Since the sum of two Borel sets is in general not a Borel set, it is
clear that »?*»?' need not be a Borel set if jj, jj' e ^B). The following
lemma ensures that this will cause no problems.
Lemma4.5.8. For convex bodies K,K'eJC" and Borel sets j) C Nor it,
T]' C Nor it', the set rj* pr}' is a Borel set for v-almost all pe SO{n).
Proof. Let R denote the set of all rotations p e SO{n) for which it and
pit' are not in singular relative position. Let peR. For {y,u.)e
Nor(it + pit') we have y = x •\- px' with suitable {x, u) 6 Nor it and
(pa:', m) 6 NorpAT'. Since peR, the points x and x' are uniquely

4.5 Integral-geometric formulae 243
determined. Hence, we can define maps
f:NoT(K + pK')-*NoTK by f(y, u) := (x, u),
f:NoT(K + pK')-^NotK' by f(y, u) := (x',p-^u).
It is easy to see that these maps are continuous, hence
ri * pri' = rHri) 0 f-\ri')
is a Borel set. By Corollary 2.3.11 we have vE0(«)\/?) = 0. This
proves the assertion. ■
Proof of Theorem 4.5.7. Let K,K'€X" and e>0 be given. We are
going to show that the set N^K, K', ?j, ?j') defined in Theorem 4.5.7 is
measurable for arbitrary Borel sets jj, t}' e ^B). The following
proposition will be useful.
Proposition 1. Let (p be a map from ^(X) into the set of all subsets of
G„ that has the following properties:
(a) If r)x, 7j2 e '^(.2) and JJi H 7J2 = 0, then (K'Ji) ^ (pi^i) is of measure
zero;
(b) If r]i e ^(I) for / e f^, then q{\JieN^i) = DieNfpimh
(c) If ?j e ^(.2) is closed, then q)(T}) is measurable.
Under these assumptions, the set q)(T}) is measurable for each ?j e ^(.2).
In fact, it follows from (a), (b) and (c) that the system of all sets
jj e ^(.2) for which q)(T]) is measurable is a a-algebra in Z that contains
the closed sets; hence this system coincides with ^B). This proves
Proposition 1.
Now let ?j, 7j' C 2' be closed and write
N := N,(K, K', T), 7)') U No{K, K'),
where Nq(K, K') is the set of all g e G„ for which gK' touches K
(which means that K and gK' intersect, but can be separated by a
hyperplane). We want to show that N is closed. Let (g,),-6i«g be a
sequence in N that converges to some g e G„. Then the sequence
(8if^')ieN converges to gK'. Since each body g/K' can be separated
from it by a hyperplane, the same is true of gK'. Further, for each
/ e f^ there exist points Xj ebd K, x- ebd K' such that
\xi - giX'il = d{K, g,K') S e,
U, u{K, g^K')) e 7),
{giX'i,-K{K,giK'))egin'\
iid{K, g,K') >0.
Since bd it, bd it' are compact, there exists a sequence (ik)keN such that
ixi^)keN converges to a point x ebdK and (A:,'j)/t6i«g converges to a point

244 Curvature measures and quermassintegrals
x' ebdK'.lt follows that
\x - gx'\ = d(K, gK') ^ e,
(x, u(K, gK')) e jj.
, , ,r. r.,u ,| ifd{K,gK')>0.
(gx ,-u(K, gK')) € gri'l
Thus g e N, which shows that N is closed.
Now define q>(T]) '■= N^K, K', t], tj') for t] e ^B), where ?j' C 2' is a
fixed closed set. Let jJi,jj2£^('^ ^^ disjoint and suppose that
g ^ (p(Vi)^ (piVi)- For / = 1. 2 there is a point Xi e K at distance
d{K, gK') from gK' such that (xj, u(K, gK')) e ?j,. Since JJi n tjj = 0,
necessarily atj ^ X2; hence it and git' are in singular relative position.
From Corollary 2.3.11 it follows that the set (K^Ji) ^ (pir]2) is of measure
zero. If T] is closed, then N^K, K',r],T]')\J No{K, K') is closed, as
shown above, and since Nq(K, it') is of measure zero (by the argument
used after D.5.4)), it follows that Nf(K, it', jj, jj') is measurable. Thus
the map q) satisfies the assumptions of Proposition 1 (where (b) is
satisfied trivially). Hence, for each Jje (B(Z) the set N^K, it', jj, jj') is
measurable. Now we fix Tje 7^B) and define Vi'?') •= ^df^y ^'^ ^^ ^')-
For Jj' closed, we have shown that V('?') 's measurable. Proposition 1
yields that Nf(K,K',r],r]') is a measurable set, where now tj,
T]' € ^B) may be arbitrary Borel sets.
We can now apply Fubini's theorem and obtain
^NAK, K', ,, ,')) = m^^^^dX" d.(p), D.5.16)
where
Tip) := {t e E"|g,,p e N,(K, K', jj, jj')}.
This set can be described in a different way. Suppose that t e T{p).
Then there exist points A:eit,A:'ep'it' + f such that
0< |a: - a:'| = d(K,pK' + t)^ e,
{x, u{K, pK' + 0) e ri,
(x', -u(K, pK' + t)e pr]' + t.
The vector u = M(it, pit' + t) satisfies {x, u) e Nor it and (a:', -m) e
Nor(pit' + t). Writing x' = py' + t, we have
{x, w) e jj n Nor it, (py', u) e p[r]' 0 Nor it'].
With d = d{K, pK' + t) we get x' = x + du and hence t =
X - py'+ du. It is obvious that p{K - pK', t) = x - py' and
M(it - pit', 0 = M, thus
{p{K - pK', t), u{K - pK', t)) = (x - py', u)
e [77 n Nor it] * -p{r]' D Nor it'].

4.5 Integral-geometric formulae 245
which shows that
t e M,(K - pK\ [j] n Nor/C] * -p[r]' fl Nor AT']) D.5.17)
(with Mf as in Section 4.1).
If we now assume that K and -pK' are not in singular relative
position, then we can reverse the argument and conclude that D.5.17)
implies t e T{p). Thus we have
T{p) = M,{K - pK', [jj n NotK] * -p[jj' n Nor/<:'])
for a set of rotations peSO(n) which by Corollary 2.3.11 has full
measure. Lemma 4.5.8 tells us that for almost all p the set [jj n Nor K]
*-/o{?j'n Nor it'] is a Borel set, so that D.2.4) can be applied to
compute the measure ^"(r(p)). Now D.5.16) yields
n ,=0 \ ' /
xf 0,(/C - pK', [r] n Nor/C] * -p[r]' n Nor/C'])dv(p). D.5.18)
To compute the integrals, we first consider strictly convex bodies. Let
TTi.^" 'X- S"~^ -* S"~^ denote the projection onto the second factor. If K
is strictly convex and r}CZ, then
7j n Nor /<: = [E" X 7T2(r] 0 Nor K)] 0 Nor K,
since for u e S"~^ there is exactly one x eE" for which {x, u) e Nor it.
Using D.2.6) and the definition of S„, we get, for ?j e '2B),
0„(K,r]) = 0„(K,r}nNorK)
= 0„{K, [E" X 7T2(r} n NotK)] D NotK)
= 0„{K,l" X ;r2(jj n Nor/t))
= S,„{K,7r2(VI^^OTK)).
Now let K, K' eJC" be strictly convex; then K - pK' is strictly convex
for all p e SO{n). Let r], tj' g ^25B). Observing that
[jj n Nor it] * -p[r]' n Nor it'] C Nor (it - pit'),
n2{[r] n Nor it] * -p[jj' n Nor it'])
= n2{r) n Nor it) fl -p7r2(?j' fl Nor it')
and 0„(-it', -jj') = 0„{K', t)'), we deduce from D.5.18) and D.5.14)
that D.5.15) is true.
Equality D.5.15) in the general case is now proved by approximation.
Again let it, it' e 3:!" be general convex bodies.
Proposition 2. For fixed tj' e 'B(Z), the function niN,(K, AT', •, t]')) is
a finite measure on '3B).

246 Curvature measures and quermassintegrals
For the proof, only the a-additivity has to be shown. For tjj,
?j2 e ^(•S) with Jji n 7j2 = 0 we have seen above that
(KNAK, K\ Jj,, T]') n N,{K, K\ JJ2, rj')) = 0,
hence the additivity of n implies that n(Nc{K, K', -,»?')) "s additive.
Let (jj,),€N t»e a sequence in ^B) such that ?j, i 0. Then
N,{K, K', 7,,, 7J') i N := f>,(/C, /C', TJy, »?')
for i ^ 00, hence
\imn(N,iK,K',r],,ri')) = H(N)
by the 0-additivity of fx. Let ge N. For each /'e Fy there is a point
XjSbdK at distance d{K,gK') from git' such that {Xi,u(K, gK')) e
T}j. Since MygMJjy = 0, not all the points Xj coincide, thus K and gK' are
in singular relative position. By Corollary 2.3.11, n(N) = 0. Thus
H{Nc{K, K', •, 7)')) is 0-continuous and hence a-additive. This proves
Proposition 2.
Propositions. If the sequence {Ki);^\^ in JC" converges to K, then
{xmMn{N,{Ki, K', r), r]')) S yi{N,{K, K', jj, jj'))
(■-.»
for all open sets t], tj' C Z, and
iim//(N,(A:„ a:', s, Z)) = //(N,(/(:, a:', s, Z)).
For the proof, let ?j, ?j' C 2' be open and denote by N%K, K',t],t]')
the set of all g e N,(K, K', r], r]') for which d(K, gK') + e and /C and
gK' are not in singular relative position. Let g e N%K, K', tj, tj'). For
each i e fy choose points atj e bd it; and x-ebdgA^' such that
\xj - AT'l = d(/Ci, gK'). Let a: g bd it, a:' e bdgit' be the unique points
for which \x - x'\ = d(K, gK'). It is easy to see that at; -» a: and a: • -* a:'
for i—> 00. It follows that 0<d(Ki,gK)< e for all i sufficiently large.
Since {x,u{K, gK')) er}, (x',-u(K, gK')) e gr]' and r], r]' are open,
we also have (x,, u(Ki, gK')) e jj and (a:-, -M(it,, gA^')) ^ 8^' for almost
all i; thus g e Nc(Ki, K', r], r]') for almost all /. We have proved that
N%K, K', T], Jj') C lim\nfn{N,{Ki, K', r), r)'))
/—♦«
and hence that
yi{N,{K, K', T], T]')) = niN'iiK, K', jj, jj'))
^ Mliminf N,(A:„ a:', jj, t]'))
l-.cc
^ llminf//(N,(A:i, K', r), jj')).

4.5 Integral-geometric formulae 247
which is the first assertion of Proposition 3. Since NdK-,, K',Z,Z) is
the set of all g e G„ for which gK' meets it, + eB" but not Kj, its
measure can, by means of the principal kinematic formula, be expressed
in terms of Minkowski functional, from which its continuous
dependence on Kj is clear. This proves Proposition 3.
Of course, results similar to Propositions 2 and 3 are valid with the
roles of the pairs (K,rj) and (K',r]') interchanged.
We are now in a position to prove formula D.5.15) for general
convex bodies. Let us write the assertion D.5.15) in the form
n-l
(i(NAK, K', T}, V')) = 2 ars0r(K, vHs(K\ »,'), D.5.19)
r,s=0
where e is fixed and the coefficients a„ are those resulting from
rearranging D.5.15). Then equality D.5.19) is already established if K
and K' are strictly convex bodies and jj, jj' e (B(I).
Let it e 3;" be strictly convex and it' e JC" arbitrary. Let (it-);^,!^ be
a sequence of strictly convex bodies converging to it'. By Proposition 3,
liminifi{N,(K, KU r], Z)) S n(N,(K, K', r], Z))
for open ?j C 2' and
iim^,(N,(K, K'j, Z, Z)) = n{N,{K, K', Z, Z)).
Thus ti(N,(K, K'j, •,Z))^t4N,(K,K', • ,X)) for i-* «>. Since D.5.19)
can be applied to it and it,', it follows that
ti(N,(K, K', •,Z)) = 2ars0r(K, • Hs(K', Z). D.5.20)
r,s
Let it and it J be as before and let r]C Z be open. By Proposition 3 we
have
\iminin(N,(K, K'j, r), r)')) S niN^K, Kj, jj, jj'))
for open jj' C Z, and from D.5.20) we obtain
\\mn{N,{K, K'j, T], Z)) = n{Ne{K, K', r], Z)).
(■-.ec
Thus n{N,{K,K\,r),-))^H{N,{K,K',r),-)) for /-*«>. From
D.5.19) and the weak continuity of the generalized curvature measures
we infer that
lx{N,{K, K', 7), ,,')) = S««0.(A:, r)H,{K', jj') D.5.21)
for all T]' e '3B). Since for fixed K, it', r)' both sides, as functions of j],
are measures the equality holds for arbitrary Borel sets r).
Now let K, K' e X" both be arbitrary. Choosing a sequence iKi)ieN

248 Curvature measures and quermeissintegrals
of strictly convex bodies converging to it, we deduce from Proposition 3
that
t4Ne(Ki, K', •, X)) A ti(N,{K, K', •, I))
and then from D.5.20) that
KNe{K, K',-,I)) = 2ars0r(K, • Hs(K', I).
r,s
Here the roles of K and K' may be interchanged, hence
H(NAK, K',Z, ■)) = 2ars0r(K, ZHs(K', •)• D.5.22)
r,s
Let Kj be as before, and let tj' C2' be open. From Proposition 3 and
from D.5.22) we see that
ti{N,{Ki, K', ■, jj')) ^ ti(N,{K, K', ■, ,,')).
From D.5.21) and the weak continuity of the generalized curvature
measures we get
H(Ne{K, K', Jj. T]')) = 2ars0r(K, r]Hs(K', t}')
r,s
for Jje'^(r). Since both sides are measures as functions of jj', we
deduce the equality for arbitrary t}' e Oi(Z). This completes the proof of
Theorem 4.5.7. ■
Comparing the coefficients of e"~' in formulae D.5.15) and D.5.18),
we obtain the following theorem as a corollary.
Theorem4.5.9. Let K, K'eJC" be convex bodies, let ?jCNor/C,
t}' C Nor it' be Borel sets and let j e {Q, . . . , n - \). Then
L(n)®>^^ ■*■ P^'' "^ * Z^'?')''^'^) = "i; tll)&k(K, vHi-k(K', 7J').
D.5.23)
This formula generalizes D.5.14). Another special case of D.5.23) is the
equality
f q(K + pK', p + pP')dxip) = -^ t{i]ck(K, P)q.,{K', n,
JSO(n) (o„ k^o\kl
D.5.24)
which is valid for Borel sets /3Cbdit, j8' Cbdit'. This follows from
D.5.23) since C„{K, P) = 0„{K, ^ x 5""') and
(/3 + pP') X 5"-' = (/3 X 5"-') * p(p' X 5"-').

4.5 Integral-geometric formulae 249
From D.5.23) we shall now derive integral-geometric formulae for
projections.
If £ C E" is a linear subspace, we denote by x\ E the image of a: e E"
under orthogonal projection onto E, and we use a similar notation for
subsets of E". For sets tj C 2' we define
r]\E := {{x\E, u)\{x, u) e r] and u e E}.
Theorem4.5.10. Let K eJC" be a convex body, let r] C Nor K be a Borel
set and let EC ^" be a k-dimensional linear subspace, where k e {1,
...,«-!}. Then
^^^^0'{K\pE, »j|p£)dv(p) = -^ 0i(K, 7J) D.5.25)
L
for j e {0,. . . , k - 1), where 0y is the generalized curvature measure
taken with respect to the subspace pE.
Special cases of D.5.25) are the formulae
f S'{K\pE, (o n p£)dv(p) = ^ Sj^K, m) D.5.26)
for we ^E"-') and
l^^^C){K\pE, /3|p£)dv(p) = -^ q{K, P) D.5.27)
for Borel sets /^Cbd/C.
Proof of Theorem 4.5.10. We choose, an (« - A:)-dimensional convex
body K' CE^ with a5""*(/<:') = 1 and let ??' be the set of all support
elements {x, u) of K' with x e relint K'. From D.2.4) it is then clear
that
©iiK', r,') = {
^^)
for j = n - k
D.5.28)
0 for y =t= « - A:.
For given e > 0, let a: e M,(K + K', r]* tj'). Then
0< d(K + K',x)^ e,
(p(K + K', x), u(K + K', x)) er]* t}'.
Writing u .= u{K + K', x), we have p{K + K', x) = y ■{■ y' with {y, u)
€ T)C Nor K and {y',u) e r)' C Nor it'. This implies u e E by the choice
of T]' and (y\E,u) e r]\E. Obviously we have |a:|£->'|£| =
d(K + K',x), p{K\E,x\E) = y\E and u{K\E, x\E) = u; thus

250 Curvature measures and quermassintegrals
x\E e Mc(K\E, r]\E), where M'^ is defined with respect to the subspace
E. Vice versa, one readily verifies that each point in M'c(K\E,t]\E) is
the projection of a point in M^K + K', ?j* jj'); thus
Me{K + K',r}* r}')\E = M',{K\E, r}\E).
If K and K' are not in singular relative position, then the set of points
in Mc(K + K',r]* r]') that project into the same point of £ is a
translate of relintit'; furthermore, ?j|£ is a Borel set. Hence, Fubini's
theorem gives
X"{M,(K + K',r}* T}')) = X^M'.iKlE, ri\E)),
and D.2.4) yields
-2e''--i"]e.(K + K',r)*r)') = \ 2 £*"'( M0;(/C|£, 7j|£).
" ,=0 \' / k j=o MI
For v-almost all p, this can be applied to pE, pK', pr)' instead of E,
K', T]'. If we then integrate over all rotations p, use D.5.23) and
D.5.28) and compare the coefficients of equal powers of e, we complete
the proof of formula D.5.25). ■
Theorem 4.5.7 has a counterpart where the moving convex body K'
is replaced by a moving A:-flat. Let K eX" be a convex body and
E € A(n, k) be a A:-dintensional affine subspace of E", where k - {0,
.. ., n - \}.\i X e K and a:' e £ are points between which the distance
is a minimum, we write d{K, E)'•=\x - x'\ and u{K,E)--
{x' - x)l\x' - x\.
Theorem 4.5.11. Let K e JC" be a convex body and let t] e 3B!) be a
Borel set; let k e {0, .. . , n - I}. For e>0. Let N^iK, tj) be the set of
all k-flats E e A(n, k) for which there exist points x e K and x' e E such
that
0< |a: - a:'| = d(K, E) ^ e,
{X, u{K, £)) e T}.
Then
fidN'AK, ri)) = ^ S e'(""*H„_,_,(/f, v)- D.5.29)
Proof Some of the subsequent arguments, not surprisingly, are similar
to those in the proof of Theorem 4.5.7. The convex body K and the
A:-flat E are said to be in singular relative position if some translate of E
supports K (that is, meets K and lies in a supporting hyperplane of K)
and contains more than one point of K. We fix a A:-dimensional linear

4.5 Integral-geometric formulae 251
subspace L/^ of E" and use the fact that the invariant measure v/^ can be
written as the image measure of A„_/t i8> v under the map
(r,p)'->p(L/t + t) from Lk x SO(n) into A(n,k), where A„_/t is the
restriction of X"''' to the Borel sets in Lj^. For Nq(K), the set of all
A:-flats supporting K, we therefore obtain
JSO(n)JT(p)
with
Tip) := {< 6 L;^|p(L, + t) e N^K)}.
Since T(p) is the relative boundary of the image of p~^ K under
orthogonal projection onto Lj;, we have 35"~*(r(p)) = 0 and thus
We complete the measure space {A(n, k), ^{A{n, k)), fi^), and
measurability refers to this completion. First we show that N*(/l, jj) is
measurable. Let r]CS be closed. Then N := N^iK, n){J No(K) is
closed. In fact, let (£,)i6M be a sequence in N converging to some A:-flat
E. Then d{K, E) S e, and if d{K, E) = 0 then E lies in a supporting
hyperplane of K. For each / e Fy there exist points Xj ebdK, x'j e £,
such that
k - x;| = d{K, Ed ^ e,
(Xi, u(K, £,)) e jj if d(K, Ei) > 0.
If d(K, E)>0, then by compactness some subsequence of ((a:,,
u(K, £',)),6M converges to an element (x, u) e jj, and clearly x e K and
M = u(K, E). Thus E e N, which shows that N is closed. Since
Hk(No{K)) = 0, the set N^K, jj) is measurable.
Let JJ,, jj2e^(r) be sets with jji n Jj2 = 0. Suppose that Ee
N'^(K,r]i)r\ Nc(K,T}2). Then there are points Xi,X2e K at distance
d(K,E) from £ and such that (a:,, m(/l, £■)) e jJj for /=1, 2. Since
JJi n jjz = 0, we must have a:, =t= a:2. Thus the flat E - d(K, E)u(K, E)
supports K and contains more than one point of K; in other words, K
and E are in singular relative position. From Corollary 2.3.11 it follows
that N*(/C, jji) n Ni{K, Jjz) has jU/t-measure zero.
By an argument analogous to that in the proof of Theorem 4.5.7 we
now conclude that Ni{K,t}) is measurable for each jj e "B(I).
By Fubini's theorem,
MN.U,^)) = LJ,(^,d^'-^dv(p),
where now
T(p) .= {t e LtlpiLk + 0 e N'^K, jj)}.

252 Curvature measures and quermassintegrals
Let t e T(p). There are points x e K, x'e piLi^ + t) such that
0<\x -x'\ = d(K, piLk + t))^e and (x, u(K,p{Lk + t))) e t]. Write
d(K, p{Lk + t)) = d and u(K, p{Lk + t)) = u. From x' = p{y' + t) with
y' e Lk and x' = x + du we obtain
t = p~^x + dp~^u - y' = z + dp'^u
with z '■= p~^x - y'. Because u is orthogonal to p(L/t + t), we have
p~^u e Lj; and thus z e L^. Since y' e Lk, the point z is obtained from
p~^x by orthogonal projection onto Lj;, thus zep~'/L|L^. From
{x, m) e Jjn Nor it we get (p~^x,p~^u) e p~^r] D Norp~'/L and hence
(z, p-^M) 6 (p-'»j|Lj^) n Nor(p-'A:|Lj^).
Thus r = 2 + dp~^u satisfies
re M;(p-'A:|L^p-'jj|Lt),
where M'^ is constructed in L^. If we assume that it and p{Lk + 0 are
not in singular relative position, then the argument can be reversed and
we arrive at
r(p) = M'Ap-'K\Li,p-'n\Lt).
We will show that the projection p~'?j|L^ is a Borel set for almost all
p. Let R be the set of all pe SO(n) for which K and pL^ are not in
singular relative position. By Corollary 2.3.11, vE0(«)\/?) = 0. Let
peR. For (y, w) e Nor(it|pL^) we have y = x\pLJ; with suitable
(x, u) € Nor it. Since pe R, the point x is uniquely determined. Hence
we can define a map
/:Nor(A:lpL;^)-*NorA: by/(>',«) := (x, u).
This map is continuous, hence the set r][pLk = f'^iv) is a Borel set.
For almost all p, we can apply D.2.4) to compute ^"~*(r(p)).
Integrating over SO(n) and applying Theorem 4.5.10, we obtain
= ^"Y e-^->[" - ')L^„e',iK\pLt, ,\pLiN^p)
This completes the proof of Theorem 4.5.11. ■
As a consequence of formula D.5.29) we note the limit relation
lim- HkiN'AK, V)) = "TT^ 0n-k-i(K, v)- D-5.30)
It can be viewed as an integral-geometric interpretation of the
generalized curvature measures.

Notes for Section 4.5 253
Notes for Section 4.5
Integral geometry. General reference works for integral geometry are the
books by Blaschke A937a) (see also Volume 2 of his collected works,
Blaschke A985a)), Santal6 A953), Hadwiger [18] {Chapter 6), Stoka A968)
and Santal6 A976). In style and attitude these books differ considerably
from each other. In contrast with other treatments of integral geometry,
which consider smooth submanifolds and use differential-geometric methods,
this section of the present volume is restricted to convex bodies (and finite
unions of them) and follows a measure-theoretic approach. It thus tries to
achieve a synthesis of the viewpoints of Hadwiger [18] and Federer A959).
Intersection formulae for curvature measures. Federer A959) developed a
theory of curvature measures for sets of positive reach and in this general
case proved the formulae of Theorems 4.5.2 and 4.5.5. A short proof was
given by Rother & Zahle A990). For convex bodies, simpler approaches are
possible. Schneider A978a) proved D.5.1) as a consequence of the
characterization theorem for curvature measures quoted in Section 4.2, Note 9. A
slightly simpler proof was given in Schneider A980b). The proof presented
above is taken from Schneider & Weil A986) and Lemma 4.5.1 is taken
from Goodey & Schneider A980).
As an example of recent developments leading to very general versions of
the kinematic formula (and requiring considerably deeper techniques) we
mention the work of Fu A990).
The classical principal kinematic formula. The specialization to total
measures, that is, formula D.5.2) for intrinsic volumes or its equivalent
formulation in terms of quermassintegrals, includes the convex case of the
classical principal kinematic formula of Blaschke, Santal6 & Chern (see
Santal6 1976). For convex bodies and sets of the convex ring, Hadwiger
A950a, 1951, 1956, [18]) developed an elegant method of proof. It goes
back to ideas of Blaschke and uses Hadwiger's characterization theorem for
quermassintegrals, quoted as Theorem 4.2.6. This method allows us to prove
a general version of the principal kinematic formula (Hadwiger [18], 6.3.5,
with different notation):
Theorem. Let q> be an additive and continuous real functional on X", and
define associated functional by
<Pn-kiK) := -^ f MK n £)dM*(£)
for K € JC.Then
i<(iK n gK')dpLig) = — i ~~<Pn-,iK)V,iK')
for K, K' eX".
A short proof of formula D.5.3) (which extends to a proof of D.5.2), but
apparently not of D.5.1)) was given by Mani-Levitska A988).
Convex cylinders. The method by which Theorem 4.5.5 was deduced from
Theorem 4.5.2 is taken from Federer A959). By the same method,
intersection formulae for a convex body and a moving convex cylinder can be
obtained. Let q e {0,,.., w}, let £, C E" be a qr-dimensional linear sub-
space and let E^ be its orthogonal complement. Then for KeX",
fi e TiiE"), a convex body C c £^, a Borel set r)(z E^ and / e {0, ... , «},
the formula

254 Curvature measures and quermassintegrals
n
k=,
with a„ii, as in Theorem 4.5.2, is valid (Schneider 1980b).
5. Translative integral geometry. The simple form of the kinematic formulae
D.5.1), in particular the fact that on the right-hand side the convex bodies
K and K' appear separately, is a result of the integration over the full group
of rigid motions. If one integrates only over the translations, one still has a
formula of the type
l^iiK n (K' + t),pn (p- + t))dt = 'Z0'^\K, K', p X p') D.5.31)
with finite measures 0V'(^, K', •) on 7i(E'' x E"), where
0)'\K. K\ px P')= 0jiK, pH„iK', p'),
<P^M> K\ pxp')= <P„iK, PHfiK', p'),
but the other mixed functional 0^' do not simplify in a similar way.
Theorem 4.5.3 contains an explicit representation for them in the case of
polytopes. From that theorem, one can derive the existence of a
representation D.5.31) for general convex bodies and can prove some properties of the
functions 0*/' such as weak continuity, additivity and homogeneity in each of
the first two arguments; see Schneider & Weil A986), where one may also
find analogous results for cylinders and flats. The cases j = n and j = n - I
of D.5.31), where no proper mixed measures 0*/' occur, can be deduced
from more general results in measure theory; see Groemer A977b, 1980a)
and Schneider A981b).
With the notation
0fiK, K', E" X E") = n.U,_,(/:, K-),
the case /3 = /?' = E" of D.5.31) can be written in the form
ly^iK n {K' + t))dt
rt-l
= VjiK)V„iK') + 2 vV,Uy-^^, K') + V„iK)VfiK'). D.5.32)
Special cases are
/,
j^V„iK niK' + t))dt = V„iK)V„iK'),
V„.iiKniK' + t))dt= V„.iiK)V„iK') + V„iK)V„.^iK-).
J I"
Further results can be formulated in terms of mixed volumes. The latter will
be treated in Chapter 5; we use here the results and notation of that chapter
without explanation. For j = 0 the left-hand side of D.5.32) is equal to
?C,''i{t 6 E"\K n (K- + t)^ 0}) = V„iK - K'),
hence
j^VoiK n iK' + t))dt = t(l)ymkl -K-[n - k]). D.5.33)
For the remaining cases, 1 s y g « - 2, the expressions appearing in D.5.32)
can be expressed as Crofton-type integrals involving mixed volumes, namely

Notes for Section 4.5 255
Ai)
(/ )
= ' f V{K n E[k - j], -K'[n + j - A:))dM„-y(£). D.5.34)
For dimensions two and three, translative integral geometry was treated
by Blaschke A937b), Berwald & Varga A937) and Miles A974) (see also
Firey 1977). For D.5.33) and extensions see Groemer A977b); for D.5.34)
and other representations of V*/'„+ ^ in the case of centrally symmetric
convex bodies, see Goodey & Weil A987). These authors also study the
counterpart to D.5.32) for the case where K' is a convex cylinder. The
special case of a flat reads
\^^y,{Kr\{E, + t))A^>-'-'{t)
= -!-(" \v{K[n + j - k], B,[k - j]) D.5.35)
for KeX" and y e {0, . . ., k). Here * e {1, . . . , n-\), Ei, is a
^-dimensional linear subspace of E" and Bk is a A:-dimensional unit ball in
E^. A proof of D.5.35) appears in Schneider A981a); see also Section 5.3.
6. Iterations. The kinematic formulae D.5.1) can clearly be extended to the
intersections of a fixed convex body and a finite number of independently
moving convex bodies. In particular, from D.5.2) one deduces by induction
that
f ■ ■ f Vi(K^ n g,K, n...n g^K,)An(g,y ■ dftig^)
Ao kp = f
ko+... + kp=pn + j
*" '" r.icA^'.icnr
for Ko, Ki,. . . , Kp e X", p eN, and / 6 {0,. . . , «}. The possibility of
iterating stems from the fact that the same series of functional appear on
the left-hand side and on the right-hand side of D.5.2). See also Streit
A970).
A similar iteration of the translative formula D.5.32) can be obtained if
either n = 2 (Blaschke 1937b, Miles 1974) or j= n or « - 1 (Schneider
1970b, Streit 1973, 1975). In the other cases, new classes of mixed
functional appear; see Blaschke A937b) and Berwald & Varga A937) for
« = 3 and Weil A990a) for a thorough study of the general case (short
surveys are given in Weil A989b, c)). However, explicit descriptions of the
mixed functional that appear are still known only in special cases.
Applications of intersection formulae in stochastic geometry. The integral-
geometric formulae for the intersection of a fixed and a moving set have
remarkable applications in stochastic geometry, where they yield expectation
formulae for the functional densities that can be associated with certain
random sets or particle processes. We first explain the relevant concepts and
formulae and then giVfe some hints about the literature.
A random closed set X in E" (see Matheron 1975) is a random variable (a
measurable map from some probability space) with values in the space of
with

2S6 Curvature measures and quermassintegrals
closed subsets of E" that has the topology of closed convergence and the
induced Borel structure. It is called stationary (isotropic) if t o X and X have
the same distribution, where t is a translation (rotation) of E". The extended
convex ring LUAC") consists of all sets M in E" that are locally finite unions
of convex bodies, in the sense that M n K e U(JC) for each K e X". For
K g U(X"), there exist meN and convex bodies Ki,..., K,r> such that
K = KiV ... U K„, and we denote by N(K) the smallest number m for
which such a representation is possible. For the purposes of this note, we
now say that A" is a standard random closed set if it is a stationary random
closed set with values in LU(X") and satisfies
T^2mxr,K) < 00 for all K e X". D.5.36)
Here E denotes mathematical expectation. D.5.36) is a type of integrability
condition, ensuring the existence of the expectations appearing below. To
show this, a theorem of Eckhoff A980) can be used.
For a standard random closed set X, densities corresponding to the
intrinsic volumes (or the quermassintegrals) can be introduced in several
equivalent ways. First, one can show that
EVXX n rK)
exists for each convex body K e X" with o 6 int /^ and is independent of K.
Second, let /3 c E" be a bounded Borel set. Choose a convex body K with
^ C int a:. Then 0;(A', p) := 0y(A' n K, j8) is independent of K and defines
a random variable. One can show that
E0/^, p) = D,{X)X''ip). D.5.38)
Finally, let
C- {(§„...,§„) 6 ElOg^.gl, /■ = !,...,«},
B*C := {(I,, ...,§„) 6 Elmax{§„ ...,§„} = !}
(observe that 3* C is an element of the convex ring U^X")). Then
E[Vj(X n C) - Vj(X n 5*C)] = Df(X). D.5.39)
The densities DjiX) appear in the following counterpart to the kinematic
formula D.5.2).
Theorem. Let X be an isotropic standard random closed set in E", let
KbX" and let y 6 {0, . . ., n). Then
n
EVi(K nX)^ J,ani,V,iK)D„^,.,(X). D.5.40)
*=/
The Crofton intersection formula has the following analogue.
Theorem. Let A" be an isotropic standard random closed set in E", let
L e A(n, k) be a ^-flat and let / 6 {0, ..., fc}. Then
Di(X n L) = (XnikDn^^-k{X). D.5.41)
Of particular importance are the standard random closed sets arising as
union sets of stationary Poisson processes in the space X" (the so-called
Boolean models). A stationary Poisson process Y in X" (to be interpreted
as a random field of convex particles) is determined by a positive number y,
its intensity, and by Pq, the distribution of the typical particle or grain
distribution for the process Y\ Po is a probability measure on the space X"^
of convex bodies with circumcentre (centre of the smallest circumscribed
ball) at the origin. For a bounded Borel set A 6 'li{X''), the probability that

Notes for Section 4,5 257
exactly k particles of Y belong to A is then given by
prob{card Y Pi A = k) = e-^<'^' ^^^^.
where A (the intensity measure of the Poisson process Y) is the image
measure of yPo ® ^'"^ under the map {K, ()>-* K + t from T" x E" onto
Ji".
For a stationary Poisson process Y on X", the V^-density can be defined
by
It satisfies
4;(y) := YJ^VfiK)dPo(K).
'""—vj;]^)— = ^'(^)'
r—•«:
for each convex body K b X" with o 6 int /^, and also analogues of D.5.38)
and D.5.39).
If the process Y is isotropic, it satisfies versions of the kinematic and
Crofton formulae that are analogous to D.5.40) and D.5.41).
The union set of the process Y is defined by Xy ■= Ul^yL. Assuming a
suitable integrability condition, Xy is a standard random closed set, and it is
of interest to know how the densities Dy(A'y) and ^/(Y) are related. Due to
the Poisson assumption, the following general result can be obtained. If / is
a translation invariant valuation on [/(X") and K e X" is a convex body,
then
EfiK n Xy)
= i(-i)*"' jrix, ■ L/^^' '^" ■ ■ ■' ^*)'*''°(^')- ■ •'•''°(^*)
where
F{K, Ku.-, K,)
.= 1^- ■ -j J(K n(Ki + xi)n ...n (a:* + xi,))dxr ■ dx^.
Evidently, further evaluation requires iterated intersection formulae as
discussed in Note 6. In the isotropic case, one can thus obtain explicit
expressions for D/iXy) in terms of Aj(Y),.. . , A„(Y). For n=2, for
example, these relations are
D2iXY)= l-e-^^<''',
D,(^y) = e-^^<''L,(y),
Do(Xy) = e-^^<>')[4o(y) - — 4,(yJ].
Now we give some hints to the literature. Much of the motivation for the
considerations described above came from the need to establish
mathematical models and solid foundations for some methods and formulae of
stereology. For an introduction to stereology, see Stoyan, Kendall & Mecke
A987), Chapter 11, and for a survey emphasizing the connections with
integral geometry, Weil A983a); see also Weil A983b). All the results
mentioned in the present note can be found with proofs in Weil A984) and
Weil & Wieacker A984). These papers unify and extend previous results of
varying degrees of generality obtained, using different methods and partly
under stronger assumptions, by Matheron A975), Miles A976), Davy A976,

258 Curvature measures and quermassintegrals
1978), Wieacker A982), A.M. Kellerer A983, see also 1985) and H.G.
KeUerer A984). Weil & Wieacker A984, 1988) show (in a refined version)
that each random set with values in LU(X") can be written as the union set
of a point process Y on X". Extensions of the considerations in this note,
for instance to cylinders, non-isotropic random sets and particle processes,
are treated by Weil A987). The approach to stochastic geometry as
described here is further developed (but mostly without proof) in Mecke,
Schneider, Stoyan & Weil A990). Similar investigations for more general
classes of random sets and point processes are due to Zahle A982, 1986b).
More recent objects of study are LU(X")-va\ued random sets, which are
stationary, but not necessarily isotropic; see Weil A988, 1989b, c, 1990a, b)
and Betke& Weil A990).
Rotational mean values. Theorem 4.5.6 is due to Schneider A975f). The
proof given here is essentially that of Schneider A986b), In that paper, the
more general version of Theorem 4.5.9 is obtained. The special case D,5.24)
was proved by Weil A979c). Also, the projection formula of Theorem 4.5.10
appears in Schneider A986b). Its special case D.5.26) had previously been
proved, in Schneider A975f), and case D.5.27) in Weil A979c). A certain
extension of the projection formula D.5.25) to sets of the convex ring, for
which multiplicities have to be taken into account, is treated in Schneider
A990c), A more general integral-geometric formula derived from Theorem
4.5.6 is applied in Papaderou-Vogiatzaki & Schneider A988) to a question
on geometric collision probabilities.
The proof of Theorem 4.5.7 is adapted from Schneider A986b) and
Schneider A978d). The special case ?)= /3x 5"-' of Theorem 4.5.11 was
first proved in Schneider A978a); the proof given above is an immediate
extension. The integral-geometric interpretation D.5.30) of generalized
curvature measures was mentioned in Schneider A980a); its specialization to
area measures is due to Firey A972) and its specialization to Federer's
curvature measures appears in Schneider A978a).
Distance formulae. Integral-geometric intersection formulae such as the
principal kinematic formula and its generalizations involve integrations over
the set of rigid motions g for which a moving set gK' intersects a fixed set
K. If the distance between K and gK' is taken into account, one can also
obtain formulae for non-intersecting sets. Let K, K' 6 X" be convex bodies
and let g e G„ be a rigid motion for which K r\ gK' =0. Let x e K,
x' e gK' be points between which the distance is minimal. Before Theorem
4.5.7 we defined diK, gK') ;= \x' - x\ and u(K,gK') := (x' - x)/\x' - x\;
we also put x{K, gK') := x if the pair {x, x') is unique, which is the case for
//-almost all g.
Theorem. Let /: @, «) x bd /^ x bd /^' ^ IR be a measurable function for
which the integrals in D.5.42) are finite. Then
L
n.K-^J^''^'^' «*^">' ''^^- «^'>' S-'x(gK', K))An(g)
Let /; @, oe) X 5"-' x 5"-' ^ R be a measurable function for which the
integrals in D.5,43) are finite; then

Notes for Section 4.5 259
Ln K- J^''^^' ^^'^' "^^' ^^'^' 8S'u(gK', K))dK8)
X I £,.J^,.,/(r, ", u')dSdK, u)dSi-,{K', u')r"-'-' dr. D.5.43)
Here we avoid some complicated coefficients by using Q, 5* instead of
0k, ^k. A common generalization of D,5.42) and D.5.43) can be
formulated, if support elements and the generalized curvature measures 0* are
used.
The following result is obtained as a special case. Let /: [0, <»)^@, <»)
be a measurable function for which /(O) = 0 and
M,(/) •= k\^ fir)r''-' dr < » for A: = 1, . , , , n.
Then
ifid(K,gK-))df4g) = -^t t {"]{'' )M,,^.„(fWk(K)Wj(K').
This formula was first proved by Hadwiger A975a). Analogues for
moving flats are due to Bokowski, Hadwiger & Wills A976), and various
generalizations to Groemer A980b). Special formulae of the type D.5.43)
were first obtained by Hadwiger A975b); see Schneider A977b) for a short
proof. A thorough study leading to the results above, extensions, translative
versions and analogues for flats was made by Weil A979b, c, 1981a).
10. Contact measures and touching probabilities. The consequences of Theorems
4.5.7 and 4.5.11 can be interpreted in terms of touching or collision
probabilities for convex bodies. To explain this, let us first formulate an
intuitive and unprecise question. Let convex bodies K, K' e X", and subsets
pGbdK, p'GbdK' of their respective boundaries, be given. A random
motion g is applied to K' in such a way that gK' touches K. What is the
probability that K and gK' touch at a point of pngp'7 Of course, this
question does not make sense as long as no underlying probability space is
specified. For this, one would need a 'natural' probability measure on the
set Go(K, K') of rigid motions g e G„ for which gK' touches K. Such a
natural measure can, in fact, be deduced from the Haar measure, in the
following way.
Let K, Le X" be disjoint convex bodies. There is a unique translation,
denoted by t{K, L), by a vector of length d{K, L), such that
K n z(K, L)L ^ 0. For a Borel set ex 6 ^^(Gn) and for f > 0, define
A,(K, L, (X) ;= {g e G„|0 < d{K, gL) g e, t{K, gZ.) o g 6 *},
Then Af(K, L, a)isa Borel set, and
ti(AAK, L, a)) = j- le"-'!"]! q(K - pL, r(*, p))dy{p)
with
Tia, p) := {t 6 E''|y(f, p) e a (^ Go{K, L)}.
It follows that
(fiK, L,a) := \im — n(Af(K, L, a)) = f Cn-i(K - pL, T(a, p))dv(p).
This defines g>(K, L, ■), a finite measure concentrated on the set Go(K, L)

260 Curvature measures and quermassintegrals
of rigid motions bringing L into a contact position with K; it is called the
contact measure of K and L.
Assuming now that convex bodies K, K' e X" and Bore! sets
p, p' e^iJE") are given, we can choose the completion of the probability
space
(GoiK, K'), -JKG^iK, K')), (fiK, K', ■ )/(p{K, K\ Go{K, K')))
as a model for the question posed at the beginning. Then it follows from
Theorem 4.5.7 that the probability for a collision at a point of /3 n g/3' is
given by
/=0\ J I
p)C„-,-t(K', p')
K)C„-i.,iK\ K-)
The contact measure as defined above was introduced by Weil A979b, c).
Touching probabilities have been found in several cases, including the case
where bodies touch at prescribed sets of normal vectors, the case where flats
or cylinders touch and the case where there is inner contact (Firey 1974a,
1979, McMullen 1974a, Molter 1986, Schneider 1975e,f, 1976, 1978d, 1980b,
Schneider & Wieacker 1984, Weil 1979b, c, 1981a, 1982a, 1989a). See also
Burton A980b) for a result on associated measurability problems.
11. Absolute curvature measures. Integral-geometric formulae for absolute
curvature measures of sets of positive reach are treated by Rother & Zahle
A992).
12. Mean measure of shadow boundaries. An interesting integral-geometric
result on convex bodies, of an entirely different type, was proved by
Steenaerts A985). In spirit, it is related to the boundary behaviour
considered, from various aspects, in Chapter 2. For K € X^ and a vector
u e S""', let Z{K, u) be the shadow boundary of K in direction u, that is,
Z{K, u) := {x e bd K\x + Xu $ int K for all A e R).
Define
aiK) ;= — |^.,a5''-2(r(A:, u))d^''-'(")
and
piK) := — j^^_^?0"-^{Telbd K\u^)dX''-\u).
Then Steenaerts proved, confirming a conjecture of McMullen, that
p(K) ~ inc„.i
for smooth /^. If /^ is a ball, the left equality sign holds, and for K a
polytope the right equality sign is valid.
13. Mean intersection bodies. In Crofton's intersection formula D.5.8) one
integrates an intrinsic volume of the intersection K r\ E over all ^-flats
E 6 A(n, k), where the integration is with respect to the invariant measure
Hk' One may also integrate the intersections themselves (in the sense of
Minkowski addition): For K 6 X" and k e {I,. . . , n - 1}, there is a unique
convex body Mit(K) for which
h{M,(K), ■) = \ h(K n £, ■)d//*(£).

4,6 Local behaviour of curvature measures 261
This '^th mean intersection body' M^iK) was investigated by Goodey &
Weil A991+). In particular, they showed that Mi(K) is always a ball,
whereas MiiK) determines K uniquely, up to a translation.
4.6. Local behaviour of curvature measures
The notion of curvature, in its various forms existing in geometry, is
designed to describe and measure, in one sense or other, local geometric
shapes. In the present section we ask for the information of this kind
that can be gained from knowledge of the curvature measures or area
measures of a convex body.
A natural first question to ask is the following. What does it mean
geometrically if the mth curvature measure C„(K, P) vanishes on a
relatively open subset j8 of the boundary of Kl The answer is given
below by the description of the support of C„(it, ). The support of a
Borel measure ^, denoted by supp/^, is the complement of the largest
open set on which the measure vanishes.
Theorem 4.6.1. Let K e Xq and me {0,...,«-!}. The support of the
mth curvature measure C„{K, •) is the closure of the m-skeleton of K:
suppC„(/L, •) = cX&ximK.
Proof. Let K e JLq and m e {0,...,«- 1} be given. First we show
that
C„{K, bd K\txx„ K) = 0. D.6.1)
Put j8„ ■= bd A:\ext„ K (which is a Borel set by Section 2.1). From
D.2.21) we have
Co(A:, )8o) = 7C."-\a{K, )8o)) = 0,
where the latter follows from Theorem 2.2.9, since u e a{K, Pq) implies
that F(K, u) contains a segment; hence w is a singular normal vector of
K. Now we use a special case of Theorem 4,5.5 that can be written as
C„(K, P) = a„J CoiK n E, l3)dfi„.„(E) D.6.2)
JA(n,n-m)
with a positive constant a„„. For each E e A{n, n - m)v/e have
P„n E C relbd(A: fl £)\exto(A: fl E),
since a point a: e j8„ D £ is the centre of an (m + l)-dimensional ball
contained in K and- hence is the centre of a segment contained in
a: n £, It follows that
CoiK n E, P„) = Co{K n E,p„n E) = 0,

262 Curvature measures and quermassintegrals
by the result proved above, applied to K 0 E. From D.6.2) we deduce
that
C„{K, bd K\ext„ K) = 0 D.6.3)
and hence that
suppC„(/L, •) C clext„ K.
For the proof of the oppposite inclusion we need a quantitative
improvement of the case m = 0.
Proposition. Let j8 C E" be an open ball with centre x e K and radius
p. If Cq(K, P) = 0, then x is the centre of a segment of length 2p/n
contained in K.
For the proof we first show that
a: e conv(/<: n bdj8), D,6.4)
Suppose this were false. Clearly it fi bd j8 =t= 0, since otherwise K C P
and hence Co(K,P)>0. Thus conv(/L fi bdj8) is a convex body not
containing x. By Theorem L3.4, there exists a hyperplane H strongly
separating x and K Dbdp. Let // "^ be the open halfspace bounded by
H that contains x. Since x e K C] H^, it is clear that the outer unit
normal vectors of K at points of K C\ H* fill a neighbourhood of the
inner normal vector of H* on S"~^. From D,2.21) it follows that
Co(K, j8) ^ Co(K, bdK n H*) > 0,
a contradiction. Thus D.6.4) holds.
By Carathdodory's theorem, there are k'^ n + 1 points yi,. . . ,
yk € K nbdP such that
k k
X = ^ajy/ with ary > 0 (/ = 1, . , . , k), X«^/ = !•
From
k ^ k
we deduce that there exists an index i for which
o^i ^ 1
For
then
aiyi + . .
e convlyi, .
1 - a-, ~ k -I'
X - arj',
. + Uj-iyi-i + a,.+,y,.+, + , .
1- or,
..,yk)CK.
■ + akyk

4.6 Local behaviour of curvature measures 263
Since x = ajyi + A - ar,)y,
\x - yil = p,\x - y\ = — \x - yi\ ^ —,
1 — OTj n
the proposition is proved.
Now let P C E" be an open set for which C„(K, P) = 0. From D.6.2)
we have
f Co(A:n£, )8)d/z„_„(£) = 0,
JA(n,n-m)
hence Co{K C] E, P) = 0 for /in_„-almost all planes E e A(n, n - m).
Let X e K C[ P and let E e A{n, n - m) be an (n - m)-plane through
X that meets intK. There exists a sequence (£/)y6i«g of (n - m)-planes
converging to E such that Cq{K fl Ej, /3) = 0 for all /. For / sufficiently
large, we may choose Xj e K O Ej fl /3 such that Xj -^ x for / -* », and
we may further assume that, for some fixed p > 0, the open ball with
centre Xj and radius p is contained in j8. By the proposition, Xj is the
centre of a segment [oy, bj] of length 2p/(n - m) contained in it fl Ej.
By selecting suitable subsequences, we infer that x is the centre of a
segment [a,b] C K 0 E. We have proved that every (n - m)-dimen-
sional plane through x that meets int K, meets it in a segment of length
2p/(n - m) with centre x. This is only possible if x is the centre of an
(m + l)-dimensional ball contained in K. Thus x ^ ext„ K, and we have
proved that
E"\suppC™(/<:, •) C l"\ext„K,
hence
clext„it CsuppCm(it, •)■
This proves Theorem 4.6.1. ■
In a similar way, we can obtain a counterpart to Theorem 4.6.1 for
area measures. Since the proof is slightly more complicated, we
formulate part of it as a lemma. For u e S"~^ and 0 < < < 1 we define
D(u, t) := {v€ S"-'\{u,v) > t);
this is the open spherical cap with centre u and spherical radius arc cos t.
Lemma4.6.2. Let K e JC^, n^2, ue S"-\ t e @,1) and x e F(K, u)
and suppose that
' S„_,iK, Diu, 0) = 0.
Then there are normal vectors u\, Ui^ N(K,x) fl S"~^ such that
u e relintpos{Mi, Ui}

264 Curvature measures and quermassintegrals
and {u, M,) S c{n, t) for i = 1,2, where c{n, t) <l is a constant
depending only on n and t.
Proof. First we prove that
M e pos(N(/<:,A:) nbdD), D.6.5)
where D = D{u, t), and bd D denotes the boundary of D relative to
5""'. Suppose this were false. Then there exists a hyperplane through o
that strongly separates u and com{N{K,x) fi bd D). Hence there are a
vector u e 5"~' with {u,v)>0 and a number 7j>0 such that
{w,v)<-r] for all w e N(K,x) D bd D. All elements w of the
compact set {w ebd D\{w,u)^ -T]} satisfy w $ N(K,x) and hence
{x.w)<h(K,w). Therefore, we can choose a number £>0 with
{x, w) < h{K, w) - e for all w e bd D satisfying (w, u) ^ -t}. Put
y := X + au + pv with a '■= , 6 := ,
^ T] + t T] + t
then {y, u) > {x, u) and hence y $ K. Let w e bd D. If {w,v) < -tj,
then
{y, w) = {x, w) + a{u, w) + ^{v, w)
< h(K, w) + at - jSt}
= h(K, w).
If (w, u) = -?j, then
{y, w) = {x, w) + a{u, w) + ^{u, w)
< h(K, w) - £ + a- + j8
= h(K, w).
Let w g 5"~'\cl D and w =}= -w. We can choose Wq e bd D and A, /^ > 0
such that Wq = Am + fiw. If {y, w) ^ h(K, w), we obtain
h{K, Wo) > {y. Wo) = A(>', u) + niy, w)
> Xh{K, u) + nh{K, w)
^ h{K, Aw + nw)
= h(K, Wo),
a contradiction; hence {y,w) < h{K,w). For w=-m this holds
trivially.
Let fi C int it be a ball and put
A := conv(S U {y}) 0 bdK.
For each w e S"~hD we have proved that {y,w) < h{K,w), hence
F{K,w)nA=0. This shows that Ac:t(K,D) and hence, by
D.2.24), that

4.6 Local behaviour of curvature measures 265
5„_,(A:, D) = X"-\x{K, D)) ^ :}C"-\A) > 0,
contradicting the assumption of the Lemma. Thus D.6.5) is true.
From D.6.5) and Carathdodory's theorem we deduce the existence of
k^ n vectors Uj, . . . , U/t e N{K, a:) fl bd D such that
k
u = ^ajUi with OTy > 0 (/=!,..., k).
/=i
From {uj, w) = < for / = 1, .. ., A: we get (ari + ... + ai^)t = 1 and
thus kt^{ai + . . . + al) ^ 1, hence there exists a positive constant Ci =
Ci(k, t) <l for which
k k
/=! /='
We deduce the existence of a number / e A, . . ., A:} for which
{u, u - a;Vi) 1 - or/
|m - aiVi\ Vl - lOit + aj
Put
M - ar/U,
M, := Vj, U2 ■= ] :
\u - or/U/l
and
c(n, t) := max{c,B, t), . . . , Ci(n, t), t}.
Then u^ e N(K,x)n S"-\
\u - aiVi\u2 = cvi«i + . . . + cv,-i«,-i + a^,+i«,+i + . • • + (Xk^k
e N(K,x)
and thus wz e N(K,x) n S"~^; finally
M = ar,Mi + |m - aiVj\u2 e relintpos(Mi, M2}
and (m, Mj) = r ^c(«, 0. (w. ) = <^i(*. 0 = c(«, 0- This completes
the proof of Lemma 4.6.2. ■
We can now describe the support of the mth area measure.
Theorem 4.6.3. Let K e J^.^ and m g @, ...,«- 1}. The support of the
mth area measure Sm(K, •) is the closure of the set of all (« - 1 - m)-
extreme unit normal vectors of K.
Proof. Let E e G(n, m + 1) be an (m + l)-dimensional linear subspace
of E". As a special case of D.5.26) we have
S„iK, (o) = b„„j^^^^S'„(K\pE, 0, n pE)dxip) D.6.6)
for we 2E""') with a positive constant b„m', here 5^ is computed in

266 Curvature measures and quermassintegrals
pE. Let wC 5""' be an open set for which S„(K, (o) = 0. Then D.6.6)
shows that
S'„{K\pE, mn pE) = 0
for v-almost all rotations peSO(n). Let u e (d and let peSO{n) be
such that u € pE. We can choose a sequence (py)/gM of rotations
converging to p such that
S'„(K\pjE, 0) n pjE) = 0 for / e f^.
Writing Uj - PjP'^u for / e N, we have w/-* u for /-»<» and Uj e pjE.
Since co is open, we can assume that D{uj, t)C a> for all j eH with
some fixed t <l. Choosing Xj e F{K, uj) for / e N, we have
XjiPjE e F{K\pjE, uj).
By Lemma 4.6.2 applied to K\pjE, there are vectors
Vj, Wj e NiK\pjE, Xj\pjE) n E""' D p;E)
= N{K, Xj) n 5""' n PjE
such that
Uj € relintposivj, Wj} D.6.7)
and
{uj, Vj) %c{m + I, t), {uj, Wj) S c(m + 1, 0- D-6.8)
After suitable selection of subsequences and a change of notation, we
may assume that Xj—*x, Vj-^v, w,-^ w for /'-* », with a certain point
xehdK and vectors u,veS"~^. Since UjE N(K,Xj), we have
h(K,Uj) = {xjyUj) and hence h(K,u) = {x,u); thus ueN(K,x).
Similarly we see that v,w e N(K,x). From Oi,WjepjE we get v,
wepE. Relation D.6.7) implies w e pos(j;, w}, and from D.6.8) we
obtain {u,v) ^c(m + I, t), {u,w)^c(in + \,t)<l; hence w e relint
pos{t;, w}. Thus we have proved: Any (m + l)-dimensional linear
subspace pE through the given vector u e (d contains a two-dimensional
subcone of N(K,x), where x e F(K, u), which contains u in its relative
interior. In other words, each (m + l)-dimensional linear subspace
through u intersects the normal cone N(K, F(K, u)) in a cone of
dimension at least two containing u in its relative interior. This is only
possible if the face T{K,u) of N(K,FiK,u)) (see Section 2.2)
containing u in its relative interior has dimension at least n + I — m. By
definition, this means that u is not an (« - 1 - m)-extreme normal
vector of K. Denoting the set of A:-extreme normal vectors of K by
extn/t it, we have proved that any open set (oC S"~^\suppS„{K, •)
satisfies (dC 5"~'\extn„_i_„ K, hence
clextn„_i-„/: Csupp5„(/<:, • )•

4.6 Local behaviour of curvature measures 267
Vice versa, let w be an open subset of 5"~'\extnn_i_„ K. Then each
vector u e (t) satisfies (i\mT{K,u)^ n + I - m. If m = n-l, we
deduce from D.2.24) and Theorem 2.2.4 that
5„_i(A:, m) = X"-\r{K, «)) = 0,
since t(K,o}) contains only singular points of K. Using this resuh and
formula D.6.6), we obtain S„(K, oS) = 0 for m = 1,...,«- 2. This
completes the proof of Theorem 4.6.3. ■
In special cases, the support of the area measure S„(K, •) tells us
even more about the structure of K. If Pe'J)" is a polytope, then
formula D.2.18) shows that the area measure S„{P, •) is concentrated
on the union of the spherical images of the relative interiors of the
m-faces of P. The following theorem implies that only polytopes can
have area measures with supports of this kind.
Theorem4.6.4. Let K e JCq and m e {I,. . . , n - I), and suppose that
the support of the area measure S„(K, •) can be covered by finitely
many (« - 1 - m)-dimensional great spheres. Then K is a polytope.
Proof. First let m = « - 1. Then S„-\{K, •) is concentrated on finitely
many points Ui, ... , u^e S"~^. Let
k
P := n'^'(^. ".)•
1=1
Then F is a polytope containing K. Suppose P has a vertex x that is
not a point of K. The set (o of unit normal vectors (pointing towards x)
of the hyperplanes strongly separating x and K is disjoint from (wi,
... , M/t) and satisfies
S„.,{K, CO) = a5"-'(T(A:, CO)) > 0,
a contradiction. Hence each vertex of F is a point of K and we
conclude that K = P.
We may, therefore, assume that m^n-2. Let Ai,...,Ak be
(« - 1 - m)-dimensional great spheres covering supp S„{K, •). We put
k
A := [JAi and co := S"-\A;
(=1
then Sm{K,co) = 0. From formula D.6.6) we deduce
S'„{K\pE,con pE) = 0 D.6.9)
for v-almost all rotations p e SO{n), where E e G(n, m + 1) is arbitrary
but fixed.
Let i e {I,. . . , k) and suppose that p e SO(n) is a rotation for which
pE meets A; in more than two points. Then pE meets lin^l,- in a

268 Curvature measures and quermassintegrals
subspace of dimension at least two. Since dimlin>!; = «-m, this
means that pE and lin^l,- are in special position. From Lemma 4.5.1 we
know that this happens only for the rotations p in a set of measure zero.
Hence we deduce that for v-almost all p e SO{n) the subspace pE
meets A in at most 2k points. Together with D.6.9) this shows that, for
v-almost all p, the support of S'„(K\pE, •) contains only 2k points and
hence (by the result proved above, applied to K\pE) K\pE is an
(m + l)-polytope with at most 2k facets.
If p e SO{n) is arbitrary, we can choose a sequence (Py)/6i«g in SO{n)
converging to p such that each projection K\pjE is an (m + l)-polytope
with at most 2k facets. Since K\pjE-^ K\pE (in the Hausdorff metric)
for /—»<», it follows that K\pE is an (m + l)-polytope with at most 2k
facets. By a well-known theorem, which we do not prove here (see Klee
A959a), Corollary 4.4), P is itself a polytope. ■
Another aspect of the local behaviour of curvature measures is
expressed by the fact that they cannot be positive on sets where the
Hausdorff dimension is too small. For instance, for m > 0 the measure
C„(K, •) cannot have point masses. This is a consequence of the more
general estimates expressed in the following theorem. Its proof requires
some facts from geometric measure theory; for these (as well as for the
notion of rectifiability) we refer to Federer A969).
Theorem4.6.5. Let K e X" and m e {\, . . . , n - 1}. Then
C„(K, P) ^ a,a5'"()8) D.6.10)
for every Borel set j8C E", with some constant a, depending only on n
and m, and
S„{K, oj) ^ a2^^"-'"-\oj) D.6.11)
for each (:^"~'"~^«-m-l) rectifiable set wC5"~', with some
constant ai depending only on n, m and K.
Proof. From D.6.2) we have
C„{K, P) = a„J Co(K n E, )8)d/z„.„(£)
J A{n,n — m)
^ a„„Co(K n E, E")f card(£ n )8)d/z„_„(£)
^ a,a5'"()8)
by Federer A969, 2.10.16 and 3.3.13). Similarly from D.6.6) we obtain
(with Ee G{n,m -t- 1))

Notes for Section 4.6 269
S„(K, 0)) = b„„j^^^^S'„(K\pE, (o n p£)dv(p)
S b„„m&xS'„{K\pE, S"-' n pE)l card(« n p£)dv(p)
£ jiU(n)
^ a2^"-'"-'(a>)
by Federer A969, Theorem 3.2.48). ■
Notes for Section 4.6
1. Theorem 4.6.1 was proved by Schneider A978a). Theorem 4.6.3 was
conjectured by Weil A973), p. 356, who showed it for m = 1, and proved in general
by Schneider A975f). The latter theorem implies, in particular, that
supp5,(/^, •) <^ supp5p(/^, •) ioT g > p,
which had been conjectured by Weil A973); see also Firey A975). Theorem
4.6.4 is due to Goodey & Schneider A980); for m = 1, Weil A973), Satz
(Theorem) 4.4, proved a stronger result. A special case of the estimate
D.6.10) appears in Federer A959), p. 489. The general case and the estimate
D.6.11) were noted in Schneider [43], §8. Firey A970a) proved the inequality
5„(A:, D(u, cos a)) g ADiK)" sin"-'"-^ aseca
for K eJC", /n e A,...,«- 1} and a e @, n/2), where D{K) denotes the
diameter of K and ^4 is a constant depending only on n and m. From this
one can also deduce an inequality of the type D.6.11), namely
Sr,iK, CO) g A„„D(Kr^^''-'"-\o))
foTa)e73(S''-^).
2. In R', Aleksandrov A942b) considered the specific curvature Co(K, •)/
CiiK, •), defined on the Bore! sets /3 with CziK, p) > 0, and its influence on
the local shape of bdK. He proved: If the specific curvature of a convex
body K e ^o'* bounded on a neighbourhood of jc 6 bd K, then either bd K
is differentiate at jc, or jc is an internal point of a perfect 1-face of K.
Consequences and further results in this spirit can be found in Busemann
[12], Section 5.
3. Weil A973, Satz 4.7) showed: If the convex body K e .T" satisfies
Si(K, ■)^co with some constant c, where o is the spherical Lebesgue
measure on S""', then K is a summand of some ball (the converse assertion
is trivial); in particular, K is strictly convex.

Mixed volumes and related concepts
S.l. Mixed volumes and mixed area measures
The concept of mixed volumes, which forms a central part of the
Brunn-Minkowski theory of convex bodies, arises naturally if one
combines the two fundamental concepts of Minkowski addition and
volume. In this section, we introduce mixed volumes and the closely
related mixed area measures, and we establish their fundamental
properties. This is most easily done by first considering strongly isomorphic
polytopes, and then using the approximation theorem, 2.4.14.
As before, we use the symbol V„ for the volume of «-dimensional
convex bodies, and we recall that Vi{K), for an /-dimensional convex
body K, is its /-dimensional volume.
Lemma5.1.1. Let P e 'Po be an n-dimensional polytope, fet F,, . . . , F/^
be its facets and let u-, be the outer unit normal vector of P at Fj (/ = 1,
..., N). Then
N
2K„_,(FX = o E.1.1)
1=1
and
1 '^
Vn{P) = - I,h{P, M,)K„-,(F;). E.1.2)
" /=i
Proof. Let z e E"\(o}. If n^ denotes the orthogonal projection onto
H^Q, then obviously
V„-,{n,P)= S <Z, Wi)V^n-l(Fi)= - E (Z, Mi)V^n-l(F,),
<z.u,)gO <z.u,)<0
hence '^^=i{z, Ui)V„-i{Fi) = 0. Since z was arbitrary, E.1.1) follows.
270

5.1 Mixed volumes and mixed area measures 271
From E.1.1) we see that the right-hand side of E.1.2) does not
change under translations of P. Since this is also true for the left-hand
side, we may assume that o g int F. Then P is the union of the pyramids
conv(Fi U {o}), /=1,...,N, which have pairwise disjoint interiors.
Formula E.1.2) is an immediate consequence. ■
In the following, we assume that c4. is a given a-type of «-dimen-
sional simple polytope, as defined in Section 2.4. By u\,. . . ,us we
denote the unit normal vectors (of the facets) of the a-type cyf,. For
P ed, we use the abbreviations
Fi := F(F, Ui),
Fit ■■= Fil^ Fy,
hi := h(P, M,).
The numbers /j,,. .., hfj, which determine F uniquely, are called the
support numbers of P. Further, we define
J := {('. /)l', / e {1, • . . , N}, dimFy = n-2}.
Observe that J depends only on the a-type cd. For (/, /) g J, let 0,y be
the angle between w, and Uj, and let Vjj 1 w, be the unit normal vector
of the (« - l)-polytope F; at its (« - 2)-face Fy. We write
hij := /j(F,-, Oil).
Using the obvious relation
Uj = M,cos0y + Vij sin 0ij,
we take the inner product of Uj with some vector in F,y and obtain
hji = hjcosec ©ij - /j,cot0y. E.1.3)
Lemma 5.1.2. The volume of P ec4 can be represented in the form
V^n(P) = Sfly,... A • • ^»' E1-4)
where the sum extends over /i, ...,/„ g A,..., N} and where the
coefficients aj^, j^ are symmetric and depend only on the a-type d.
Proof. We use induction on n. For « = 1, the assertion is trivial. We
assume that n>\ and that the assertion has been proved in dimension
n-\. Let ig{l,...,N}. By the induction hypothesis, there is a
representation
V^n-i(^/)= S a/t^;,^/t„-,^vt, • • • ^vt,-r E1.5)
Here the coefficients a/t^,'^,/t^_, depend only on the a-type cAi of F, and

272 Mixed volumes and related concepts
thus, by Lemma 2.4.10, only on the a-type c4.. By E.1.3),
hik, = A/t,cosec0,/t^ - hi cox 0ik,. E.1.6)
Inserting E.1.6) into E.1.5) and the latter into E.1.2), we obtain a
representation of the form
Vn{P) = 2«/„„A • • • \.
Introducing additional zero coefficients, we may assume that the
summation extends formally over all /,,...,/„ e {1,..., N}. Clearly the
coefficients can be assumed to be symmetric in their indices. Since the
numbers fl/t^'.^jt^.,, as well as the angles 0y, depend only on <^ the same
is true for the coefficients fly,.,.;^. ■
Now we assume that strongly isomorphic polytopes Pi,. . ., P„ eeyl
are given. We define F^K F^p, hY\ h'jp for F, in the same way as Fj,
Fij, hi, hij were defined for P. Then we introduce the mixed volume of
Px,...,P„\>y
V{P,,...,Pn)--=J.ai,...ihf:>---ht\ E.1.7)
where the coefficients fl/, ,./„ are those of E.1.4). Thus K(Fi,. .. , F„) is
symmetric in its arguments, and
V{P, ...,P)= V„{P). E.1.8)
Let polytopes P^,. .. , P„ecA and numbers Ai,...,A„^0 with
^A, > 0 be given. Then AjF, + . .. + A„F„ e c^ by Lemma 2.4.10 and
the definition of strongly isomorphic polytopes, and
/»(AiF, + . . . + A„F„, u) = Aj/jI') + . . . + X„h^r\
From E.1.4) and E.1.7) we immediately obtain
m
K„(A,F, + . . . + A„F„) = S A,, • • • kiVi^Pi^, ..., Pi). E.1.9)
ii....,(„=i
The following lemma expresses the mixed volume explicitly in terms
of Minkowski sums.
Lemma 5.1.3. For P^, . . . , P„eci,
K(Pi, ...,F„) = ^i (-1)"+* 2 K„(Fi, + . • • + P„). E.1.10)
Proof. Denote the right-hand side of E.1.10) by /(Fi,. . ., Pn),
including the case where one of the P, is equal to {o}. For Aj,. . ., An > 0 it
follows from E.1.9) that /(AiPi,. . . ,KPn) is a homogeneous polyno-

5.1 Mixed volumes and mixed area measures 273
mial of degree n in ki,. .. ,X„. Now
(-l)"^'«!/({o}, F2, ...,F„)
= 1 V„(P,)-
ISiSn
2 K„({o} + F^) + X V„(P, + Pi)]
S V^„({0} + Pi + Pk)+ S V^n(P, + P/ + Pk)
2Si<kSn 2Si<i<kSn
= 0;
thus /@Fi,A2F2. •••' ■^n^'n) "s identically zero for all A2, ...,An.
Hence, in the polynomial f(?^\P\,. . ., k„P„) all monomials Aj^ • • • A;^
with I $ {ii,. .. ,i„) have zero coefficients. Since 1 can be replaced by
each of the numbers 2, ...,«, we conclude that only the monomial
A] • • • An has a non-zero coefficient, and this is obviously equal to
v{p„...,p„). m
We note that E.1.10) implies, in particular, that V(Pi,. . ., P„) does
not change under arbitrary translations of any of the F,.
Formula E.1.2) extends to mixed volumes. To see this, let Fj,. . . ,
Fn_i be strongly isomorphic {n - l)-polytopes. They lie in (« - l)-di-
mensional affine subspaces that are all parallel, say to H. We can define
v(P'i, . . . , P'„-\) as the mixed volume, relative to H, of arbitrary
translates of Fj,. . ., P'„-i contained in H. By the translation in-
variance property just noted, this mixed volume is well defined.
Similarly, we can define the (« - 2)-dimensional mixed volume t;*""^^ for
strongly isomorphic {n - 2)-polytopes.
Lemma5.1.4. For Pi, . . . , P„ e cyf.,
nF„ . . . , F„) = - ih\%{F?\ ..., f\"\ E.1.11)
n ,=1
Proof. We use induction over n. The case « = 1 being trivial, assume
that n> I and the assertion is true in dimension n - I.
Denote the right-hand side of E.1.11) by W(Fi, ..., Fn). Applying
E.1.2) to AiFi -t-... + AnFn and using E.1.9) in dimension n - I, we
obtain
n
K„(A,F, + ... -t-A„F„)= 2 A,, ■ • ■ Ai^W(F;,, . . ., F;„)
.■„••■.'-='
for arbitrary A,, . . . , An > 0. Hence, it suffices to show that W(Fi, . . . ,
P„) is symmetric in its arguments. By the inductive hypothesis.

274 Mixed volumes and related concepts
W(Fi, ...,P„)
= i^M^'irT 2 /.?V"-«(F?\..., F^)
r<s
(for « = 2, the expressions u^"~^^ have to be replaced by 1). By E.1.3),
h9^h?s> + h'-Ph^l'' = /.(')[/.?)cosec0„ - /.?>cot0„]
+/j (')[/»(/> cosec 0,,-;j(^) cot 0„],
and this expression is symmetric in the upper indices 1 and 2. Thus
W(Fi, P2, Pi, ., Pn) = W(P2, Pi, F5, . . ., Pn), and since W(P,, P2,
Pi,. . ., Pn) is symmetric in its last n - \ arguments, as follows from its
definition and the symmetry of mixed volumes in dimension n - \, the
proof of the lemma is complete. ■
Mixed volumes are closely related to mixed area measures, which we
now define for strongly isomorphic poly topes. Recall that the area
measure Sn-\{K, •) of a convex body K e Xq is, by Theorem 4.2.5, the
Borel measure on the unit sphere 5"~' for which
S„.i(K, 0)) = X"-\t(K, (d)) for w e ^E""^).
In particular, for a polytope P (not necessarily with interior points) with
normal vectors ui, . . ., u^ "we have
5„_i(F, «) = S V^n-i(F(F. «,))• E.1.12)
For strongly isomorphic polytopes Pi,..., P„_i e cA we define their
mixed area measure by
S(Pu ..., P„_i, m) := S v{Ff\ ..., F^'V E.1.13)
U,E(ll
Thus 5(Fi,. .. , F„_i, •) is a finite measure on (B(S"~^). For P,,. . .,
P„ e ai and for A,,. .. , A„ ^ 0 we obviously have
Sn-iihPi + ... + A„F„, •)
m
= 2 Ai, . . . A,„.,5(Fi,, . . . , F,„.,, •), E.1.14)
'1 '/.-i=i
in analogy to equation E.1.9) and as a consequence of it, applied in
dimension n - I.
In the same way as Lemma 5.1.3 was proved (or from that lemma
and from E.1.13)) one obtains the following result, which expresses the
mixed area measure as a linear combination of area measures of
Minkowski sums.

5.1 Mixed volumes and mixed area measures 275
Lemma5.1.5. For Pi, ... , P„_i g <^,
S{Pu ...,P„-i, •)
n-l
= 7-^ 2 (-1)"^*-' 2 5„_,(F„ + . . . + F,, •).
We note that formula E.1.11), valid for Pi,.. ., P„ ec4, can be
written in the form
V(Pi, .. ., F„) = 1 L ,/»(Pi, u)dS{P2, ...,P„, u). E.1.15)
It is now easy to extend all this to general convex bodies.
Theorem 5.1.6 (and Definition). There is a non-negative symmetric
function V:(J^")" —*U, the mixed volume, such that
m
K„(A,/C, + ... + A„/C„)= 2 A,, •••A,„K(/C,.,...,/C,„) E.1.16)
for arbitrary convex bodies Ki,. . . , K„ ^JC and numbers Aj,. . . ,
A„ ^ 0. Further, there is a symmetric map S from {X")"~^ into the space
of finite Borel measures on 5""', the mixed area measure, such that
m
5n-l(^lJ^l + • • + ^m^m^ ' ) = Zi ^h ' ' ' '^'n-i'^^^'i' • • • ' ^'n-i' ' )
i,,...,1,-1 = 1
E.1.17)
for Ki,...,K„eJC" and A,,. . . , A„ ^ 0, where S(Ki,. . . ,
K„_i, •) := 5(A:„ . .. , A:„_i)( •). The equality
V{Ki, ...,K„) = - L,/»(A:„ u)AS{K2, ...,K„,u) E.1.18)
ft J^
holds for Ki, ..., K„eJC".
Proof We define ^
V(Ki,...,K„):=-^2(-l)"^'' 2 V^„(/f„ + . . . + /^U E119)
n'- k=i i,<...<n
for arbitrary Ki,. .. , K„ e X", which by Lemma 5.1.3 is consistent
with the definition already given for the mixed volume of strongly
isomorphic polytopes.
From the continuity of Minkowski addition and the volume functional
we deduce that K is a continuous function on {X")". Let it,,...,
K„ g X" be given. By Theorem 2.4.14 we can find sequences (Fi,),6i«g,
• • • > {Pni)i€N of polytopes such that P„-* it, for /-* <» (r = 1,... , «)
and such that, for each i eN, the polytopes Py,. . . , P„j are strongly
isomorphic. From E.1.9) and the continuity of volume and mixed
volume we deduce that E.1.16) holds.

276 Mixed volumes and related concepts
Similarly, we define
S(Ki, . . . , K„-i, •)
•= (r, ^ ni 2(-l)"^*'' 2 5n-i(/^„ + • •. + /^,,, •) E.1.20)
for i^i,. .. , i^n-i e ;^", which again is consistent with the former
definition, because of Lemma 5.1.5. Thus 5(it,,. ..,/l„_i, •) is a
signed measure on 7iE"~^). From the weak continuity of the area
measure, a special case of Theorem 4.2.1, we see that the map S thus
defined is weakly continuous; that is, Kn—* itj,.. . , K„-ii-^ K„_i for
i -* 00 implies
S(Kii, . . . , K„-ii, •) —» S{Kx, . . . , i^n-i, •).
An approximation argument analogous to that above now shows that
E.1.14) leads to E.1.17).
Formula E.1.18) is obtained from E.1.15) by approximating it,,... ,
K„ by strongly isomorphic polytopes and using the continuity of the
mixed volume, the weak continuity of the mixed area measure and the
fact that support functions are continuous.
It remains to show that V and S are non-negative. That V is
non-negative on strongly isomorphic polytopes follows by an obvious
induction argument, using E.1.11) (where o e Fi and hence hf =0 can
be assumed). The non-negativity of the mixed area measure on strongly
isomorphic polytopes is then a consequence of their definition E.1.13).
Finally, approximation by strongly isomorphic polytopes yields the
non-negativity of V and S in general. ■
We remark that equality E.1.13) and Lemma 5.1.4, valid so far for
strongly isomorphic polytopes, can now be shown to hold for arbitrary
polytopes. Let F,,. . . , F„ e'P". Applying E.1.12) to F = A,F,-t-
.. . + An_]Fn_] with Aj,. .., An_i ^0 and using E.1.14) on one hand
and E.1.12), E.1.9) (in dimension n - I) on the other, we find,
comparing the coefficients of Aj •• • An_j, that
5(Fi, . . . , F„_i, «) = ^v{F{P,,u),...,F{P„_„u)) E.1.21)
for coe "Ji^S"'^). Here the sum extends formally over all w e w, but in
fact only over the finitely many normal vectors of the facets of
F, + ... + Fn_i contained in (o. In view of E.1.21), formula E.1.18)
reads
V{P„ . .., F„) = ^ 2 h{Px, u)v(F(P2, u), . . ., F(P„, u)),
n usS-i
E.1.22)

5.1 Mixed volumes and mixed area measures 277
where the sum extends over the normal vectors of the facets of
We have already shown and used in the proof of Theorem 5.1.6 that
the mixed volume is continuous and the mixed area measure is weakly
continuous. We shall now establish some further properties of these
maps.
From E.1.19) and E.1.20) it follows that the mixed volume and the
mixed area measure do not change under an arbitrary translation of any
of their arguments. Further, it follows that
V(K,...,K)= V„(K),
V{aKu...,aK„)= V(K,,...,K„)
for any volume-preserving affine map a: E" -* E", and that
S(K,...,K, •)=5„_,(A:, •),
S(pKi, ..., pK„.u p(o) = S{Kx, ..., /<:„_,, (o)
for any rotation p e SO(n).
Let K, L eJC" be convex bodies such that K C L, and let Ki, ■ ■ ■ ,
K„ e JC" be arbitrary. From E.1.18) we obtain
V(K,K2,...,K„)- V(L,K2,...,K„)
= - Lxh(K, u) - h{L, u)]dS{K2, ...,K„,u)
ft ■'•^
SO,
with equality if and only if
h{K, u) = h{L, u) for each u e suppS(K2, . . . , K„, ■).
Thus
KCL=> ViK, K2,..., K„) ^ V{L, K2,..., K„). E.1.23)
By symmetry, the mixed volume is monotone in each of its arguments.
Unfortunately, no geometric description of the support of the mixed
area measure S{K2, ■ ■., K„, •) is known (see, however. Conjecture
6.6.13). Therefore, the complete characterization of the equality cases in
E.1.23) is an open problem. A special case is treated in Theorem
6.6.16.
Easier to decide is the question of when the mixed volume is strictly
positive.
Theorem5.1.7. For Ki,. . . , K„ e JC", the following assertions are
equivalent:
(a) V{K„...,K„)>0;
(b) there are segments 5, C AT, (i = 1, . . . , n) with linearly independent
directions;

278 Mixed volumes and related concepts
(c) dim (Ki^ + .. . + Kji) ^ k for each choice of indices 1 ^ /j < . . .
< i/t = « and for all k e {I, . . ., n}.
Proof. Suppose (a) holds. We prove (b) by induction. The case n = I
being trivial, suppose that n> I and that the assertion is true in smaller
dimensions. Since a convex body can be approximated by polytopes
contained in it and since the mixed volume is continuous, there are
polytopes PjCKi (i =!,...,«) such that V{Pi,. . . , P„)>0.
By E.1.22),
0 < V(P„ . . . , F„) = -^ 2h(Pu uMnPi, ",)> . . . , F(P„, M,)),
where the sum extends over finitely many unit vectors w,. Assuming that
o e relint Pi, all the summands are non-negative; hence there is some /'
for which
h{Pi, Uj) > 0,
V(F(P2, My), . . . , F{P„, Uj)) > 0.
By the induction hypothesis, the latter implies the existence of segments
Si C F{Pj, Uj), i = 2, . . . , n, with independent directions. Since
h(Pi, Uj) > 0, there is a segment Si C Fj that is not orthogonal to Uj.
Thus SjC Ki for i = I,. . . , n, and the directions of Si,. . . ,S„ are
linearly independent. Thus (b) holds.
Suppose that (b) holds and SjC Ki (i= I,... , n) are segments with
independent directions. The monotoneity of the mixed volume in each
argument implies V(Ki,... ,K„)^ V(Si,..., S„). By E.1.19),
nlV{Si, ...,S„)= V„(Si + ... + S„)>0,
which shows that (a) is true.
It is trivial that (b) implies (c). Vice versa, if (c) holds, we may
assume that o e relint it; for i = 1,..., «, and then the truth of (b) is an
immediate consequence of the subsequent lemma. ■
Lemma5.1.8. Let Li,. . . , L„ be linear subspaces of E". //
dim (Li, + ... + Li)^ k
for each choice of indices \^ ii< . . .< ii^^ n and for all k e
{1,. .. , n), then there are lines GiC Lj (i = 1,. .. , «) such that
dim(G, + . . . + G„) = «.
Proof. For each m-tuple (I'l,... , I'm) froi" {2,... , n) we have either
(a) L, C L;, + . . . + Li^ or (b) dim(Li n (L,-, + .. . + LJ) < dim L,.
Since Li is not covered by finitely many lower-dimensional subspaces.

5.1 Mixed volumes and mixed area measures 279
we can choose a line Gj C Li such that in case (b) we always have
G, 9t L,- + ... + L;„.
Let A2, •. •, ik) be a (A: - l)-tuple from {2,... , n}. Then either Li C
L,j + . .. + L;^ and hence
dim(Gi + L,j + .. . + L,^) = dim(Li + L.^ + .. . + L,-^) ^ A:,
or Li 9^ Ljj + .. . + L,^ and thus
dim(Gi + L,j + . . . + L,^) = 1 + dim(L,j + . .. + L,-^)
^ I + (k - I) = k.
Repeating the procedure with L2, and so on, we complete the proof of
the lemma. ■
An important property of the mixed volume is its Minkowski linearity
in each argument: for K, L, K2, ..., K„ e JC" and A, ju ^ 0 we have
V{XK + yiL, K2,..., K„)
= kV(K, K2...,K„) + nV{L, K2,..., K„). E.1.24)
This follows from E.1.18). By symmetry, V is Minkowski linear in each
of its arguments. The same holds for the mixed area measure:
S(kK +nL,K2 K„.u •)
= kS{K, K2,..., K„.u •) + tiS(L,K2,..., K„_u • )■ E.1.25)
For polytopes, this is clear from E.1.21) and the linearity of the mixed
volume (in dimension n - 1); the general case follows by
approximation.
The polynomial expansion E.1.16) can be written in a more concise
form. Introducing the abbreviation
V{K, K„...,K,,...,K,)=: V{K,[r,], ..., K,[r,])
and the multinomial coefficient
n\
n \ ■_ J —r^. r '^ 2// = " and rj e @, 1, .. ., «},
\r\ ... Tm;
•= < ri\ ■■■ rJ /=,
0 otherwise,
we find, using a standard combinatorial argument, for i^i,...,
/:„e3:"andA,,...,A„^0,
K„(A,a:, + ... + X„K„)
= 2 ( " ) AI'• • • A;"K(/:,[r,] K„[r„]). E.1.26)

280 Mixed volumes and related concepts
Similar polynomial expansions are obtained for the mixed volume if
some of its arguments are held fixed. Let an integer p e {I,..., n) and
convex bodies Cp+i, ..., C„ e X." be given. Then
K(AiA:, + . . . + Xn,K„[p], Cp^u ...,C„)
= 2 ( ^ ]a;'--a;"K(a:,[m], ..., A:4r„], Cp+,, ...,c„)
r„...r„=0\ri •■■ rmj
E.1.27)
For the proof, we develop both sides of the identity
K„(//(A,A:, + ... + X„K„) + tip+iCp+i + ... +
= V„{nXiKi + .. . + (iX„K„ + jUp+iCp+, + . .. + n„C„)
for ju, jUp+i, ..., ju„ ^ 0, using E.1.26), and compare the coefficients of
n''(ip+i •• • jUn in the two expressions thus obtained.
It is clear that the polynomial expansion E.1.17) leads to exactly
analogous results for the mixed area measure; for these, we use the
analogous notation. The case p = 1 of E.1.27) and the corresponding
formula for the mixed area measure expresses the known fact that both
maps are Minkowski linear in each argument.
A further additivity property is to be noted. For given p e {I,. . . , n}
and convex bodies C^+j,. .., C„ e JC", the function / defined by
f{K) := V{K[p], Cp+,, . .., C„), K e JC"
is additive on 3:!"; thus
V{K u L[p], Cp+„ ..., c„) + K(a: n L[p], Cp+,,..., c„)
= V{K[pl Cp^,, . .. , C„) + V(L[p], Cp^u ...,C„)
for K,Le X" such that K U L e X". In fact, for such K, L and for
jU, jUp+i, ..., ju„ ^ 0, we have
V„{n{K U L) + //p+,Cp+, + . . . + n„C„)
= V„{nK + Hp+xCp+x + . . . + n„C„)
+ V„(nL + (ip+iCp+i + . . . + (i„C„)
by Lemma 3.4.10. Developing both sides according to E.1.26) and
comparing the coefficients of n''(ip+i ■ ■ ■ n„, we arrive at the assertion.
Similarly, from the additivity of the area measure (a special case of
Theorem 4.2.1) we infer that the map
K >-^ S(K[p], Cp^u ■ ■ ■ , C„-i, •)
is additive on 3:!".
Finally, we note that the mixed area measure, when considered as

Notes for Section 5.1 281
defining a mass distribution on the sphere, always has its centroid at the
origin:
l„.,udS(Ku...,K„.uu) = o. E.1.28)
In fact, from E.1.18) and the translation invariance of the mixed
volume we have
0 = V(Ki, ..., K„-u K„ + t)- V{Ku ..., K„_u K„)
for arbitrary t e E", from which E.1.28) follows.
Notes for Section 5.1
1. The theory of mixed volumes was created by Minkowski A903,1911). A
presentation of its early development and history can be found in Bonnesen
& Fenchel [8]. The theory attained a more general and elegant form when
the mixed area measure was introduced, independently by Aleksandrov
A937a) and by Fenchel & Jessen A938).
If the mixed volume has previously been defined, then formula E.1.18)
can serve as a starting point for the definition of the mixed area measure: Let
Ku ..., Kn-i 6 X" be given. For K 6 X\ let h := Hk and f{h) := V(K,
Ki, ..., K„-i). This defines an additive functional / on a subset of CE''"'),
the real vector space of continuous real functions on 5""' with the maximum
norm. It can be shown (see Section 5.2) that / has a unique extension to a
continuous linear functional on C(S"~^). By the Riesz representation
theorem, there exists a unique measure S(Ki,..., ^„-i, •) on ^E""') for
which, in particular,
V(K, Ku-.., Kn-i) = - L M^' ")d5(/:i, .... ^„-i, ")
n Js '
for all K e X". The properties of S can be deduced from those of V. By this
method, the mixed area measure was introduced by Aleksandrov A937a),
and in a similar way by Fenchel & Jessen A938). For a geometric
interpretation, Aleksandrov first defined S„-i(K, •), essentially by formula D.2.24),
and then used his definition of the mixed area measure to prove the
expansion E,1.17). Later, Aleksandrov A939b) simplified part of his
reasoning by working with weak convergence. Fenchel & Jessen A938) used the
formula
ViL, K,...,K) = ^ L.''(^' u)dSn-i{K, u)
to define S„-i(K, •) and later obtained an intuitive interpretation by proving
formula D.2.9).
2. Mixed volumes can be introduced in different ways, but at some point, a
symmetry property such as that of W(Pi,.. ., P„) in the proof of Lemma
5.1.4 has to be shown. For the case of polytopes, one may compare
Minkowski A911), §21, and Aleksandrov B], p. 411; for analytic proofs, see
Bonnesen & Fenchel [8], p. 57 and the references on p. 61.
3. Lemma 5.1.8 is taken from LeichtweiB [23], pp. 176-7. A refinement is
proved and used in Schneider A988a).

282 Mixed volumes and related concepts
4. Polynomial expansions for valuations. The existence of the crucial polynomial
expansion E.1.16) can be shown for functions that are considerably more
general than the volume:
Theorem. Let cp be a real-valued translation invariant valuation on "p". Then
for Pi,..., P^e 'p" and rational Ai,.. ., A* S 0, q>{^iP\ + ... + A^P*) is a
polynomial in Ai, ..., A^ of degree at most n. The coefficient of Ai' • • • A" is
a translation invariant valuation in Pj that is homogeneous of degree ry (y = 1,
...,k).
If, moreover, q> is defined and continuous on X", then the expansion
extends to XiKi + ... + X/^Kk with Ki,..., K/^e 7^" and real Ai,.. .,
A* = 0. The question whether such a result might hold was posed by
McMullen in 1974. He developed a theory yielding the above theorem and
more general results (McMullen 1975c, 1977); see also McMullen & Schneider
[40], §10. Using different approaches, McMullen's question was answered by
Meier A977) and Spiegel A978).
Meier's approach to mixed valuations was via 'mixed polytopes'; for these,
see also McMullen & Schneider [40], pp. 200-1, and, as part of a quite
general theory unifying many previous results, McMullen A989).
5. The mixed volume as a distribution. The representation E.1.18) exhibits the
linearity of the mixed volume in its first argument, but not the symmetry of
the mixed volume. To obtain a more symmetric representation, Weil A981b)
expressed the mixed volume as a distribution. More generally, such a
representation is possible for multilinear and continuous functions on JC".
The following theorem is due to Goodey & Weil A984):
Theorem. Let q>: (X")"—* U be multilinear and continuous. Then there is a
unique distribution 7" on E""')" that can be extended to a Banach space of
functions on E""')" containing the tensor products h^,® ... ® hic„, for all
Ki, . . ., K„e X", in such a way that
In fact, there are real functions /^ of class C°° on E""')" for y = 1, 2, ...
such that
L, ,• • • L-.''(^i' "')••• ''(^'» "<■)/;("•. • • •' "n)do{ui) ■ ■ • do(u„)
as y —» », uniformly for all Ki in any fixed ball (a is the spherical Lebesgue
measure).
6. Characterization theorems. As mentioned in Section 4.2, Hadwiger's
characterization theorems 4.2.6 and 4.2.7 for linear combinations of quermass-
integrals were the starting point for several similar investigations.
Characterizations of functionals related to mixed volumes by means of their additivity
properties require strong additional assumptions. F^ry A961) characterized,
for a given convex body U e X", the functionals
n
q,:K^ J,c,V(K[il U[n - /]),
1=0
where to,..., c„ 6 R, as the translation invariant, continuous valuations q>
satisfying (p{K) = <fiL) whenever ViK[i], U[n - /]) = V{L[i], U[n - /]) for
/ = 0,...,«. Firey A976) showed that an increasing Minkowski linear
function q>:X"—*U that is zero on one-pointed sets must be of the form
qiK) = V(K, L[p - 1], S,^u ...,S„)

Notes for Section 5.1 283
with an essentially unique convex body L and pairwise orthogonal unit
segments 5p+i, ..., 5„ that span the orthogonal complement of the affine
hull of L.
The following theorem was proved by McMullen A980): Let cp be a
continuous, translation invariant valuation on X" that is homogeneous of
degree n - \. Then there is a continuous function g: 5""' -»IR such that
<K^) = /j,.,g(")d5„-i(^, u) for K 6 X\
From this result, McMullen deduced that for any such valuation <p there exist
sequences {Lj)j^hi, (Mj)je\ in JC" such that
(p(K) = \im[V{K[n - 1], L,) - V(K[n - 1], M^)]
for K 6 X".
The following counterpart is due to Goodey & Weil A984): A functional
<p: X" —»R is a continuous, translation invariant valuation that is
homogeneous of degree 1 if and only if there are convex bodies Lj, Mj for y = 1, 2,
... such that
(p{K) = lim[V(A:, L\n - 1]) - V(K, M,{n - 1])]
uniformly for all A: 6 {A: 6 I^i::''! A: C mB") and for any m > 0.
McMullen A990) showed: If (p-.JC^R is a monotone, translation
invariant valuation that is homogeneous of degree « — 1, then there is a convex
body Ls X" such that
qiK) = V(K[n - 1], L)
for all Ke X".
An identity for mixed volumes. If K, L, K^, . . ., Kn^ X" are convex bodies
such that K U L is convex, then
V(K, L, Ki, ..., K„) = V(K U L, K n L, Ki, . . . , Kn).
This follows from the additivity property of the mixed volume. It was noted
by Groemer A977a), p. 160, and proved in a shorter way in McMullen &
Schneider [40], p. 179. An abstract version was given by McMullen A989),
Theorem 15.
Mixed area under separate rotations. The mixed volume V{Ki,. . ., K„) is
invariant under simultaneous rigid motions of the arguments, but not, of
course, under separate motions applied to the arguments. Gortler A937a)
investigated the pairs K, L in X^ for which V{K, L) is invariant under
arbitrary rotations of one of the arguments.
Characterizations of convex bodies in terms of mixed volumes. A convex body
is determined, up to a translation, by the values of its mixed volumes with
sufficiently many other convex bodies. As an example, the following result
was proved in Schneider A974d): Let /' 6 {1,.,.,«- 1} and let K, K' 6 X"
be convex bodies of dimension S / + 1. If
V{K[i\, B''[n - i - 1], L) = V(K'[i], B-ln - i - 1], L)
for all two-dimensional convex bodies L, then K is a translate of K'. In fact,
it is sufficient to assume the equality only for those convex bodies L that are
congruent to a fixed triangle having at least one angle that is an irrational
multiple of IT.
In vie^v of such uniqueness results, it is not suprising that it is possible to
characterize certain classes of convex bodies in terms of relations satisfied by
mixed volumes. For example, Weil A974b) showed that L 6 ^'' is contained
in some translate of /^ 6 :^'' if and only if

284 Mixed volumes and related concepts
V(L, K2,..., K„) S V(K, K2,..., K„)
for all K2, ..., K„e X". Furthermore, each of the following conditions (a),
(b), (c) is equivalent to the fact that L is a summand of K:
(a) (Weil, 1974b) For each y 6 {1, ...,«- 1}, the real functional
Tk.l.j = 2, (^)(-!)'-* V(^[*]. m - k], B-ln - 1 - y], •)
is monotone on X".
(b) (Matheron, 1978b) For each y 6 {0, ..., «} and A 6 @,1),
V(K~- XL[n - j], L[j])
=!(%-'>-
l)*A*V(/:[n - k- j], L[k + /]).
(c) (McMullen, 1990) For all polytopes P, Q b X" with PCQ,
V{Q[n - 1], K) - ViQ[n - 1], L) - V{P[n - 1], K)
+ V{P[n - 1], L) g 0.
Other results in a similar spirit concern zonoids, generalized zonoids and
centrally symmetric convex bodies. For K e X" and u 6 5""', let Vn-iiK")
denote the (n - l)-dimensional volume of the orthogonal projection of K
onto the hyperplane Hu.o', here K" = K\Hu,o- Weil A976b) proved that a
centrally symmetric convex body K isa zonoid if and only if
V{K, L[n - 1]) g V(K, M[n - 1])
for all centrally symmetric L, M e X" satisfying Vn-iiL") g Vn-i(M") for all
1/6 5""'. Goodey A977) showed that a centrally symmetric convex body
K 6 X" is a generalized zonoid if and only if there exists a constant C such
that
\ViK, L[n - 1]) - ViK, M[n - 1])| S C sup |V„-,(^'') - V„_,(M")|
for all centrally symmetric L.MeX". He also showed that K e X" is
centrally symmetic if and only if
ViK, L[n - 1]) = ViK, M[n - 1])
for all L,Me X" satisfying Vn-^L") = V„_i(M'') for u 6 5"-'. Similar and
more general results were proved by Weil A979a) and by Goodey A984a, b);
for some necessarv modifications of the latter results, see §6 of Goodey &
Weil A991).
5.2. Extensions of mixed volumes
The mixed volume, as defined in the preceding section, is a function on
w-tuples of convex bodies that is Minkowski-linear in each variable. In
the present section we shall collect some information on the possibilities
for extending the mixed volume to other kinds of argument, while
preserving or generalizing its linearity properties.
Our first observation is a negative one: under any extension to
non-convex compact sets, the mixed volume would lose its essential
properties. More precisely, the following result holds (Weil 1975a).
Theorem5.2.1. Let X' d^" be a class of compact sets containing X"
and closed under Minkowski addition. If there exists a function

5.2 Extensions of mixed volumes 285
V:{JC')" ^■U that is Minkowski-additive in each variable and for which
V{K,. .. , K) = X"(K) for K e X', then X' = X".
Proof Suppose that X' and V satisfy the assumptions. Since V(-, Ki,
. . . , K„) (where K2,. . . , K„ e X' are fixed) is Minkowski-additive, the
observation at the end of Section 3.4 shows that
V{Ku K2,..., K„) = K(conv K^, K2, . .. , K„)
for Ki e X'. Similarly,
V(con\ Ki, Ki, Ki, . . . , K„) = V(con\ Ki, com K2, K^, . . . , K„),
and so on. In particular, for M e X' we have ^"(M) = ^i"(conv M). If
dimconvM = «, this implies M = convM. If dimconvM<«, we
choose a convex body K in an affine subspace complementary to aff M
such that dim (it + convM) = n. For M' ■= K + M we have
^^"(M') = :^^"(con\M') and thus K + M = M' = com M' = K +
convM. Since K and conv M lie in complementary subspaces, this
implies M = conv M. Thus each element of X' is convex. ■
If, however, we consider the mixed volume not as a Minkowski
multi-additive function on convex bodies, but as a multi-additive
functional defined on support functions, then a natural and sometimes useful
extension is possible. In the following, we identify convex bodies with
their support functions, restricted to 5""', and hence also write
V(K„...,K„)= V(h^^,...,hf:^) E.2.1)
ioT Ki,..., K„eX".
The real vector space C(S"~^) of real continuous functions on the
sphere 5""' contains as a subspace the vector space D{S"~^) spanned
by restrictions of support functions. If g e DE""') has the
representations g = hK-hL = hf:-hi- with K, L, K', L' e X", then K + L'
= K' + L; hence it follows from the additivity of the mixed volume that
V{g, K2,..., K„) := V{K, K2,..., K„) - V(L, K2, . . . , K„)
does not depend on the special representation. Clearly V{-, K2, ... ,
K„) thus defined is a linear functional on DE""'). Similar extensions
are possible in the other arguments. Explicitly, for g, e DE""') with
the representations g, = hf - h], where hf, h] are restrictions of
support functions (i = l,...,«), we can define the mixed volume of
gi,- • ,gn by
V(gu-,gn)= 2 (-iy'*-*''"V{h\\...,h)p).
v„...,v„e{0,l)

286 Mixed volumes and related concepts
Then V is symmetric and «-linear on D(S"~^), and satisfies E.2.1).
Similarly, the mixed area measure has an (« - l)-linear extension to
D(S"~^) given by
S{gu . . . , g„-„ •) = 2 i-iy* ^""-S{h\\ ..., hU , •).
V, v„.,e{0.1)
Formula E.1.18) extends trivially to give
V(gl, ■■■, gn) = — J^„.,5l(w)d5(g2, . . ., gn,u)
forg,,...,g„6DE"-').
For given convex bodies K2,.. ■ , K„ e X", we can immediately
obtain an extension of the functional V( •, K2,. . . , K„) to the space
CE"~') (with the maximum norm) by defining
V(f, K2,..., K„) := -^ j^J{u)dS{K2, ...,K„,u) E.2.2)
for / e CE""'). Then V{ •, K2, ■ ■ ■ , K„) is a continuous linear
functional on CE""'), with norm V{B", K2,. .., K„). The mixed area
measure S{K2,. . . , K„, •) turns out to be the unique measure
representing this linear functional according to the Riesz representation
theorem. (This remark has played a role in the literature; cf. Section
5.1, Note 1).
It should, however, be pointed out that a further extension of the
mixed volume to a multilinear continuous functional on CE"~') is not
possible. It suffices to show this for n = 2.
Theorem 5.2.2. There is no bilinear function K; CE')^ —»R that is
continuous in each variable and satisfies V(hK, h^) = V{K, L) for
K,L€ X\
Proof. Suppose V exists. We identify 5' with [0,2n] via (cos a,
sin a) i-> a. For a continuous function / on [0, 2tt] and a convex body
K e 7K} formula E.1.18) extends to give
V{f,hK)-\\j{oc)AS{K,oc).
For the proof, we approximate / uniformly by differences of support
functions (which is possible by Lemma 1.7.9) and use the linearity and
continuity of V in its first argument.
Now choose /(a) = 1 - |sinar|'^ and let 0<e<l. Let K be the
segment with endpoints ±@, a), where a = 1/e. The measure S{K, •) is
concentrated at the points or = 0 and or = tt and associates a weight 2a
with each of these points. Let L be obtained from AT by a rotation

5.2 Extensions of mixed volumes 287
around its centre by an angle q>, where sin q>= e^. Then K and L are
Hausdorff distance e apart. On the other hand,
V{f, Hk) - V(f, ht) = 2a- 2a(l - Vsinq?) = 2.
Thus the function g^ '.= h^ - hi satisfies \\ge\\ = e and V(f, g^) = 2,
which for e —»0 contradicts the continuity of V in its second variable. ■
If convex bodies are identified not with their support functions but
with their characteristic functions, then a different extension problem
arises for the mixed volume. We shall describe briefly the theory
developed by Groemer A977a) for this case. For K eX", let K* be the
characteristic function of K, and let V(J^") be the real vector space
spanned by these functions for K e DC". (Compare Section 3.4, Note 7.
Since Groemer's notation K* is used only in that note and in the
present section, no confusion with the polar body should arise.) The
first step is to show that there exists a unique bilinear map
t/»: V{X"Y -'■ViJC") such that
V</C*, L*) = {K + D* for K, L eJC".
The construction of this map uses the Euler characteristic on V(JC") as
established by Groemer A972, 1974). The vector space V(J^") together
with the multiplication x defined by / x g = v(/, g) is a commutative
algebra over U with unit element. The Euler characteristic is an algebra
homomorphism. To every affine map a: E" —»E" there exists a unique
linear map a: V(J^")^V(jiC") such that a(K*) = (aK)* for K e J^".
In the special case ax = Xx with A ^ 0 one writes a{f) : = A o / for
/ e V(X"). Obviously there is a unique linear functional V„ on V(J^")
such that V„(K*) = V„(K) for K e JC". Groemer showed that E.1.26)
can be generalized as follows. For non-negative integers ri,... , r„ with
ri + . . . + r„ = n there exists exactly one «-linear map
Vir, .„):n:^")"-R
such that
Vir, rjiKl .■.,Kt)= V(K,[r,], . . . , K„[r„])
for Ki,...,K„€j^". For /i,...,/„e V(J^") and Aj,. . . , A„ ^ 0 we
have
^n((A,o/,)X ... X(A„o/„))
'Ar,
For the further investigation of these generalized mixed volumes we
refer to the original article of Groemer A977a).
Finally we consider a very special situation where the mixed volume

288 Mixed volumes and related concepts
can be extended in a natural way. This extension will be particularly
useful for the proof of the Aleksandrov-Fenchel inequality in
Section 6.3.
Let c4. be a given a-type of strongly isomorphic simple ^-dimensional
polytopes. As in Section 5.1, we let w,, ... , W/^, be the normal vectors
of the facets of any P ec4., and we use the notation of the first part of
Section 5.1. The N-tuple
is called the support vector of P. We recall that the support numbers of
the facet F; are given by
hij = hjCOsecOji - hj cot ©ij for (/, /) e J,
and furthermore that
nF„...,F„)=2«..../„A?--•^l"^ E.2.3)
v(Ff\...,Fr'')=2ait,„.,h^---ht:' E.2.4)
for Pi,. .. , P„ ec4, where the coefficients are symmetric and depend
only on the a-type ct/.. The summations in E.2.3) and E.2.4) can be
considered to extend respectively over all j\,. . ., )„ and all k^,.. .,
k„_i, from 1 to N, if zero coefficients are introduced and, say, hy :- 0
if (/, /) ^ J.
We can consider V, as given by E.2.3), to be defined for w-tuples of
support vectors, and we now extend this to arbitrary N-tuples
Xr = {x\'\ . . . , x^;,^) e U"", r = 1, ...,«,
by putting
V^(^i,...,^„):=2«w„^i:^---^^ E-2.5)
Then V is an N-linear function on U'^. Furthermore, for AT = (a:,, ...,
x^) e R'^ we define
[jrycosec 0;y - Jfjcot 0/; if (i, /) e J,
""'' ' @ if (/, /) ^ J
and
AiX ■= (xn, . . . , XiN), E.2.6)
so that Aj-. U'^ ^ R'^ is a linear map. Then we put
v(A,X„ . .. , A,Z„.,) := 2a>^^..,„_^'ll^ ■ ■ ■ xt!'- E-2.7)
Lemma 5.1.4 now extends to give
V{X„ . . . , Ar„) = - ^x^^v{AiX2, ..., A,X„). E.2.8)
n (=1

5.3 Special formulae for mixed volumes and quermassintegrals 2S9
This follows from the facts that the equality is true for support vectors,
P + eXr is a support vector '\i P ec4 and e is in a suitable
neighbourhood of 0 (as follows from Lemma 2.4.12), and both sides are «-linear
functions.
Note for Section 5.2
The multilinear extension of V and S to differences of support functions appears
in Aleksandrov A937a). It was further investigated (and used for
characterizations of convex functions within the vector space of differences of support
functions) by Weil A974a); see also Weil A974b).
The assertion of Theorem 5.2.2 was first proved, in a different way, by Meier
in 1982 (unpublished). The construction used above is taken from Schneider
A974c), in a modified form.
5.3. Special formulae for mixed volumes and
quermassintegrals
The purpose of this section is to collect a number of special formulae for
mixed volumes and mixed area measures and, in particular, for
quermassintegrals. These formulae relate mixed volumes to the notions of
Section 2.5, that is, to local curvature functions in the case of convex
bodies of class C+, and also to the integral-geometric considerations of
Chapter 4. Moreover, some special representations are treated that are
valid in restricted cases or for particular convex bodies, such as zonoids.
In the first part of this section, it is supposed that the convex bodies
K, L, Ki € Xo are all of class C+. First we recall that in this case, by
D.2.19) and D.2.20), the curvature measures and area measures have
the explicit representations
S„{K,m)=\s„dX"-' E.3.2)
for /3e^(E"), ojeliiS"-^) and m = 0,1,...,«-1. Applying E.3.2)
for m = n - I to the body K = XiKi + . . . + X„.iK„.i with Aj,. . . ,
A„_i U 0 and using E.1.17) and B.5.33), we find
S{Ku ■■■, K„.u (o) = \j{K^, ..., K„.u u)<iX"-\u). E.3.3)
Formula E.1.18) for the mixed volume now reads
V{K„ ...,K„)
^ 7 L-^^^" "^^^^^2' ■■■'^"' u)dX"-\u). E.3.4)

290 Mixed volumes and related concepts
Since the mixed volume is symmetric in its arguments, this yields
integral formulae such as
l^.M^^u u)s{K2, Ki,..., K„, u)dX"-\u)
= l^.Mf^2, u)s{Ku Ki,..., K„, u)dX"-\u). E.3.5)
The area measures of lower order are obtained from the mixed area
measure by taking a special case: in analogy to formula B.5.35), namely
SjiK, u) = s(K, ..., K, B",..., B", u), E.3.6)
J n-l~ j
we have
Sj{K, 0)) = S(K, ...,K,B\...,B", (o). E.3.7)
i n-\-j
This follows immediately from E.3.3), or (for general convex bodies)
from a comparison of the local Steiner formula D.2.16) for m = « - 1,
that is,
5„_i(A: + pB\ (o) = 2p"-i-'(" T M Si(K, (o),
1=0 \ J I
with the polynomial expansion E.1.17). Consequently, the quermass-
integrals or Minkowski functional Wj,. . . , W„, which in Section 4.2
were introduced by
n
turn out to be special mixed volumes: by E.3.7) and E.1.18),
Wi{K) = V{K, . . . , K, B", . . . , B") E.3.8)
n - I
for i = l,. . . ,n, but also for i = 0, since Wq(K) = V„(K), by
definition. Of course, E.3.8) is valid for arbitrary convex bodies K.
From E.3.2) and E.3.1) we see that
W.
<^> = i/s"-.'"-'^^'""'' (^•3-9>

5.3 Special formulae for mixed volumes and quermassintegrals 291
W.
w,
for / = !,..., n. Using E.3.5), we obtain another representation,
namely
'i(K) - ^ j^jKSn-i-idX"-' E.3.11)
for f = 0,...,«- 1. From E.3.10) we see again that nWi(K) is the
surface area of K, and the case i = n-1 of E.3.11), together with
^kM + hj({-u) = w(K, m), shows that
W„.i{K) =-^ b(K), E.3.12)
where b{K) is the mean width of K (by approximation, this extends to
general convex bodies).
Using B.5.30) and B.5.28), we can transform formula E.3.11) into
where qic(x) := hK(v(x)) = {x, v(a:)) for x ebd K.
In differential geometry, the equalities resulting from E.3.9), E.3.11)
and from E.3.10), E.3.13) respectively,
t-Ad^^""' = /,„.,A^^/-id^^"-'. E.3.14)
and
for / = 1,...,«- 1, are known as Minkowskian integral formulae.
The case i=n — 2 of E.3.11) can be given a form that is quite
useful. By B.5.23) we know that
5, = /ja: + ^sh^:,
n — v
where A5 is the spherical Laplace operator on 5""'. This yields
W„.2{K) = ^\^.M[hK + -^^sh^<i^'"-'. E.3.16)
More generally, it follows from E.3.4) and E.3.6) that
V{K, L,B",...,B")
= iL.^^^ + -^^shL]d^^"-'. E.3.17)

292 Mixed volumes and related concepts
Using Green's formula on 5""', we can rewrite E.3.16) as
W„.2(K) = 7|,„., (a'/c - -~y Vs;,;,)d^"-', E.3.18)
where Vj denotes the second Beltrami operator (square of the gradient)
on 5""'. Taking a local parametrization N: M-* E" of the sphere 5""'
and using the notation of Section 2.5, in particular X = ^o N and
Cij = {Nj, Nj), we have, for h = hj(<> N,
n-l
with
One easily finds
n-l
h' := 2e^'hr
/=i
n-l
X = hN + ^ h'Ni
and thus
= \X - hN\^ = \XV - h\
Thus (^s^k)(^) 's equal to the squared distance between the point
where the support plane H(K, u) touches K and the foot of the
perpendicular to H{K, u) through o.
Heil A987) pointed out that formula E.3.18) holds for arbitrary
convex bodies K. Observe that in E^ this is a formula for the surface
area.
In the second part of this section we now treat the relationship of
mixed volumes and of mixed area measures to the integral-geometric
considerations of Section 4.5. In particular, we collect some observations
on translative integral geometry and on rotational mean values. The
convex bodies K, K', Ki occurring in the following are arbitrary.
First we recall the principal kinematic formula (the case / = 0 and
j8 = /3' = E" of Theorem 4.5.2), written in terms of quermassintegrals:
\ XkK n gK')dii{g) = — i (") W,{K)W„.,{K'). E.3.19)
If here the motion group G„ is replaced by the group of translations,
which may be identified with E", we obtain
/,
jdK n (K' + r))dr = Z^"{t e E"\K n (K' + r) + 0}
= V„{K - K')
V{K[k\, -K'[n - k]). E.3.20)
-10

5.3 Special formulae for mixed volumes and quermassintegrals 293
Thus mixed volumes appear inevitably in translative integral geometry.
(See also Section 4.5, Note 5.)
Since the proof given in Section 4.5 for the principal kinematic
formula first treated the translative case, it yields, incidentally, a new
representation for the mixed volume in the special case where the
arguments are two polytopes. Suppose that P, P' e 'J)" are polytopes.
Theorem 4.5.3, for / = 0, ^ = j8' = E", gives
l„xiP n (F' + t))dt = xiP)Vn{P') + v„{P)xiP')
n-l
Using E.3.20) for K = P, K' = P' and then replacing P by XP in both
equations and comparing the coefficients of equal powers of A, we
obtain
(:)
V(P[k], -P'[n - k])
= S S y(F,F',P,P')[F,F']V,{F)V„.,{F'). E.3.21)
At present it is not clear whether this is a special case of some formula
valid for general convex bodies.
Another translative integral formula, but now referring to subspaces,
is obtained as follows. Let E^ C E" be a A:-dimensional linear subspace,
k e {I,.. ., n - I}, and let U/^ be a convex body contained in E/^. By
t;**\ W^*\ 5**\ 5^*^ we denote respectively the mixed volume, /th
quermassintegral, mixed area measure and area measure of order /,
each computed in a A:-dimensional affine subspace. For A ^ o we have
t{"]^'V(K[n-j],Udj])
= V„{K + XU,)
= j^/MK + W,) n (£, + 0)d5S5"-*(r)
= \^yk([K n (£, + t)] + w,)<iX"-'{t)
By comparison, the following two consequences are obtained. Choosing
for Uk the A:-dimensionaI unit ball Bk = B" f) E^, we find
l^^wf\K n (£, + r))d^."-*(r) = (^j(*) V{K[n - /], Bdj])
E.3.22)

294 Mixed volumes and related concepts
for /■ = 0,. . . , A: - 1. This is another example where a formula in
translative integral geometry requires mixed volumes. The case j = k
yields
(:)
V(K[n - k], Uk[k]) = Vk(Uk)V„.k(K\Et)
Here we choose Vk{Uk) = 1, replace K by ^^jKi and compare the
coefficients. Interchanging the roles of E^ and £^, we arrive at
o^'\K,\Ek, ..., Kk\Ek) = l^j V(K„ ...,Kk, (/„-,{« -
k]h
E.3.23)
which is valid if U„-kC Ej; and V„.k{U„-k) = 1. Thus A:-dimensional
mixed volumes of orthogonal projections onto E^ can be expressed by
mixed volumes in E".
We turn to rotational mean values. If in E.3.20) we replace K' by
-pkK' with pe SO(n) and A^Q, integrate over SO{n) with respect to
the invariant measure v, use E.3.19) and finally compare the
coefficients of equal powers of A, we end up with the equality
l^^^V{K[m], pK'[n - m])dv(p)
= — V(K[m], B"[n - m])V(B"[m], K'[n - m]) E.3.24)
for m = 0,.. ., n. Here we replace K, K' by S^^i ^""^ Si"/J^>'
respectively, expand both sides and compare the coefficients to obtain
(after changing the notation)
= — V{K„ ...,Kr„, B"[n - m])V{B"[m], K„^„ ..., K„).
E.3.25)
A mean value formula for projections is obtained as follows. Let Ek be
a A:-dimensional linear subspace of E" (k e {I,. .., n - 1}) and choose
a convex body U„-k C Et with V„.k{Vn-k) = 1- Then
l^^^v^'KK^lpEk, ..., Kk\pEk)dxip)
= L .il) ^(^" • • •' ^*' P^n-*l« - k])Mp)
JSO{n) \k I
V(Ku ...,Kk, B"[n - k])V(B"[k], {/„_,[« - A:]).
(:)

5.3 Special formulae for mixed volumes and quermassintegrals 295
Here the formulae E.3.23) and E.3.25) were used. Since
V{B"{k\ {/„_,[« - k\) = W,([/„_,) = -^ K„-/t([/„-/t) = 7^
by D.2.26), we arrive at the formula
L ^i;<*>(A:i|p£,, ..., A:,|p£,)dv(p)
K,
As a special case, we note that
= ~V{Ki,..., Kk, B"[n - k]). E.3.26)
Lin)^^'-'^^^^^'^'^'^^^ = ^ ^n-/(/^) E.3.27)
for j = 0,.. . , k, which is known as Kubota's integral recursion. The
further specialization A: = / = « - 1 is Cauchy's surface area formula.
Similar formulae for mixed area measures are obtained if we use the
equality (a special case of Theorem 4.5.6)
l„^^S„.i{K + pK', (o n p«')dv(p)
O^n k^0\ k j
valid for Borel sets m.co' e^(S"~^). In the usual way, by 'mixing', we
deduce
f S(Ki, ..., K„, pK„,^x, ..., pK„.i, (o n pta')dv(p)
= — S(Ku ..., K„, B"[n - 1 - m], w)
X S(B"[ml A:„+„ .. . , K„.u m') E.3.28)
for m = 0, ...,«- 1.
For projections onto a A:-dimensional linear subspace £/t, formula
D.5.26) states that
j^^^^s]'\K\pE,, o> n p£,)dv(p) = -^ S,{K, (o) E.3.29)
for (oeOiiS"'^) and / = 0,.. ., A: - 1. From the case j=k-\ we
deduce, again in the usual way, the rotational mean value formula
isOin/'^^^'^P^'^' • • • ' ^^-'l/^^*' «^ ^ pEk)Mp)
= --^S{Ku ■■■, Kk.u B"[n - k], a>). E.3.30)
0}„

296 Mixed volumes and related concepts
Of particular importance are projections onto hyperplanes. For these,
we introduce some simplified notation. For K e J!." and a unit vector
M g 5""' we denote by K" the image of K under orthogonal projection
onto the linear subspace orthogonal to w; thus K" = /l|//„o. For the
(n - l)-dimensional mixed volume in a hyperplane we write v instead of
{,(«-!) If (/ is a segment of unit length parallel to u, then
v(Kl ..., K"„-i) = nV{Ku ..., K„.„ U) E.3.31)
by E.3.23). We may assume that U has centre o; then h(U,v) =
\\{u, v)\ and hence, by E.1.18),
o(Kl . .. , /C"„_i) = i l^Jiu, v)\dS(Ku .. . , K„.u v). E.3.32)
By integration we immediately obtain
\^.,o{Kl ..., K"„.,)d^."-\u) = nK„.,V(K„ . . . , K„.u B"),
E.3.33)
which is a special case of E.3.26) in a slightly different formulation.
Formula E.3.32) shows that the function a*-* u(K'i,.. ., K"-i) is
always the restriction of a support function to 5""'. In particular, for
it e .Z" there is a convex body YIK e X" for which
hiUK, M) = i j^Jiu, v)\dS„.dK, V) E.3.34)
for M e E". Thus nit is a zonoid (see Section 3.5) with centre at the
origin; it is characterized by
h(nK, u) = V„.i{K") for u e 5""'.
We call TIK the projection body of K.
In the last part of this section we shall now derive formulae for mixed
volumes, mixed area measures and quermassintegrals of zonoids. It is
sometimes useful to have explicit expressions for these functions in
terms of the generating measures.
For Mj,. .. , M/t e 5""' we denote by D/t(Mi,. .., w/t) the A:-dimen-
sional volume of the parallelepiped spanned by wi,... , w/t; thus
D/t(Mi, ... , M/t) = K/t(conv {o, M,} + .. . + conv {o, w/t}).
For a linearly independent (n - l)-tuple (wj,... , w„_i) e E""')""' let
T(ui,. .. , w„_,) e S"~^ be the unit vector orthogonal to Wj,. .. , m„_i
and such that the «-tuple (w,,.. ., u„-i, T(ui,. .. , u„-i)) has the same
orientation as some fixed basis of E". Then T is continuous and hence
can be extended (trivially) to a Borel measurable map from E""')""'
into 5""'. In particular, for a Bore! measure q? on E""')""' the image
measure T(JD„-x d(p) of the indefinite integral JD„-i d(p is well defined

5.3 Special formulae for mixed volumes and quermassintegrals 297
by
T(/D„.,d<p)(c.) = /^.,^^D„.,d,p
for we ^E"-').
Theorem5.3.1. For i = I,. . . , n, let Zj e X" be a zonoid with
generating measure p, (and centre o); thus
h{Zi, u) = IJ{u, v)\dpi{v) for us E".
Then
nz„...,z„)
= -^ \^„., ■ • ■ j^-^D„(uu ..., M„)dp,(Mi) • • • dp„(u„) E.3.35)
and
5(Z,, ... , Z„_i, •) = ^^^\y t{J D„_, d(p, <8) ... ® p„_,)J.
E.3.36)
Proof. We first assume that Zj = ... = Zn = Z and Z is a zonotope,
say Z = Si + .. . + Sk, where
Si = conv (or/t;,-, -ar,t;,} with Vj e 5""', Uj ^ 0.
Then
MZ, ') = 2aM-,Vi)\ = \U-,v)\dp(v),
where p is concentrated at ± u, and assigns mass ar,/2 to each of these
points. For an even function / on 5""' one has jfdp = 2^i/(t^;) ^^d
hence
Js-i ■ ■ ■ L-i^"^"'' • • • ' "")''^"«) ■ ■ • ^f^""^
k
= S D„{Vi^, . .., Vijaj^ ■ ■ ■ aj^
i,,...,i„=i
ig.,<..,<.„gfc
2„ .^. S_.V^nE,. + ... + 5,„).
On the other hand
V„{Si + ... + S,)= 2 V„(Si^ + ... + S^. E.3.37)
lSii<...<i„Sk
This is easily proved by induction if one uses the relation
V„{S, + ... + 5;t+,) = V„(Z + 5/t+i) = V„(Z) + \S,,^i\V„-i{nZ),

298 Mixed volumes and related concepts
where |5/t+i| denotes the length of the segment Sk+i and n is the
orthogonal projection onto a hyperplane orthogonal to Sk+i. Thus the
formula
Vn(Z) = ^ |^„., • • • |^.,D„(Ui, . . . , M„)dp(Mi) • • • dp(M„) E.3.38)
is proved for a zonotope Z with generating measure p.
The area measure of Z is, according to E.1.12), given by
5„-i(Z, «) = 2 V„.,(F{Z, u))
uew
for OJ e 3E""'). The face F(Z, u) is a translate of the sum of the faces
of the segments Si,...,Sk that are orthogonal to u, hence from
E.3.37) in dimension /j - 1 we obtain
5„-i(Z, a>) = S 2 K„_iE,- + . . . + 5,;.,)
ueo) lSii<,..<i„.iS;c
ii,,,...,i>,^_,lu
= 2"-' 2 2 D„_,(i;,., . . . , i;,„.,)(V,. • • • ar,„.,
u€(u lSii<...<i„_iS/c
ll„,-..,ll,..,lU
2" *
(« - 1)!
2 Dn-iivi^, ..., «,„.,)ar„ • • • ar,„.,,
'1 'n-l
7'(l', .'',..,N«'
hence
5„_,(Z, c«) = ^^ ^_" ^^, r(| D„_,dp"-'j(c«). E.3.39)
Now let Z be a general zonoid. We can approximate Z by a
sequence (Z,)igfig of zonotopes whose generating measures p, converge
weakly to the generating measure p of Z (compare the proof of
Theorem 3.5.2). Then the product measures p" converge weakly to p".
Hence, equality E.3.38) extends to arbitrary zonoids.
Formula E.3.39) can also be extended in this way. For any
continuous real function / on 5""S
|^„.,/(x)dr||D„_,dpf-i)(x) =
L"-'" ■ ■ L-/^-^^"!' ■ • • ' "n-l))J5n-l(Wl, • ■ • , W„_,)dp,(w,) • • • dp,(M„_i),
and the function /(r( •,..., • ))D„-i{ •,...,•) is continuous on
(S"~^)"~^, since Dn_i is continuous and vanishes where T is
discontinuous. Hence, the sequence (T(JD„-idp"~^))i^f^ converges weakly to
r(|Dn_idp""'). Since En_i(Z,, •))j6m converges weakly to
Sn_i(Z, •), equality E.3.39) follows.

5.3 Special formulae for mixed volumes and quermassintegrals 299
Finally, if Z; has generating measure p,, then X1Z1 + . .. + k„Z„
(Aj^O) has generating measure A,p, + .. . + AnPn. Inserting this into
E.3.38) and E.3.39) and comparing the coefficients, we complete the
proof. ■
If some of the Z, in E.3.35) are balls, one can reduce the number of
integrations, using the following formula. Here we denote spherical
Lebesgue measure on a A:-dimensional unit sphere by o^.
Lemma5.3.2. If LCE" is a k-dimensional linear subspace @< k< n)
and cpi^u) is the angle between u e E"\(o} and L, then
sin<KM)da„_,(M) = -=-^^ . E.3.40)
■'■^ (^n-k + l
Proof For x e B" let y := p{L, x) and s{x) := \x - y\. Then
f s(x)<iX"{x) = f f s{x)<iX"-''(x)d^j''(y)
JB" ^ JtnB" JB(y) ^ ' ^ ^^
with B{y) := {L^ + y)^ B", which is an (« - A:)-dimensional ball of
radius /? = A - fP')^^, p = \y\. With an obvious transformation,
A - p2)("-*+l)/2,
)B(y)
n - k + 1
hence, with a similar transformation.
l*>«"('> = 7^TT-i L lo'c - »'y'-'">'V-' ■""">.->
O^n-k
f^n + l-
On the other hand,
\^s{x)d76''{x) = \j,x\s\n<fix)d'X"{x)
= J^., J^rsin<p(M)r"-^drda„_,(w)
= ^ ^ ^ J^.,sin<Kw)da„_i(w),
from which the assertion follows. ■
This allows us, in particular, to simplify the formula for the
quermassintegrals of a zonoid.

300 Mixed volumes and related concepts
Theorem 5.3.3. IfZCE" is a zonoid with generating measure p, then
W,{Z) = —^ /^„., • • • 1^., D„_y(M„ . . . , M„-,.)dp(M,) • • • dp(M„_;)
E.3.41)
for j = 0, . . . , n.
Proof. By E.3.8) and E.3.35),
W:(Z) = ^ / i ,•••[ ,D„(w,, ..., w„)
"•"■n-1
X dp(M,) • • • dp(M„_y)da(M„_;+,) • • • da(M„),
since B" is a zonoid with generating measure o/2k„_i, where o = o„-i
is spherical Lebesgue measure on 5""'. Now
Dk+i{ui, ... , Uk+i) = Dii(ui, . .. , Uk)sin(p
if (J? denotes the angle between w^^+i and lin(ui,.. . , w^^}. Repeated
application of E.3.40) now yields the assertion (the Legendre relation
(m - 5)!m! = 2"^'r'^Bm)! can be used to simplify the numerical
factor). ■
Notes for Section 5.3
1. For dimensions two and three, many of the special formulae of this section
can be found in Blaschke [5); see also Bonnesen & Fenchel [8]. For the
integral-geometric formulae, references are found in the notes for Seaion 4.5.
Theorems 5.3.1 and 5.3.3 are taken from Weil A976a), where they are
proved for generalized zonoids; special cases appear in Blaschke [5], Weil
A971) and Matheron A975). Weil A976a) also gave a formula for the density
of 5(Zi,..., Z„_i, •) in the case where the generating measures pi,...,
p„_i have densities (with respea to o).
2. Differentiation of mixed area measures. Let Ki,. . . , K„-i 6 X" be convex
bodies of class C+. If u 6 5""' and (<w,)/6m is a sequence of Borel subsets of
5""' with ^^""'@),) > 0 shrinking to u, then E.3.3) shows that
(-« 35" '(ft),)
Aleksandrov A939c) extended this to general convex bodies Ki,..., K„-i:
Almost everywhere on 5""' their support functions are simultaneously twice
differentiable. At each point u where this happens, 5(^1,..., K„-i,u) can
be defined by B.5.36), and then the limit relation above is valid for suitably
chosen sequences (ft),),^^.
3. Minkowskian integral formulae. The formulae
and
|^^.,syd5r.'-' = j^^_hKSi-idX''-' E.3.14)

Notes for Section 5.3 301
are the prototypes for a number of integral formulae that have proved quite
useful in global differential geometry. We give a simple example of a typical
application. Suppose that K is a convex body of class C+ whose ^th
elementary symmetric function of the principal radii of curvature is constant,
say
Sk = 1,
for some ^6{1,...,«-1}. Newton's inequalities for elementary symmetric
functions yield
if ^ = 2, hence s*-1 = So. and
5, g 5'/* =1=5*.
We may assume that oeint/^, hence hK>0. Applying E.3.14) twice, we
obtain (integrating over 5""' with respect to X"'^)
\ ^k = j flKSk-l = J HkSo = J Si g J S*,
hence si = s]/''. If ^ = 1, we have
S, = Sf g S2
and
J *i = J *2 = J h^Si = j hK = j si.
hence si = S2^^. In both cases, we deduce that every boundary point of K is
an umbilic, and this is known to imply that K is a ball. Similarly, E.3.15) can
be used to show that K (of class C+) must be a ball if //* is constant for
some ^ 6 {1, ...,«- 1}.
Apparently this simple argument was first used (in the more general setting
of relative differential geometry) by Siiss A929).
Formulae E.3.14) and E.3.15) express the symmetry of the mixed volume
in special cases. As mentioned in Section 5.1, Note 2, this symmetry can be
proved analytically. In differential geometry, the usual way to prove such
integral formulae is to apply Stokes' theorem to a suitably chosen differential
form. Various such choices yield useful generalizations and analogues of the
Minkowskian integral formulae. A systematic exposition of the derivation and
application of integral formulae in global differential geometry can be found
in Chapter 3 of Huck et al. A973). To illustrate the wide applicability of
generalized Minkowskian integral formulae we mention a few papers only:
Chem A959) (pairs of convex hypersurfaces), Simon A967) (a systematic
study), Schneider A967d) (affine differential geometry), Katsurada A962)
(Riemannian spaces).
The extension of the Minkowskian integral formula E3.15) to convex
bodies without differentiability assumptions, also in spaces of constant
curvature, was studied and applied by Kohlmann A988).
Other extensions of the Minkowskian integral formulae yield
representations of the quermassintegrals of the types E.3.9) and E.3.11), but involving
higher powers of the support function (and other expressions). General
versions of such formulae were derived by Bokowski & Heil A986), who
extended and unified earlier results of Simon A967), Shahin A968), Yano &
Tani A969) and Firey A974d). The formulae are complicated, but Bokowski
& Heil A986) made an elegant application to inequalities for
quermassintegrals.

302 Mixed volumes and related concepts
4. Quermassintegrals of bodies of constant width. The quermassintegrals Wo,
IVi,..., IV„ of a convex body of constant width b satisfy [(« + l)/2)
independent linear relations, namely
2Wn-k = E(-l)' (*) b''-'V/„., for OS A: S n, A: odd.
/=o \' /
For « = 3, this includes Blaschke's relation
Vj = \bF - \Ttb\
where F is the surface area. Proofs for the general case were given by
Dinghas A940b), Santald A946) and Debninner A955).
5. Wilk' functional. The special linear combination of quermassintegrals given
by " In\ \
,=0 \ I I Ki j=0
has some remarkable properties; for instance,
W(K) = |^„e-"''<*^'^>'dji:
and, hence, W(K ® L) = W(K)W(L) for an orthogonal direct sum K^ L.
For these and further properties, see Hadwiger A975c). The functional IV
attracted considerable interest after Wills A973) had conjectured that
G(K) S W(K) for K 6 JC", where G{K) denotes the number of points with
integer coordinates (with respect to some orthonormal basis of E") in K. The
conjecture is easily established for « = 2 and was proved for « = 3 by
Overhagen A975), but Hadwiger A979) found counterexamples for n §441.
Related results and many connections between discrete and convex geometry
can be found in the survey article of Wills A990a). Much of the work
described there is concerned with relations between lattice point inequalities
and quermassintegrals, topics both of which were created by Minkowski.
Wills A990b) establishes inequalities between the zeros of the polynomial
2y=o^/(^)(~^)' and the successive minima of K, as defined by Minkowski
in the geometry of numbers.
6. Further integral-geometric formulae. Goodey & Weil A987) and Weil A990b)
obtained some Crofton-type formulae for mixed volumes, of which we
mention the following. Let y 6 {1, ..., « - 1} and K, L e X". Then
f V(K n £[/], L[n - n)d^t(E) = -p^ V„{K)V„.,(L).
JM",/) I n Y
\ 7 *""
If K and L are centrally symmetric, then
«(«-l)a„o(,>o^,(ni^^nL,fi"....,B''),
;)
where 11K denotes the projection body of K.
A formula for mixed volumes of two polytopes. For polytopes P, Q e'p" and
for k e (I,..., n - 1}, the mixed volume V{P[k], Q[n - k]) can be
represented in a way similar to E.3.21), but depending on a vectorial parameter.
Suppose that x 6 E''\{o} is a vector in general position with respect to P and
Q, which means that there do not exist faces F o( P and G of Q such that
dimF + dimG> n and
X € lin[N(P, F) + N(Q, G)].

5.4 Moment vectors and curvature centroids 303
Then, as proved by Betke A992),
(?) V(Pik], Q[n -k])= ^ 2 [f. G]V,(F)V„.,(G).
xeN(P,F)-N(Q,G)
Here, as indicated, the summation extends only over those pairs F, G for
which X is in the difference of the corresponding normal cones. By
integration over all unit vectors x (almost all of which are in general position with
respect to P and 0), formula E.3.21) is again obtained.
5.4. Moment vectors and curvature centroids
In combining the notion of volume with the Minkowski addition of
convex bodies, one is led naturally to mixed volumes and, as a special
case, to quermassintegrals. In a similar way, a theory of mixed moment
vectors can be developed, which by specialization leads to curvature
centroids. We give a brief outline of such a theory, but without proofs.
The latter can essentially be modelled on those for mixed volumes, and
can also be found in Schneider A972a).
For K € JC" and for r e {dim it, ...,«} we define
Zr^AK) ■.= \^xdVC/{x)
and call Zr+i(Jt), for r = dim it, the moment vector of K. This vector
depends on the position of K relative to the origin; in fact
Zr^,{K + 0 = Zr^l{K) + V,{K)t
for t € E". If P € 'Vo is an «-dimensional polytope with facets F\,. .. ,
Fn and corresponding outer unit normal vectors wi,. . . , W/v, then
1 ^
z„^,{P) = -—- J,h{P, M,)z„(F,.). E.4.1)
n + 1 ,=i
There is a symmetric function z.{X")"*^ -* E" such that
2n + l(Aiiti + . . . + k„Km)
m
= 2 A;, • • • A,„^.z(/t,„ . ..,/t,„^,) E.4.2)
for K^,... ,K„&JC" and A,,..., A^ ^ 0. The map z is called the
mixed moment vector. We have
z(it|, . . ., it„+i)
= r»ini S(-ir^^' 2 z„+,(a:„ + ... + A',,) E.4.3)
(« + 1)! /t=i (,<...<u
ind, with notation similar to that in Section E.1),

304 Mixed volumes and related concepts
z(A]/l, + . . . + X„K„[p], Kp+i, . . . , K„+i) =
2 ( ^ ]x[^■^■X'^z(Kl[r^],...,K„[r„],Kp^l,■■.,K„^^)
r,„...r„=0\''l • ■ • rml
E.4.4)
for 1 ^pg « + 1.
We shall list some properties of the map z. For K, Ki,.. . ,
K„+i € JC" we have
z(a:,..., a:)=z„+i(a:),
z(ar/<:,, . . ., aK„+^) = \deta\az{Ki, . .., K„+i)
for any linear map a: E" —» E". Under translations, z behaves according
to
z(Ki + <],..., K„+i + t„+i) = z(Ki, .. ., K„+i)
+ —^ [V(K2, ..., K„^i)ti + ... + V(K„ .... K„)t„^^]
n + I
E.4.5)
for t],. . . , t„+| € E". Furthermore, z is continuous, and for fixed
p € A,...,« + 1} and Kp+i,. . ., K„+i the map
K-^ z(K[p], Kp^^, . . . , K„^i)
is additive on X". If K(/<:,, ...,ki,..., /<:„+,) = 0 (/C, deleted) for
i = 1, ...,« + 1, then z(/<:,, . . . , K„+i) = o.
In analogy with the quermassintegral Wr(K)= V(K[n - r], B"[r])
we define the rth quermassvector of K by
^^(^> •= nV\- r ^^^t" + 1 - '■1' ^"[^D E-4-6)
for r = 0,.. ., n; in particular, ?o(J^) = z„+\{K). From E.4.4), we have
the Steiner-type formula
Zn^AK + AS") = i f ") A^?.(/C), E.4.7)
since z„+i(S") = o. It is consistent with the earlier definition of Zr-y\{K.)
to extend z^+i to convex bodies K with dim it > r as follows:
(:)
Zr^AK) := ^^ q„-r{K) E.4.8)
for r = 0,...,«. The vector z^+iC^) 's called the intrinsic (r + 1)-
moment of it. If P € ^" is a polytope, then
2r+i(P) = 2 V(f. P)Zr^x{F), E.4.9)

5.4 Moment vectors and curvature centroids 305
in analogy with D.2.30). From E.4.8) and E.4.9) we obtain a
representation for qr{P), which in view of D.2.17) can be written in the form
qriP) = j^j^„xdC„.,{P,x).
Using approximation and the weak continuity of the curvature
measures, we deduce that
qr(K) = -^ l^x dC„.,(K, X) E.4.10)
for arbitrary K ej^".
Let 3:; := {KeJC"\dimK^n-r}. For r = 0, . . . , « and K e X"
we define
., ^r(K) ^ (n + l)z(K[n + 1 - r], B"[r])
^'^ ' Wr{K) (« + 1 - r)V{K[n - rj, B"[r])'
Thus po(K) is the ordinary centroid (centre of gravity) of K, and for
r = l,...,n,
xdC„.,(K,x)
Pr(^^)=^Vr^—nr^ E.4.11)
JdC„_,(A:, x)
is the centroid of the mass distribution given by the curvature measure
C„_r(it, •)• The points pi(K),. .., p„{K) are, therefore, called the
curvature centroids of K. In particular, Pi{K) is the surface-area
centroid of K. The centroid of the Gaussian curvature measure
Co(K, •) is the Steiner point, that is,
p„(K) = s{K) E.4.12)
for K e JC". This can be seen as follows. First let K be of class C\. We
use the notation of Section 2.5. By B.5.8), ^(u) = gTadhK(u). We
integrate this over the unit ball B". (More precisely, we apply the
divergence theorem to each of the vector fields huej, where (e^, . ..,
e„) is an orthonormal basis of E", and to the domain
{a: € E"|r ^ |a:| ^ 1}, and then let r > 0 tend to 0.) The result is
jj{u)du = l^_^h^{u)udX"-\u) = K„s(K),
according to the definition A.7.3). On the other hand, using
§(Aw) = |(w) for A > 0,
lsM<^)du = j^l„J(u)dX"-\u)
= — f xH„.i(x)dZ^"-\x)
by B.5.30). This yields
K„s{K) = — IxdC^^K, X) = q„{K),
n J

306 Mixed volumes and related concepts
which by approximation extends to arbitrary K e DC" and thus proves
E.4.12).
We mention some properties of the curvature centroids. From
E.4.11) it is clear that
P,(A:) e relint/C. E.4.13)
For each similarity a: E" ^^ E", we have pXaK) = apr(K). The map Pr
is continuous on X", and WrPr = qr is additive. For A = 0,
t("y\^'-''WiiK)Pi(K)
Pr{K + AB") = '='1 j , E.4.14)
so that Pr(K + XB") is a convex combination of Pr{K),... , p„{K).
In analogy with Hadwiger's axiomatic characterization of the quer-
massintegrals, quoted as Theorem 4.2.6, one can characterize the
quermassvectors of the curvature centroids by some of their properties.
Proofs of the following theorems appear in Schneider A972b).
Theorem 5.4.1. Let n ^ 2. If the map cp: DC" —> E" is additive, equivariant
under rotations, continuous and such that (p{K + t) - (p(K) is parallel to
t for all t el." then there are real constants Cq, . . . , c„ such that
n
(piK)= ^CrqAK) forK e X".
r=0
Theorem 5.4.2. Let n^2 and r e {0,. . . , n}. If the map cp: :^; ^ E" is
equivariant under rigid motions, continuous and such that W^cp is additive
on XI then
cp{K) = pXK) for K € X"r.
Notes for Section 5.4
1. Steiner point and curvature centroids. The point later called the Steiner point
was first introduced, as a curvature centroid, by Steiner A840a), for planar
convex bodies that are either a polygon or sufficiently smooth. The
characterization given by equation (A.7) in the Appendix appears, for n — 2, in
Kubota (I9I8) and implicitly in Meissner A909), p. 3.11. (A.7) is essentially
the same as A.7.3). Equality E.4.12), that is, the equivalence of definition
E.4.11) for r = n with A.7.3), was proved for some special cases by Bose &
Roy A935c), Gericke A940a) and Shephard A966a). Bose & Roy A935c)
and S. N. Roy A936) have similar integral representations for the other
curvature centroids in E'. An extension of the Steiner point to not necessarily
convex hypersurfaces was treated by Flanders A966). Some related integral
identities are derived in Hadwiger & Meier A973).

Notes for Section 5.4 307
2. Extremal properties of the Steiner point. Originally, Steiner was led to the
Steiner point by considering certain extremal problems. Other proofs oi the
extremal properties established by him as well as further such properties are
found in Ferrers A861), Meissner A909), Nakajima A921), Hayashi A«24),
Su A927a, b) and Arnold A989).
3. Curvature centroids of parallel bodies. Equation E.4.14) appears in special
cases (mostly for n = 2) in Kubota A918); Blaschke A933,1937a) and Jose
& Roy A935a, b,c). Related observations on the relative positions of the
various curvature centroids of a convex body and its parallel bodies vere
made by Duporcq A896,1897), Kubota A918), Nicliborc A932), Blaschke
A933) and Bose & Roy A935a, b,c).
4. Curvature centroids of bodies of constant width. In analogy with Note 4 of
Section 5.3, the curvature centroids po, ■. ■, p„ of a convex body of constant
width b satisfy
2lV„_,p„_, - EC-l)' f *| b'^-'W„..j}„.i = o
for IS k ^ n, k odd. For n =2 this states that the Steiner point and
perimeter centroid coincide; this was noted by Meissner A909) and proved in
a different way by Bose & Roy A935a). For n =3 one finds that the Steiner
point coincides with the mean curvature centroid (see Bose & Roy 1935c) and
that the centroid, surface-area centroid and Steiner point are collineai- A
proof of the general relations appears in Schneider A972a).
5. Characterization theorems. As a counterpart to Theorem 3.4.2, the follo\ving
characterization of the Steiner point, with Minkowski additivity replaced by
additivity, was proved in Schneider A972b):
Theorem. Let n § 2. If the map (p: X" -» E" is additive, equivariant uiider
rigid motions and continuous then (p is the Steiner point map.
This result was used to prove Theorem 5.4.2 in Schneider A972b) and to
deduce Theorem 5.4.1 in Hadwiger & Schneider A971) and in Schneider
A972b).
The special case of Theorem 5.4.2 referring to the ordinary centroid states:
If q>: Xo —»E" (where n § 2) is a continuous map commuting with rigid
motions and such that V„<f is additive, then (p(K) is the centroid of K, for
each K e Xq. Another characterization of the centroid, in this case for
polyhedra, was proved in Schneider A973): Let <p: U{''PS)--E" be a
translation equivariant map such that (p(P) e conv P for Pe U(fl)(f) and V„ip is
simply additive. Then (dP) is the centroid of P, for each P e 17('PJ)-
6. Integral-geometric formulae for quermassvectors. The special case P' = E' of
the integral-geometric formula D.5.1) states that
I 0,(/C n gK\ p)dfi(g) = t<^nikV„.j-kiK')<l>,(K, P)
for /3 6'3(E''). As functions of /3, both sides of this equality define finite
measures on "^(E"). Integrating the identity function on E" with respect to
these measures and observing D.2.10) and E.4.10), one obtains integral-
geometric formulae for the quermassvectors, which are analogous to D.5-2).
In a similar way. Theorem 4.5.5 and formula D.5.24) (with /3' = E") lead to
formulae for quermassvectors or, equivalently, for curvature centroids. in a
different way, such formulae were derived by Hadwiger & Schneider A971)
and by Schneider A972b); special cases for low dimensions appeared ear'ier
in Blaschke A937a) and H. R. Muller A953).

308 Mixed volumes and related concepts
7. Position of the curvature centroids. Let K 6 .^o be a convex body with
centroid o. Then a result going back to Minkowski A897) (cf. Bonnesen &
Fenchel [8], p. 53) states that
—^ w(K, u) g h(K, u) S —^ w(K, u)
n + I n + I
for u e S""'. For a convex body with centroid c this implies, in particular,
that
Similar estimates for the position of the curvature centroids seem to be
unknown, except for the surface area centroid; see G. v. Sz.-Nagy A949).
Rather precise information on the position of the curvature centroids is
available for convex bodies with curvature restrictions. Let K 6 JC" be of
class C\ and assume that there are constants Ri, Rj such that
0 < Ri § p = /?2 whenever p is a radius of curvature of bd K. Then
B(c, Ri) C K C B(c, R2)
whenever c is one of the curvature centroids pi(K),. . . , p„(K). This was
proved by Schneider A988b), generalizing a result of Goodman A985) for the
perimeter centroid in the plane.
8. Centroids in stereology. Possible stereologic applications of curvature
centroids in dimensions two and three were discussed by Davy A980a, b, 1981).

Inequalities for mixed volumes
6.1. The Brunn-Minkowski theorem
Studying the behaviour of volume under Minkowski addition, one is led
to the notion of mixed volumes on one hand and to the
Brunn-Minkowski theorem on the other. In its simplest form, this theorem says that for
two convex bodies K, L e JC" each of volume 1 the Minkowski sum
\{K + L) has volume at least 1, and its volume is equal to 1 only if K
and L are translates. This theorem is the starting point for a rich theory
of geometric inequalities, with several applications to extremal,
uniqueness and other problems. The present section is devoted to proofs,
extensions and analogues of this important theorem.
Theorem6.1.1 (Brunn-Minkowski). For convex bodies Kq, K^ e X"
and for O^X^l,
V„((l - k)Ko + XK.yl" ^ A - X)V„{K^)'l" + AV„(A:,)'/". F.1.1)
Equality for some A e @,1) holds if and only if Kq and AT j either lie in
parallel hyperplanes or are homothetic.
If ATo, Ki e X" are given, we often write
A:^ := A - A)A:o + A/C, forOgA^l. F.1.2)
For cT, T € [0,1] and 0 ^ A ^ 1 we have
A - A)A:„ + AA:, = (l - a)Ko + aKi
with a = A - k)o + At. Applying Theorem 6.1.1 to K„ and K^, where
CT, T 6 [0,1] are arbitrary, we deduce that the function
A^ v„((i-a)a:o + aa:,)'/''
is concave on [0,1], and linear only in the cases described in Theorem
6.1.1. (As is usual in this context, we say 'linear' instead of 'affine'.)

310 Inequalities for mixed volumes
The first proof we shall give for the Brunn-Minkowski theorem is a
classical version due to Kneser & Siiss A932), which is also reproduced
in Bonnesen & Fenchel [8). It still has some advantages.
Proof of Theorem 6.1.1. If Ko and K^ lie in parallel hyperplanes, then
Kx lies in a hyperplane and hence V„{K)) = 0 for O^ASl. Thus
equality holds in F.1.1). This is also true, trivially, if ATo and K\ are
homothetic.
Suppose that dim ATg < n and dim K\< n. Then F.1.1) is true, and if
equality holds, then Ki lies in a hyperplane, hence Ko and K^ lie in
parallel hyperplanes.
If, say, dim ATq < n and Aim Ki = n, then the inclusion /C^D
A - X)x + XKi, for arbitrary x 6 ATq, implies
V„{K,) ^ V„{{1 - X)x + A/C,) = A"V„(A:,),
with equality if and only if ATq = {x). In that case, Kq and Ki are
homothetic.
Hence, from now on we can assume that dim K^ = dim AT] = n. We
can also assume that V„(/Co)= V„{Kx) = 1; if F.1.1) is proved under
this assumption and if Kq, Ki e X!a are arbitrary, we put
Ki := V„{Ki)-^l"Ki fori = 0, 1,
^.^ Av„(/:i)'^"
{\-X)V„{Koy'" + )^V„(K,f!"'
From V„((l - l)Kf, + A/C,)'/" ^ 1 we then obtain F.1.1); the results for
the case of equality can also be generalized.
Theorem 6.1.1 is proved by induction with respect to the dimension.
The case n = 1 being trivial, we assume that n ^ 2 and that the theorem
is true in dimension n-\. We choose u e S"~^ and write
/^u.a =■ W(a), H~a=: H~{a) and, furthermore, ax.= -h{Kx,-u)
and Px •= h{Kx, u). For ? e R and j = 0, 1 define
u,(C)-= V^.^iK^n HiO),
w,(C)-= V„(Ki^ W"(C)),
so that
JCf,
On (a,, /3;) the function u, is continuous, hence w, is differentiable and
^m = ViiO > 0 for a; < C < Pi.
Let Zi be the inverse function of w,; then
^Kt^) = r f w forO< T< 1.

6.1 The Brunn-Minkowski theorem 311
Writing
A:,.(r) := a:, n //(z,(r)),
zx(r) := A - A)zo(r) + Azi(t),
we have
K, n H{zx(r)) D A - X)ko{r) + Xk,{x).
We deduce that
V„iK,) = V„-,(A:, n //@)dC = V„-,(A:, n H(z,(r)))z',(r)dr
J ax JO
^ f V„_,((l - X)ko{r) + AA:,(r))[ ^"/ + /. Jdr
where in the final line we have used the induction hypothesis and
abbreviated Vi(Ziir)) by i;,-. The inequality
1 - A A
+ —
^ 1
[A - X)v'^ + Auf]^^"
I Vo vil
holds for V(,,vi,p>0 and 0<A<1. For the proof, one takes
logarithms and notes that log is concave and increasing. Equality holds
only for vo = v\.
It follows that V„{Kx) = 1, and this completes the inductive proof of
the Brunn-Minkowski inequality.
Suppose that the equality V„(Kx) = 1 holds for some A 6 @, 1). Then
voizoir)) = Vi(zi(r)) and hence Zoir) = z[(r) for O^r^l; thus
^i(^)-^o(^) is constant. We may assume that ATq and K^ have their
centroids at the origin. Then
0 = j^{x, u)dx = £V„_,(A:, n //(O)CdC = /^z,(T)dr
for / = 0, 1 and hence Zoir) = Zi{r) for 0^ t^ 1. This yields /3o = /3i,
thus h{Ko,u) = h{Ki,u). Since u e 5"~' was arbitrary, we conclude
thatA:o=A:i. ■
One of the advantages of the proof given above may be seen in the
fact that it can be modified to yield improved versions of the Brunn-
Minkowski theorem. In particular, it leads to an associated stability
result. The uniqueness assertion of Theorem 6.1.1 states that two
n-dimensional convex bodies Kq, K\ that satisfy F.1.1) with equality,
for some A e @, 1), must be homothetic. If equality holds only
approximately, can one assert that ATq and K^ are nearly homothetic? Explicit
estimates giving positive answers are usually called stability results. To

312 Inequalities for mixed volumes
formulate a more perspicuous special case, assume that V„(Ko) =
^ni^i) = 1 an'! 0 < A< 1. Then Theorem 6.1.1 reduces to the assertion
that
V„((l - X)Ko + A/C,) ^ 1, F.1.3)
with equality only if Kq and Ki are translates, hence only if
d{K(), Ki) = 0, where AT, denotes the translate of AT, having its centroid
at the origin. A stability result for the inequality F.1.3) would ensure
that V„((l - A)/Co + A/Ci) ^ 1 + f implies d{K(,, ^i) ^ /(f), with some
explicitly given function / (possibly involving limitations on the size or
degeneracy of ATq, A^i) that satisfies /(f)-*0 for f-*0. The following
result of this type is due to Groemer A988a) (it is a special case of his
Theorem 3).
Theorem6.1.2. Let Kq, K^ e J^q he convex bodies with V„(K()) =
V„(Ki) = I, and define D := max {D{K(,), D{Ki)}. If
V„((\ - X)Ko + A/C,)'/" S 1 + f F.1.4)
for some f > 0 and A 6 @,1), then
6{K,, K,) ^ rj„l^-^j^=— + zJDf'/<"+') F.1.5)
with
77„ = 6.00025n. F.1.6)
We refer to Groemer A988a) for the proof, which follows the
approach of Kneser & Siiss but introduces explicit estimates. There one
also finds some consequences of the theorem, for instance, a
formulation for convex bodies of arbitrary volumes. We remark also that
Theorem 6.1.2 can be formulated as a strengthened version of F.1.3),
namely as the inequality
V((l - X)Ko + A/C,)'/"
^1 + 1 ., Xwj/^ ,..^r"<^(^o.^i)""'. F.1.7)
.r,„[l + 2VAA - X)]D
which is valid for V„{Kn) = V„{Ki) = 1 and 0^ A^ 1. This inequality is
obtained from F.1.5) by setting f = V„((l - A)/:,, + A/C,)'/" - 1.
We shall now sketch a second proof of the Brunn-Minkowski
theorem, essentially following Knothe A957a). As above, it is sufficient
to consider two convex bodies K, L e JC" satisfying V„(K) =
V„(L) = 1. We write ;t e E" in the form x = (x^, . . . ,x„), where the Xi
are the coordinates of x which respect to a given orthonormal basis, and
define
a:., ., := {y e K\y, = Xi for 1 ^ i S/}

6.1 The Brunn-Minkowski theorem 313
for / = 1,. . . , n, similarly for L. On int K we can define a function /j
by
("x rflU)
V„.,(K,)dt=l V„_i(L,)d/.
J — <X J —00
If /e {2, . . . , n} and /i, . . . , /y_i have already been defined, we
define /y by
/.V-/(^^ v..')'!'
_ '^n-/ + l(^Ji J,-i)
r//^)
•'«-/ +lV^/i(jr) /,-i(a;).' •"
Then fj{x) depends only on ;ti,. . . , Xj, and for ;t 6 int AT we obviously
have
9-^/ Vn-jii-hix) f,(x)) Vn-J+liK^, X,.,)
and thus
n — = 1- F.1.8)
1=1 dXj
The map f: int AT ^ int L defined by F{x) = {fi{x),. . . , f„{x)) is
bijective.
Now let A 6 [0,1). Then [A - A)id + XF]K C Kx and hence
V„(/C,) ^ V„([(l - A)id + AFj/C)
= {•■■\ |Jac((l - A)id + Af lldATi • • • d;«:„
int a:
|..|n((l-A) + A|^(;t)Jd;t, •••d;t„.
(In writing down the Jacobian, we do not need to know whether dfj/dxj
exists for i </. In fact, since fj(x) depends only on Xi,...,Xj, the
transition from the first to the third line does not require the general
transformation rule for multiple integrals, but is obtained by using
successively the substitution rule for one-variable integrals.)
Now
log
"-^)-^i-^'°^if
since log is a concave function, and together with F.1.8) this gives
V„{K,) ^ 1.

314 Inequalities for mixed volumes
If equality holds, we must have
9/;
-—'- (x) = 1 for ;t e int K and / = 1, . . . , n
dXj
and thus fiix) = ;tj + fli with a constant fli. Since any coordinate axis
may play the role of the first one, we deduce that L is a translate of K.
The preceding proof of the Brunn-Minkowski theorem can be used
to obtain extensions of that theorem, with the volume replaced by the
integral of a function with suitable properties (see Note 1 below).
Notes for Section 6.1
The Brunn-Minkowski theorem for convex bodies. The theorem now named
after Brunn and Minkowski was discovered (for dimensions ^ 3) by Brunn
A887,1889). Its importance was recognized by Minkowski, who gave an
analytic proof for the n-dimensional case (Minkowski 1910) and characterized
the equality case; for the latter, see also Brunn A894). Alternative proofs
were given by Blaschke [5], §22, who used symmetrization, and, for n =3,
Hilbert A910) who applied his theory of integral equations on the sphere. An
improved version is due to Bonnesen [7], Chapter VI. The first proof
presented above was given by Kneser & Siiss A932). All this can be found in
Bonnesen & Fenchel [8]. The proof by Kneser and Siiss was extended by
Hadwiger [18], Section 6.4.5, to yield a more general result. More recently,
different proofs have been given by Gromov & Milman A987), Appendix,
and Pisier A989).
Knothe A957a) extended his proof, sketched above, to yield the following
more general result. For each K e Xq, \et p{K, ■) be a real function on K
and suppose that p is non-negative and continuous in both variables, that
p{XK + a,kx + a) = X'"p{K, x)
for A > 0 and a 6 E", where w > 0 is a given number, and furthermore that
\ogp{Kx, A - X)xo + Axi) g A - k)\ogp{Ko, xo) + AIogp(A:i, *i)
for ACoC/Co. ACiS/C], A 6 [0,1]. (An example, with w = 1, is given by
p{K, x) := d(x,bd K) for x e K, where d denotes the distance.) Under these
assumptions, Knothe showed that
If \l/(n + m)
\^j^p{K,,x)dxj
Ir \l/(«+m) Ir \l/(n+m)
s A - k)y]^p{Ko, x)dAc| + M^]^p{f^x, ^)d^)
for /Co. K.\ 6 Xo and A g [0,1]. Related results, also for non-convex sets, are
treated by Dinghas A957b).
Knothe's method is also used in Bourgain & Lindenstrauss A988b) (see
also Bourgain 1987). These authors call the map F appearing in the proof the
Knothe map. Dinghas A957b) called a closely related map the Brunn-
Minkowski-Schmidt map. He had previously employed it in Dinghas A944),
where he referred to it as an oral communication by E. Schmidt. For more
history, see Dinghas A957b), footnote 18.
Stability estimates. Alongside Groemer's stability estimate for the Brunn-
Minkowski theorem, which was quoted as Theorem 6.1.2, a similar result is
known due to Diskant A973a)-. Suppose that K, L e X" satisfy V„{K) =

Notes for Section 6.1 315
V„(L)= 1. There are constants £o>0 and c>0, depending on n and the
inradius and circumradius of K, such that the inequality
V„((l - X)K + XL)^l + e for 0 g A g 1
with some £ 6 [0, £o] implies 6(K', L)^ ce^'" for a suitable translate K'
of K.
A survey over the stability properties of geometric inequalities, including
the present one and several others, is given by Groemer A990a).
Non-convex sets. The Brunn-Minkowski inequality is not restricted to convex
sets. There are, in fact, very general versions of it, even for non-measurable
sets. Its development is connected with the names of Lusternik, Henstock and
Macbeath, Dinghas, Hadwiger, Ohmann. A description of their methods and
results would lead us too far from our topic, the geometry of convex bodies;
hence we refer only to Hadwiger [18], Dinghas A961) and Burago &
Zaigaller A988) for details and references and to Uhrin A984,1987) for some
more recent achievements.
Logarithmic concave measures. Taking logarithms on both sides of the
Brunn-Minkowski inequality
v„((i - x)K + XLyi" § A - a)v„(a:)'/'- + av„(l)'/'-
and using the concavity and monotoneity of the log function, one obtains the
inequality
V„((l - A)/C + AL) g V„{Ky-W„{L)\
Generally, a measure P defined on the Lebesgue-measurable subsets of E" is
called logarithmic concave if
P((l - K)K + XL) § P(Ky->-P(Ly
for every pair K, L of convex subsets of E" and for all Ae @,1). Prdkopa
A971,1973) and Leindler A972) established theorems, using the Brunn-
Minkowski theorem, which assert that certain integrals yield logarithmic
concave measures.
The following formulation of the Prdkopa-Leindler theorem appears in
Brascamp & Lieb A976). If/, g are non-negative measurable functions on E"
and if A: is defined by
k(x) : =
y
then
M7^)'M
ii^iii § \\f\\r\\g\\'{ F.1.9)
for A e @,1). U A, B C E" are nonempty measurable sets and if / and g are
the characteristic functions of A and B respectively, then k is the
characteristic function of A - X)A + XB. Hence, this set has (Lebesgue) measure at
least unity if A and B have measure unity. Thus F.1.9) implies the
Brunn-Minkowski inequality.
Proofs, extensions, analogues and hints about the applications of such
results are found in Brascamp & Lieb A976), Das Gupta A980) and Uhrin
A987).
5. A counterpart to the Brunn-Minkowski inequality
v„(K + Lyi" s Vn^Ky!" -f v„{Lyi"
concerns the Minkowski difference /C ~ L. If it is not empty, then
V„{K ~ Lyl" g Vn^Kyl" - V„(L)'''".
This result (which is not restricted to convex bodies) was found by E.

316 Inequalities for mixed volumes
Schmidt. It can be deduced from the Brunn-Minkowski theorem; cf. Had-
wiger A950b) and [18], p. 159. Related inequalities were studied by Ohmann
A953,1956a).
6. Applications. The Brunn-Minkowski theorem is fundamental for much of the
development in later sections. It has many other applications, either direct or
indirect, in different fields. We mention only a few examples: it has proved
useful in the investigation of (i) other geometric volume estimates (Mityagin
1969), (ii) the characterization of parallelotopes (Guggenheimer & Lutwak
1976), (iii) the equilibrium shape of crystals (Dinghas 1944) and (iv) the
shapes of worn stones (Firey 1974d). Applications to probability and
multivariate statistics are found in the surveys by Das Gupta A980) and Buldygin
& Kharazishvili A985) and in the references given there, and still other
applications are discussed in Zalgaller A967) and Vitale A990,1991).
7. Busemann's theorem. A well-known consequence (or reformulation) of the
Brunn-Minkowski inequality is the following assertion: Let K e Xq and let P
be a two-dimensional half-plane bounded by the line L. Let the hyperplane
H normal to L intersect K in the nonempty set H n K. If
V„-,(// n a:)'/(''-i) is laid off from the point H n L on the ray H n P, then
the resulting curve in P is convex.
Busemann A949a) proved an analogous theorem on the intersection of K
by pencils of non-parallel hyperplanes. In his original formulation, this
theorem reads:
In E" let K be a convex body with interior points, L an (n - 2)-dimen-
sional space that intersects K, and P a two-dimensional space normal to L at
a point 0. Any half-hyperplane H bounded by L intersects /C in a nonempty
set H n K. If the (n - l)-dimensional volume V„-,(// n K) of H n K is
laid off from 0 on the ray H n P then the resulting curve C is convex.
Busemann also pointed out the following corollary. Let K be a convex
body in E" with o as interior point and centre. If for any hyperplane H
through o the volume V„-i(H n K) is laid off from o on the normal to H at
o (in both directions), then the resulting surface is convex.
In other words: |jt|/V„-,(A: nx^) defines a norm on E''\{o}. A
generalization (to sections of higher codimension) is proved in Milman & Pajor A989),
Theorem 3.9.
Busemann's proof of his theorem (which is important for Busemann's
theory of area in Minkowski spaces) used the Brunn-Minkowski theorem as
well as ideas employed in its proofs. Barthel A959a) extended Busemann's
theorem by weakening the convexity assumptions, reformulated it as an affine
invariant inequality and clarified the equality case (see also LeichtweiB [23],
pp. 257-67). Barthel & Franz A961) gave a geometric proof without any
convexity assumptions and simplified the equality discussion.
8. Milman's inverse Brunn-Minkowski inequality. If K, L e X" are convex
bodies of volume 1, the Brunn-Minkowski inequality says that their sum
K + L has volume at least 2". The question for an upper bound does not
make sense: the volume of K + L can be arbitrarily large. However, the
following is true: If K, L e K^ are centrally symmetric bodies with interior
points, there exists a volume-preserving linear transformation * of E" such
that
V„(K + aD^I" g C[V„{Ky''' + V„{aLY'''],
where C is a constant independent of n. This theorem is due to Milman
A986). Different approaches to its proof and relations to other theorems with

6.2 The Minkowski and isoperimetric inequalities 317
applications in the local theory of Banach spaces are described in the book by
Pisier A989).
9. Lutwak A979b) interpreted the Brunn-Minkowski inequality for convex
bodies as a complementary Minkowski inequality for special functions.
6.2. The Minkowski and isoperimetric inequalities
We now have two fundamental pieces of information about the volume
of the Minkowski convex combination (\ - X)Ko + AATi of two convex
bodies Ko, K\: it is a polynomial in A, and its nth root is a concave
function. These facts taken together have consequences for mixed
volumes.
Let Ko, Ki e Jio be n-dimensional convex bodies and let K)'.=
{\-X)Ko + XK^ forO^A^i. By E.1.26),
V'(A:a) = i(")(l - X)"-'X'V;{Ko, K,)
with
Vi{K^, a:,) := V{Ko[n - i], K,\i]) =: V^,,. F.2.1)
When we use the (classical) notation V(,), it will be clear from the
context that ATq, K^ are given and held fixed. Since VXATq, A^i) has two
arguments, it cannot be confused with an intrinsic volume.
The function / defined by
/(A) .= V„{Kxy^" - A - A)V„(A:o)'/" - XV„{K,yl"
for O^A^l is concave, by Theorem 6.1.1, and satisfies /(O) =
/(I) = 0. Hence, its derivative at 0,
fulfils /'@)^0, and /'(O) = 0 only if / is identically 0. The latter
implies equality in F.1.1). Furthermore,
/"(O) = -{n - l)V(or"'^^''[^0) - ^@)V'B)l ^ 0.
This yields:
Theorem6.2.1 (Minkowski's inequalities). For n-dimensional convex
bodies K, L e XS,
V{K, ..., K, L)"^ V„{Kr-'V„{L). F.2.2)
Equality holds if and only if K and L are homothetic. Further,
V{K, ..., K, Lf^ V„(K)V(K, ..., K, L, L). F.2.3)
One should note that so far no information on the validity of the
equality sign in F.2.3) has been obtained (equality holds, of course, if

318 Inequalities for mixed volumes
K and L are homothetic). More general quadratic inequalities for
mixed volumes are proved in the next section. These again imply
F.2.2). However, F.2.3) is required for their proof.
By considering the special case where AT or L is equal to a ball,
Minkowski's inequality F.2.2) reduces to well-known and fundamental
geometric inequalities for convex bodies. Recall that
S{K) := nW^{K) = nV{K[n - 1], B")
is the surface area of K and
h(K) = -^ W„_,(A:) = -^ V(K, fi"[n - 1])
is the mean width of K. Hence, F.2.2) implies the isoperimetric
inequality, which states that the volume V„ and surface area S of an
n-dimensional convex body satisfy
V \"~^
F.2.4)
Uni Un,
with equality if and only if AT is a ball. It also implies Urysohn's
inequality for volume V„ and mean width,
h\" V
^^ ^^ F.2.5)
.2) K„
with the same equality cases. A weaker form is the isodiametric (or
Bieberbach 's) inequality
for the diameter D oi K, which follows trivially since b{K) g D{K).
The improved version of the Brunn-Minkowski theorem given by
Theorem 6.1.2 also leads to a stability result for Minkowski's inequality
F.2.2). For a convex body K e ,l^o. we use the normalization defined
by
K := V„(A:)-'''"[/C - c{K)\, F.2.7)
where c{K) denotes the centroid of K. Thus K has centroid o and
volume 1.
Theorem6.2.2. Let K, L e Xq and set D := max{D(K), D(L)). Then
V„(/C)'-V„(L) -l=D-<^(^'^^ ' (^-^-^^
where y„ = 2n/{4rf„)"'*'^ and r\„ is given by F.1.6).
Although we did not reproduce Groemer's proof of Theorem 6.12,
we give the simple argument of Groemer A990b) to deduce F.2.8).

6.2 The Minkowski and isoperimetric inequalities 319
Proof. First suppose that V„{K) = V„{L) = 1 and c{K) = c{L) = o, and
put /(A) := V„((l - X)K + AL)i/" and
Un[l +
VAA - A)
2VW^]d! ^^' ^^"'
n + 1
Then /(A) ^ 1 + u(A) by F.1.7), and /(O) = /(I) = 1. By the Brunn-
Minkowski theorem, / is concave, hence /'(O) ^[f{h) - f@)]/h for
0</t^l and in particular /'(O) ^ 2[/(^) - 1] ^2u(^). Since /'@) =
V(a:, . . . , a:, L) - 1, we obtain
V{K, ...,K,L)^1 + -~^ 6{K, L)"^'
nD
and thus
v(a:, ..., a:, l)" ^ 1 + -^ «5(a:, l)"^i.
Inequality F.2.8) for convex bodies of arbitrary volume follows
immediately. ■
Next we formulate a theorem that strengthens Minkowski's inequality
F.2.2) in several respects. It may look rather complicated at first sight,
but it has interesting consequences and admits specializations of intuitive
appeal.
Theorem 6.2.3. Let K, L e XS be convex bodies, and let QC 5""' be a
closed set such that
a: = Dfi'iK, u). F.2.9)
If L is defined by
L := fl'^'C^' «)' F.2.10)
then
v^iK, LyK"-^^ - v„(a:)v„(LI/<"-i)
S [V^{K, £)'/<"-'> - r{K, L)V„(L)'/<''-'>]", F.2.11)
where r{K, L) denotes the inradius of K relative to L.
This theorem is due to Diskant A973b); the special case Q- 5""' (in
which L = L) had been treated previously by Diskant A972). We shall
present Diskant's proof in Section 6.5.
Solving inequality F.2.11) for r{K, L) and choosing i2=5"~\ we
obtain

320 Inequalities for mixed volumes
r{K, L)
^IV,{K,L)V
~\ Vn{L) )
[V,{K, LY'^"-'^ - V„(/C)V„(L)^/<"-')]V"
v^CD^A"-) ' ^^'-^'^
which will be used in the proof of Theorem 7.2.2.
To interpret inequality F.2.11), we suppose that the assumptions of
Theorem 6.2.3 are satisfied. We may assume that r{K, L)L C K. If
X € E"\/C, there exists u e Q with x i H~{K,u) and hence
X « H-(r{K, L)L, u) = H-{r{K, L)L, u).
It follows that
r{K, L)LC K.
Further, we remark that
ViiK, L) = V,{K, L).
This is easy to prove, but we refer to Section 6.5, where the result is
obtained in the course of the proof of Theorem 6.2.3. Using Theorem
6.2.1, we obtain
V^{K, LY = V.C/C, LY ^ V„{KY-'V„{L) ^ r(K, LY^"-'W„{LY■
Hence, for the right-hand side of F.2.11) we have
V^{K, D'A"-!) - r{K, L)V„(Z;)'/<"-') ^ 0,
with equality if and only if K is homothetic to L. Since V„(L) ^ V„(L),
it is now clear that inequality F.2.11) is an improvement of Minkowski's
inequality F.2.2).
Inequality F.2.11) can be used to obtain some stability results; see
Note 1 and Theorem 7.2.2. Here we discuss only the special case
obtained by taking for L the unit ball B = B".
Let K 6 Xo be given. Denoting its volume by V„, its surface area by
5 and its inradius by r, we obtain from F.2.11) the inequality
[^J - V„E)'/<'-')V„ ^ [(I) - rV„E)'/<'-')| . F.2.13)
Two further special cases are of interest. We take Q= 5""' and use
the inequality x"~^ - y""' ^{x - yY~^ (for Jc ^ y ^ 0) to obtain
S" - n"K„V"„-^ ^ Ei/('"-i) _ ft,J,/<'-i)r)"("-i). F.2.14)
Since 0}„r"~^ is the surface area of the inball of /C, it is clear that the
right-hand side of F.2.14) is non-negative and is zero only if AT is a ball.
Thus F.2.14) estimates the 'isoperimetric deficit' S" - n"K„V"„'^ in
terms of a quantity that has a simple geometric meaning and is
obviously positive if K is not a ball.
Next, if i2 is a closed set satisfying F.2.9), we note that F.2.13)
implies

Notes for Section 6.2 321
S"/V"„-^ ^ n"V„iB), F.2.15)
with equality if and only if K is homothetic to B. One says (see Section
6.5) that the convex body K is determined by Q if F.2.9) holds. Thus,
among all convex bodies determined by a given closed set i2c5"~',
precisely the bodies circumscribed about a ball, that is, homothetic to
r\H-(B",u),
have the smallest 'isoperimetric ratio' S"/V'^~\ For finite sets Q, this is
a theorem first proved (for n = 3) by Lindelof A869) and in a different
way by Minkowski A897); for general sets Q it was proved by
Aleksandrov A938a).
The smallest closed set Q that one can take for a given convex body
K so that F.2.9) holds is the closure of the set of outer unit normal
vectors at regular boundary points of K. If Q is this set, then
K^ := f]H-{B". u)
is called the form body of K. Inequality F.2.15) now reads
S"/Vr' ^ tfV^iK^), F.2.16)
with equality if and only if K is homothetic to its form body. Using
Theorem 2.2.8, one sees that this is the case if and only if AT is a
tangential body of a ball. Inequality F.2.16) strengthens the
isoperimetric inequality for convex bodies with singularities.
Notes for Section 6.2
Minkowski's inequalities. In the plane E^ there is only one Minkowskian
inequality, namely
V(Ku K^f § V2(/C,)V2(/C2)
for the mixed area V(Ki, Ki), with equality if and only if /C, and K2 are
homothetic. There are several proofs and improvements in the literature; see
Bonnesen & Fenchel [8], Section 51, and Note 4 below.
Upper bounds for the mixed area in terms of perimeters were obtained by
Betke & Weil A990). For convex bodies K^KieX^ with perimeters
L(Kt), L^Ki) they showed that
V(K„K^)^mK,)L{K2),
with equality if and only if /C, and Ki are orthogonal segments (possibly
degenerate), and
\/3
V{K,-K)^ -^UKY.
Here the equality sign holds if K is an equilateral triangle, and probably only
in this case, but this has only been proved if /C is a polygon.

322 Inequalities for mixed volumes
For n = 3, the cubic inequality F.2.2) is a consequence of the quadratic
inequalities of type F.2.3). These inequalities were first proved by Minkowski
A903) (announced in A901a,b); see Bonnesen & Fenchel [8], Section 52, for
more information). For general dimension n, F.2.2) was proved by Siiss
A931b), using the Brunn-Minkowski theorem, and Bonnesen A932) noticed
that it can be obtained from the Brunn-Minkowski theorem in the simple
way described in the proof of Theorem 6.2.1.
The stability problem for the Minkowski inequality F.2.2) requires us to
find explicit estimates for the deviation of suitable homothets of convex
bodies K, L e Xq in terms of the 'deficit'
V„iKY-'V„{L)
Theorem 6.2.2 gives a result of this type, which is due to Groemer A990b).
The first result of this kind had already been obtained by Minkowski A903),
§6, who proved his inequality F.2.2), for n = 3, in the following sharper
form. Let K, L e JCl have coinciding centroids, and let D be the maximum of
Vv^)h(L, •)
^^/VjLJhiK, ■)
over 5^; then (inequality G6) of Minkowski 1903)
V(K, K, L) ^ ^ 1 (P - 1)*^
^V:,{KfV^(L) 2'»3''7'''^ D'
Stability estimates for the Minkowski inequality F.2.2) in dimension n
were obtained by Volkov A963) and Diskant A972); see also Diskant A973a)
and Bourgain & Lindenstrauss A988b).
Minkowski's inequality F.2.2) is a useful tool for the solution of many
different geometric extremum problems. Some examples are found in
Ohmann A958).
2. The Blaschke diagram. For a three-dimensional convex body K € X^ one
often uses the classical notation V(K, K, K) = V (volume), W(K, K, B^) = S
(surface area), 3V{K, B^ B^) = M ('integral of mean curvature'). The
quadratic Minkowskian inequalities, special cases of F.2.3), now state that
M2 ^ 4ff5,
where equality holds if and only if /C is a ball, and
S^ ^ 3VM,
where equality holds if and only if /C is a cap body of a ball (equality
conditions are discussed in Section 6.6). These quadratic inequalities imply
the two cubic ones,
S^ ^36ffV2, M3 S48ff2v.
On the other hand, the two quadratic inequalities are not a complete system
of inequalities satisfied by the three functionals V, S, M of convex bodies.
Following Blaschke, one considers in an (AC^)-plane all points with coordinates
_ 4riS_ _ 4Sn^V
* " M2' ^ ~ M' '
for all possible three-dimensional convex bodies and finds that part of the
boundary of the resulting 'Blaschke diagram" must correspond to a third
sharp inequality, of the form V ^ f(S, M), which is still unknown. This
problem is neatly di.scussed in Hadwiger [17], §§28-9. Later contributions
resulted in the non-sharp inequality

Notes for Section 6.2 323
V §
(f--)
given by Groemer A965) (extended to higher dimensions by Firey 1964) and
in the interesting conjecture, due to Sangwine-Yager A989), that
The isoperimetric inequality. The isoperimetric inequality has many facets,
and with all its versions and ramifications it can easily fill a book by itself. In
fact, isoperimetric inequalities are one of the central themes of the book by
Burago & Zalgaller A988), which a reader interested in a broad synopsis of
isoperimetric problems should consult first.
Minkowski's discovery that the isoperimetric inequality for convex bodies
can be deduced from the Brunn-Minkowski theorem has strongly influenced
the later development. Minkowski's inequality F.2.2) can be interpreted as
an isoperimetric inequality for a suitably defined notion of relative surface
area, and versions of the Brunn-Minkowski theorem for non-convex sets lead
to corresponding isoperimetric inequalities (for a brief version of the proof,
including the Brunn-Minkowski inequality, see Federer A969), 3.2.41-
3.2.43). Here the uniqueness question often poses major problems. We refer
the reader to the treatments in Hadwiger [18] and Dinghas A961). Further
references are Busemann A949b), Ohmann A952a), Baebler A957) and
Barthel & Bettinger A961,1963).
Isoperimetric problems in Minkowski spaces are the subject of papers by
Busemann A947,1949b), Barthel A959b) and Holmes & Thompson A979).
An impressive survey of the various ramifications of the isoperimetric
inequality, mainly from the viewpoint of analysis and differential geometry,
and its applications to these fields is given by Osserman A978).
For isoperimetric inequalities in mathematical physics and in analysis that
are analogous to the geometric one, we refer to the classical book of Polya &
Szego A951) and to the more recent one of Bandle A980).
Diskant's inequality. Diskant's A973b) improvement F.2.11) of Minkowski's
inequality extends and unifies many earlier (and even some later) results.
The two-dimensional case, for Q= S^, can be written in the form
Ag, - AoA, § (Ao, - rAxY F-2.17)
with Aoi := V(A:, L), /1o:=V2(A:), /li := V2(L), r := r(A:, L); an
equivalent formulation is
/lo -2Moi + rMi go. F.2.18)
If /C e JCl has area A, perimeter L and inradius r, a special case of F.2.17)
gives
O - AnA^{L -Inrf. F.2.19)
For various proofs, see the references in Note 3, but also Bonnesen &
Fenchel [8], p. 113, and Hadwiger A944, 1948).
Inequality F.2.14) was first proved by Hadwiger A949c) for n = 3 and by
Dinghas A948) for general n; a short proof is given in Hadwiger [18], Section
6.5.2. Dinghas A948) had already obtained a slightly weaker inequality of the
form F.2.13).
Inequality F.2.16) was proved in Hadwiger [18], Section 6.5.5. Several
special cases had been treated before by Bol A940), Bol & Knothe A949)
and Dinghas A940c, 1942, 1943a, b, 1949b, c,d), but the easier deductibility
from Minkowski's inequalities was not always noticed.
The improvement of the isoperimetric inequality by F.2.16) is strong if K

324 Inequalities for mixed volumes
has sharp singularities. Improvements that are strong if K has large flat
pieces in the boundary were treated by Hadwiger A949b); see also Ohmann
A958).
3. Bonnesen-style inequalities. Let n = 2, Kq, Ki e Xq, V(Kq, Ki) =: Aoi,
ViiKo) =■■ Ao, V2(Kt) := /!,, r(Ko, /Ci) := r and r(Ki, Ko) := 1/R, so
that R is the circumradius of Kq relative to Ki; in other words, KqC RK'i
for a suitable translate K[ of Ki, and R is the smallest number with this
property. With these notations, inequality F.2.18) can be generalized to
Ao-2pAoi + p^Ai^O, F.2.20)
or equivalently
Ai - AoA, ^ (/l„i - pAl)^ F.2.21)
for all numbers p in the interval [r, R]. The two cases p= r and p= R
together with (a^ + b^)/2 ^ [(a + b)f2Y yield
A
Al- AoAi^^iR- ry. F.2.22)
If Ki = B^, these inequalities read
A - pL + np^ ^ 0, F.2.23)
L^ - 4nA^(L - lupY F.2.24)
for r g p g R and
L2 - 4ff/l g ff2(/^ _ ^J. F.2.25)
where r, R are respectively the usual inradius and circumradius. The last
inequality has a sharper version, namely
L^ - 4nA ^ 4n(Ro - ro)^ F.2.26)
where ry and Rq, ry = Rq, are the radii of two concentric circles enclosing
bd Ki) such that Ry - ro is minimal.
All these results are known as Bonnesen's inequalities. Inequalities
F.2.24) and thus F.2.25) were proved by Bonnesen A921) and are
reproduced in Bonnesen [7], pp. 60-3 (see also Eggleston [13], pp. 108-10); for
F.2.26), see Bonnesen A924) and [7], pp. 70-4. An elegant integral-
geometric proof of F.2.25) due to Santalo can be found in Blaschke A937a),
§11. Osserman A979) gives a comprehensive survey of inequalities equivalent
to F.2.24) and their extensions, including non-convex curves and curves on
surfaces. New treatments of F.2.26) are given by Gage A990) and by
Fuglede A991), who considers non-convex curves.
An integral-geometric proof of F.2.22) appears in Blaschke A937a), §15,
where the inequality itself is ascribed to Bonnesen. Also related to integral
geometry are the proofs given by Hadwiger A941), Fejes T6th A950) and
Randers A968). Fourier series are' used by Bol A939) and Wallen A987). Bol
added the equality condition: Equality in F.2.22) holds if and only if Ko and
Ki are either homothetic or are parallelograms with parallel sides.
In higher dimensions, the inequality F.2.14), that is,
S" - rfK„V"„-^ ^ (S'/C--!) - i^lK'-i)r)n(.n-')
can be considered as a generalization of F.2.24) for the value p= r, the
inradius. An extension of F.2.23) for p= r is the inequality
V„ - rS + (n - l)K„r" S 0, F.2.27)
where equality holds only for the ball. F.2.27) was conjectured by Wills
A970) and proved independently by Bokowski A973) and Diskant A973c)
and in a slightly sharper form by Osserman A979). In each case, the
inequality is deduced from F.2.13).

Notes for Section 6.2 325
For n = 2, inequality F.2.27) remains true if the inradius r is replaced by
the circumradius R, giving
A - RL + nR^ ^ 0; F.2.28)
however, this does not hold in dimensions n ^ 3. Instead, one has
(n - 1)V„ - 2RS + (n + 1)k„R" ^ 0 F.2.29)
for n = 2 (note the reversed inequality sign). This was proved by Bokowski &
Heil A986), who obtained the following more general result.
Theorem. For an arbitrary convex body /C e .T", n = 2, with circumradius R
and quermassintegrals Wo, . .. , W„, the inequalities
Ci„R-W, + Ci,iR'Wi + CiiR'W, § 0 F.2.30)
hold (oTO^i< j < k^ n and c^,, := (r - q)(p + 1).
F.2.29) is a special case of F.2.30), as is the inequality
iW.-i - 2(i + l)RWi + (i + 2)R^W,^, ^ 0
for I = 1, ..., n — 1.
For n=2, F.2.30) reduces to
A - 2RL + 3nR^ § 0, F.2.31)
with equality only for circles; this is due to Favard A929). A survey over
inequalities for three functional of a planar convex body was given by
Santal6 A961).
An extension of F.2.28) to higher dimensions that does hold is the
inequality
W„.2 - fiRW„-, + (fi- l)R'W„ ^ 0 F.2.32)
with /3 = n/(n - 1). More generally, Sangwine-Yager A988b) established the
inequalities
V„(K) - nrVi(K, L) + (n - l)r^V2(K, L) g 0 F.2.33)
and
(n - 1)V„.2(K, L) - nRV„.t(K. L) + R'V„(L) g 0 F.2.34)
for K,Le X", with r = r(K, L) and R = l/r(L, K). Equality in F.2.33)
holds if K is an (n - 2)-tangential body of L, and in F.2.34) if L is an
(n - 2)-tangential body of K. The special case « = 3, L = B^ of F.2.33) has
already been treated by Sangwine-Yager A988a), where the author was able
to show that equality holds only for cap bodies of balls.
Heil A987) investigated whether F.2.32) can be improved for n^3 by
increasing /3 and, by an interesting application of the calculus of variations, he
found that this is the case, at least if the body considered is centrally
symmetric or if R is replaced by half the diameter.
Teissier A982) discussed Bonnesen-type inequalities in relation to algebraic
geometry; see also Oda A988), p. 188.
Stability estimates for the isoperimetric inequality. If the isoperimetric
inequality
S" - n"K„V';,-' ^ 0
for the surface area S and the volume V„ of an n-dimensional convex body
K e JCo holds with equality, then K is a ball. A stability estimate for this
inequality would be any result of the kind
S" - n-K^Vr' S e => A(K, Bk) S /(t),
where A is some metric on the space of convex bodies (or some other
measure for the deviation of two convex bodies), Bk is a suitable ball and /

326 Inequalities for mixed volumes
is an explicitly known function, non-decreasing and satisfying limfjo/(£)
= /(O) = 0. This function may involve constants that depend on some given
bounds for e and for the inradius and circumradius of K as well as on the
dimension.
In the plane, the Bonnesen inequalities F.2.25) and F.2.26) immediately
yield such stability estimates. The difference R — r of circumradius and
inradius is, of course, a satisfactory measure for the deviation of a convex
body from a suitable ball. Inequality F.2.25) has no immediate extension to
higher dimensions, since the example of a convex body close to a segment
shows that for n=3 the isoperimetric deficit S" - n"K„V"„'' can be
arbitrarily small while R - r = 1, say. However, one can show that
S" - n"K„V"„-^ ^ c(n)r'-<'--"l 1 F.2.35)
for K e X", where c(n) is an explicitly known constant depending only on n.
On the class of convex bodies with r = Co > 0, Cq given, F.2.35) provides a
stability estimate for the isoperimetric inequality. It was obtained by Groemer
& Schneider A991), as a consequence of more general results (see also
Section 6.6, in particular F.6.11)).
Some results in the literature can be expressed in the form
S" - n"K„V"„-' § cd{K, Bk)", F.2.36)
where B^ denotes a suitable ball, c may depend on n and bounds for inradius
and circumradius of K, and * depends only on n. An inequality of type
F.2.36) becomes sharper as its exponent * becomes smaller. The
specializations of the stability results for the Minkowski inequality due to Volkov
A963) and Diskant A972,1973a) mentioned in Note 1, as well as an
argument of Osserman A987) using F.2.14), yield estimates of this kind, all
with a= n. In Groemer & Schneider A991), an estimate with a-(n + 3)/2
was achieved, and it was shown that one cannot have a <(n + l)/2. By
analytic methods, Fuglede A986b, 1989) obtained stability estimates for the
isoperimetric inequality and under additional assumptions also for non-convex
sets, which in a certain sense are of optimal orders.
For the sharp inequality bounding the area of a plane domain of given
constant width from below, stability estimates were obtained by Groemer
A988b).
7. Improved versions of the isoperimetric inequality for restricted classes of
convex bodies. It is clear that on any class of convex bodies that does not
contain bodies arbitrarily close to balls, a strengthening of the isoperimetric
inequality must be possible (in principle). Simple examples can be found in
Hadwiger [18], 6.5.3, 6.5.4, 6.5.6. A beautiful inequality of this type was
proved by Fejes T6th A948):
Theorem. The surface area S and volume V of a three-dimensional convex
polytope with k facets satisfy
53 n k
— ^54(k- 2)tana'tDsin2a'* - 1), ** = J-j~^'
Equality holds precisely for the simple regular polytopes (tetrahedron, cube,
dodecahedron).
Further related results (and some interesting problems) are found in the
survey article by Florian [46].
Another example is Hadwiger's improvement of the isoperimetric
inequality for 'half-bodies'. A convex body K e XS is called a half-body if it
has a supporting hyperplane H such that the union of K and its image under

6.3 The Aleksandrov-Fenchel inequality 327
reflection in H is convex. Hadwiger A968b) proved that the volume and
surface area of such a body satisfy
Equality holds if K is the union of a half-ball of radius p and a right circular
cylinder of height p attached to the equator {n - l)-ball of the half-ball. A
different proof will be given in Section 6.7, Note 3.
The ratio of volume and surface area. While we are on the topic of volume
and surface area, we mention that the ratio VjS has the property of quasi-
monotoneity: \l K, L e X" and K C L, then
Vn(K) ^ V„(L)
S(K) - " S(L) '
and here the factor n cannot be replaced by a smaller number. This was
proved by Wills A970).
Quermassintegrals and other functionals. Bonnesen-type inequalities are, as
described in Note 5, a class of inequalities involving volume, surface area and
possibly other quermassintegrals, together with some other functional of
convex bodies, such as inradius or circumradius. There are several
inequalities of a similar character, that is, connecting some of the quermassintegrals
with additional geometric quantities, for example minimal width, diameter, or
the sizes of suitably chosen projections or plane sections. Some of these
results are strengthenings of isoperimetric inequalities; others stress various
different viewpoints. We refer to the survey of Santalo A961) concerning the
case of the plane and mention results by Benson A970), BoCek A983),
Bonnesen [7], Chapter 6, Chakerian A971), Favard A933a), Firey
A965b, 1966,1969), Hadwiger [18], p. 292, Knothe A957a), inequalities D1),
D3), N^denik A965,1967) and Petermann A967).
6.3. The Aleksandrov-Fenchel inequality
We have seen in Section 6.2 that the Brunn-Minkowski theorem implies
the quadratic inequality
VC/Co, /C,, . . . , K,f ^ VC/Co, A:„, /C,, . . . , K,)V„{K^).
This is merely a special case of a system of quadratic inequalities
satisfied by general mixed volumes.
Theorem6.3.1 (Aleksandrov-Fenchel inequality). For K^, K2, K^,. . . ,
V{KuK2,Ki,...,K„f
^ v{K,, a:,, a:3, ..., K„)v{K2, K2, k^,..., k„). F.3.1)
Clearly, equality holds in F.3.1) if ATj and K2 are homothetic.
However, examples show that this is by no means the only possibility
for equality. The complete classification of the equality cases is an
unsolved problem. We shall return to this question and some partial
results in Section 6.6.
The proof we shall give for Theorem 6.3.1 is due to Aleksandrov
A937b). It uses strongly isomorphic polytopes and approximation.

328 Inequalities for mixed volumes
As in the last part of Section 5.2 we assume that an a-type, c-i, of
strongly isomorphic simple n-dimensional polytopes is given. We use the
notation of Sections 5.1 and 5.2; in particular, u^,. . . ,Ufj are the unit
normal vectors corresponding to the facets of (-4, P = (hi,. . . , h/^) with
hi := h{P, Ui) is the support vector of P e <yt., and the mixed volume of
N-tuples Xi,. . . ,X„eU'^ IS defined by E.2.5). We shall often identify
PeW with its support vector P\ that is, if in V{Xi,. . . , X„) or
v{AiXi,. . . , A,Ar„_i) (defined by E.2.6) and E.2.7)) one of the
arguments Xr is a support vector P^, we replace X^ by P^ and AjXr by
F\''^= F{Pr,u,). Further, we say that Z = (Ci,. . • , U) e IR'^ «s the
support vector of a point z e E" if C, = h{{z}, «,) for / = 1, . . . , N,
thus if
2 = ({z, u,), . . . , {z, «n)).
The following theorem is a sharper version of a special case of Theorem
6.3.1, and the latter can be deduced from it.
Theorem 6.3.2. If P, P^,. . . , P„ are strongly isomorphic polytopes of the
simple a-type c -f and if Z eU^ then
Viz, P,P3,..., P„f ^ V(Z, Z,P,,..., P„)V(P, P, P,, ..., P„).
The equality sign holds if and only if Z = kP + A, where A e R and A is
the support vector of a point.
Taking for Z the support vector of another polytope in of., we obtain
inequality F.3.1) for the special case of simple strongly isomorphic
polytopes. The general case then follows from the approximation
theorem 2.4.14 and the continuity of the mixed volume. The limit
procedure is responsible for the fact that the cases of equality cause
problems.
Proof of Theorem 6.3.2. We introduce a symmetric bilinear form <P on
W^ by
<P{X, Y) := V{X, Y, Pi, . .., P„) for X, Y e W^
if n = 2 (with P-^,. . . , P„ omitted if n = 2). It suffices to prove the
following proposition.
Proposition!. If 0(Z, P) = 0, then 0(Z, Z)^0, and equality holds if
and only if Z is the support vector of a point.
In fact, suppose that Proposition 1 is true. If Z € R'^ is given, define
<KZ, P)
0{P, P)
A := -?^J^ and Z.= Z- XP

6.3 The Aleksandrov-Fenchel inequality 329
(observe that 0(P, F) = V{F, F, Pj,... ,F„)> 0). Then *(Z', P) = 0
and hence <t>{Z', Z')~0, with equality if and only if Z' is the support
vector of a point. From
*(Z', Z') = <^Z, Z) - ^^
the assertion of the theorem follows.
To prove Proposition 1, we first consider the special case
P3 = . . . = P„ = P of the theorem.
Froposition 2. The inequality
V{Z, F, ...,Ff^ V{Z, Z,F,..., F)V„{F) F.3.2)
holds. If n = 2, equality holds if and only if Z = XP -^ A, where A e R
and A is the support vector of a point.
For the proof, we note that, by Lemma 2.4.12, P + eZ is the support
vector of a polytope Qea^ if |f|>0 is sufficiently small. From
Minkowski's inequality F.2.3) we infer that
0 ^ V{Q, P,...,Ff - V{Q, Q,F,..., F)V„(F)
= e^[V(Z, P,...,Pf - V{Z, Z,F,..., F)V„{P)].
This proves F.3.2). If n = 2, then by Theorem 6.2.1 (first part) equality
in F.3.2) holds if and only if Q and F are homothetic. This completes
the proof of Proposition 2.
We prove Proposition 1 by induction with respect to the dimension.
For n = 2, the assertion is true by Proposition 2. We assume that n § 3
and that the assertion of Proposition 1 is valid in smaller dimensions.
For each i e {\,. . . ,N} we define a symmetric bilinear form cpi on
U^ by
<Pi{X, Y) := viAiX, AiY, F';*\ ..., F*"') for X, Y e U^
(with f (">, . . . , f ("> omitted if n = 3).
Propositions. ZeU'^ is an eigenvector of the bilinear form <P with
eigenvalue 0 if and only if Z is the support vector of a point.
For the proof, we first note that
<1>(X, Y) = - 'Zx^<p^iY, F,)
n ,=1
by E.2.8). Since (p,( •, P3) is linear, it is of the form

330 Inequalities for mixed volumes
thus
1 ^
<HX, y) = - S V.yy F-3.3)
" 1,/=1
Here hjj = hjj, because <P is symmetric. Stating that Z = (^j,. . . , ^/v)
=^ 0 is an eigenvector of <P corresponding to the eigenvalue 0 means that
N
S^;C> = 0 for/ = l,...,N,
or equivalently that
(Pi(Z, P3) = 0 for / = 1, . . . , N. F.3.4)
If Z is the support vector of the point z, then
(p,(Z, P3) = v{{z), fP>, . . . , F\">) = 0.
Suppose, conversely, that F.3.4) holds. By the induction hypothesis,
this implies q>i(Z, Z)^0. Without loss of generality, we may assume
that h{Pi,Ui)>0; then
1 ^
0 = -SC/<P.(Z. P3)= *(Z, Z)
n j=j
= Viz, Z, P3, . . . , P„) = V{Fi, Z, Z,P„..., P„)
1 '^
= -S/t(P3,«i)<P/(Z, Z)^0
n j=i
and hence (p,(Z, Z) = 0. By the induction hypothesis this implies that
AjZ is the support vector, relative to the a-type of F,, of a point z,.
Explicitly, this means that
AiZ = {{zi, Vii), . . . , {zi, Vif^)),
with Vij ■= o for (i, j) ^ J (where we use the notation of Section 5.1).
We choose f =|= 0 so that P3 + sZ is a support vector of a polytope
Qec'i.The equality A,(P3 + f Z) = A,P3 + eAiZ yields
h{F{Q, u,), Vij) = h{F?\ Vii) + ^B.' t^'/)
and thus F{Q, u,) = FP^ + f, with a vector tj (= fz, + a^u, for some
(V, e R). For (/,;) e 7 we conclude that the (n - 2)-face G .=
F(Q, Uj) n F(Q, Uj) satisfies G = FS/> + f, and analogously G =
f5/^ + fy, hence f, = fy. Since any two facets of F3 can be joined by a
chain of facets such that any two consecutive facets in the chain have an
(n - 2)-dimensional intersection, we conclude that f, = tj for all /, ;.
Thus Q is a translate of F3 and, hence, Z is the support vector of a
point. This completes the proof of Proposition 3.
Besides <P, we now introduce a second symmetric bilinear form *P on
R'^ by means of

6.3 The Aleksandrov-Fenchel inequality 331
nx, Y):=^i ^j^^ ;c,y, for X, Y e U'.
n ,=1 h{P, Ui)
Here we assume, without loss of generality, that h{P, u,) >0 for / = 1,
. . . , N. Since (p,(P, P3) > 0, the form "f^is positive definite.
We consider the eigenvalues A] > A2 > . . . of <P relative to V and
make use of the fact that
Xi = max{0iX, X)MX, X) = I}, F.3.5)
A2 = max {0(Ar, X)\ V{X, X) = 1
and yff{X, Y) = 0 for all Y in the A,-eigenspace}.
In analogy with F.3.3) we write
1 ^
nX y) = - S c,^,yy
" i,/=i
where
Cij ■■-
iii = j.
h{P, u.)
0 if / + j.
Then Z = (Ci, •■ • , ^n) ^ ^'^ 'S an eigenvector of * relative to 'f^ with
eigenvalue A if and only if
N
S(&,y-Ac,y)Cy=0 for / = 1, . . . , N,
/=!
or equivalently if
(pi{P, P3)
cp^iZ, P3) = A ^^|p' ^J C, for / = 1, . . . , N.
In particular, A = 1 is an eigenvalue with corresponding eigenvector
Z = P.
Proposition 4. The only positive eigenvalue of <P relative to 'f'is 1, and
it is simple.
For the proof we first assume that P = P^ = . . . = P„. Suppose
Proposition 4 were false in this case. If there is a positive eigenvalue
At =1=1, then there exists Z eU^ with V{Z,P) = 0 and 0(Z, Z) =
lx*P(Z,Z)>0. If 1 is a multiple eigenvalue, the corresponding eigen-
space is at least two-dimensional and hence contains a vector Z with
V{Z, P) = 0 and 0(Z, Z) = ^'{Z, Z) > 0. Thus in either case we
conclude from
1 ,^ wAP, P)

332 Inequalities for mixed volumes
that V(Z, P, ..., P) = 0. Since V{Z,Z,?, ..., P)=0(Z, Z)>0,
this contradicts Proposition 2.
Now let P3,...,P„e<^ be arbitrary. For ^e [0,1] let F,{&) :=
{I - ff)F + &Fr, r = 3,. . . , n. The coefficients of the corresponding
forms <P, ^ depend continuously on ^, hence the same is true for the
relative eigenvalues. By Proposition 3, the number 0 is always an
eigenvalue with multiplicity n. It follows that the sum of the
multiplicities of the positive eigenvalues is independent of ^. Since it is equal to 1
for ^ = 0, it must be equal to 1 for ^= 1. This proves Proposition 4.
Proposition 4 implies that the eigenspace corresponding to the
eigenvalue 1 coincides with lin{P} and that the second eigenvalue is not
positive, and hence that 0(Z, Z)^0 for all Z satisfying ^'{Z, P) = 0;
the latter is equivalent to 0(Z, P) = 0. Thus *(Z, P) = 0 implies
0(Z, Z) = 0. Suppose that we have equality for some Z =|= 0. Since at
Z the maximum in F.3.5) is attained, Z is an eigenvector with
eigenvalue 0. By Proposition 3, Z is the support vector of a point. This
completes the proof of Theorem 6.3.2. ■
Notes for Section 6.3
1. Proofs of the Aleksandrov-Fenchel inequality. The proof of Theorem 6.3.1
presented above is due to Aleksandrov A937b). Aleksandrov A938b) gave a
second proof, which uses bodies of class C\ and Aleksandrov's inequalities
for mixed discriminants (see Theorem 6.8.1). Although the underlying classes
of convex bodies are quite different, there are certain analogies between the
two proofs. The principal ideas can be traced back to the proof given by
Hilbert A910) for Minkowski's quadratic inequalities in three-space. The
second of Aleksandrov's proofs is reproduced in Busemann [12], and the first
in LeichtweiB [23].
It has become customary to talk of the 'Aleksandrov-Fenchel' inequality,
because Fenchel A936a) also stated the inequality and sketched a proof. We
quote from Busemann [12], p. 51: 'Fenchel's proof is very sketchy, a detailed
version has never appeared and it is not quite clear what it would involve'.
Hadwiger [18], p. 290, characterizes Fenchel's note as 'schwer verstandlich'
[hard to understand]. In honour of Fenchel, it seems justified to talk of the
Aleksandrov-Fenchel inequalities, although it is doubtful whether he had a
complete proof.
For coaxial bodies of revolution, Hadwiger [18], Sections 6.5.8, 6.5.9,
obtained improved versions of the Aleksandrov-Fenchel inequality in an
elementary way.
In recent years, surprising connections have been discovered between the
Aleksandrov-Fenchel inequality and algebraic geometry. Independently,
Khovanskii, and Teissier A979,1981), found algebraic approaches to the
Aleksandrov-Fenchel inequality, yielding a new proof via the Hodge index
theorem. A detailed version, written by Khovanskii, can be found in Burago
& Zalgaller A988), §27. Here mixed volumes of Newton polyhedra associated
with Laurent polynomials (Bernstein 1975) play an essential role. This line of
research was carried further by Gromov A988).

6.4 Consequences and improvements 333
2. Full sets of inequalities. If p § 2 convex bodies K u ..., Kp in E" are given,
there are N = ( "%''-') mixed volumes V(A:„, . . . , KJ A § ,i g ...
= i„ = p) that can be formed. The Aleksandrov-Fenchel inequality, applied
to these values in all possible ways permitted by the symmetry properties,
yields a certain system of quadratic inequalities. This set of inequalities is said
to be a full set if any given set of N quantities satisfying these inequalities
can arise as the set of mixed volumes of some system of p convex bodies.
Heine A937) for n = 2 and Shephard A960) for n^2 investigated whether
the known inequalities are a full set. The answer depends on p, but is not yet
known for all values of p.
3. Applications. Several applications of the Aleksandrov-Fenchel inequality to
the geometry of convex bodies appear in later sections. Here we mention
briefly a few uses of this inequality in other fields. It has been applied to
differential-geometric uniqueness theorems (Schneider 1966a), to extremal
problems for geometric probabilities (Schneider 1985b) and to combinatorial
questions, in particular to showing that certain sequences of combinatorial
interest are log concave (Stanley 1981, 1986, Kahn & Saks 1984).
6.4. Consequences and improvements
Since the mixed volume V{Ki, K2, ■ ■ ■ , K„) is symmetric in its
arguments, it is clear that from the Aleksandrov-Fenchel inequality
v(a:„ a:2,/Ca,..., K„f
^ v(a:„ a:,, Ki,..., K„)V{K2, K2, k^,..., k„)
many other inequalities can be deduced, by repeated application. We
first mention some of the more frequently occurring of such derived
inequalities.
Some preliminary remarks are in order. A finite sequence (aq.
fli,. . . , flm) of real numbers is called concave if
fl,_i - 2fl, -t- fl,+, ^0 for / = 1, . . . , m - 1,
or, equivalently if
flO - fll = fll - fl2 = ■ ■ ■ = flm-l - Am-
For 0= i < j < k^ m we deduce, taking arithmetic means, that a
concave sequence (flo, fli, . . . , flm) satisfies
(fl, - flj+i) + . . . -f- (fly_i - fly) ^ (oj - fly+i) -f- . . . -f- (a^_i - a^)
i - i ~ k - j
hence
fl. - fl; < fly - dk
j - i ~ k - j
and thus
{k - j)ai + {i - k)ai + (;■ - !>* ^ 0. F.4.1)
Equality holds if and only if
ar-\ - 2ar + flr+i =0 for r = / + 1, . . . , A: - 1.

334 Inequalities for mixed volumes
The sequence (oq, a^, . . . , a^) of positive numbers is called log-
concave if the sequence (logflo- ■ ■ ■ - logfl„) is concave, or equivalently
if
fl; ^ fli_ifli+i for / = 1, . . ., m - 1.
In this case, for 0^ i < j < k^ m,
af-'a'-''ai~' ^ 1 F.4.2)
or, written with positive exponents,
fl*-' ^ af-'a'r. F.4.3)
Equality holds if and only if
al = Or-xOr+i for r = / + 1, . . . , fc - 1.
If a number m e {I, . . . , n) and convex bodies Kq, K^, K^+x, ■ ■ ■ ,
K„ € X" are given, we often use the abbreviations
(^ := (a:„+i, ...,k„)
and
:= V{Ko[m - i], K,[i], A:„+,, . . . , K„) F.4.4)
for / = 0,. . . , m. This notation extends that of F.2.1) (see the remarks
made there).
If the numbers
fli := ViiKo,K„fd), i = 0,...,m,
are positive, the sequence (aq. • ■ ■ > flm) 'S log-concave by the Aleksand-
rov-Fenchel inequality, hence
v5j' ^ V(V^V'(,-; F.4.5)
iiO^ i < j < k^ m. By approximation, this inequality holds also if one
of the numbers involved is zero. The case m = n, / = 0, j - n - \,
k = n \s Minkowski's inequality F.2.2) again (but without information
on the equality case, if derived in this way).
Again taking m = n in F.4.5), but now taking AT] = 5", we find that
the quermassintegrals W^,. . . ,W„oi the convex body K^e X" satisfy
W)-' ^ W^-'wi'' F.4.6)
for 0^ i < j < k^ n. The case k = n deserves special mention; since
W„ = K„, it reduces to an inequality between only two
quermassintegrals:
k'-'W';-' ^ Wf-' F.4.7)
for 0^j<;<n. Assuming dim ATq = n -/, we have Wj>0. In this

6.4 Consequences and improvements 335
case, equality in F.4.7) holds if and only if Wl = W^-iWr+i for
r = i+l,...,n-l. We shall see later (Section 6.6) that equality in
the inequality W^_i ^ W„-2^n holds only if Kq is a ball. Hence,
equality in F.4.7) (for dim ATq = n - /) characterizes balls. In particular,
among all convex bodies of given (positive) volume, precisely the balls
have smallest yth quermassintegral for / = 1,. . . , n - 1 (case i = 0), and
among all convex bodies of given mean width (a fortiori, of given
diameter) precisely the balls have greatest /th quermassintegral for i = 0,
. . . , n -2 (case j = n - 1).
Next, we derive from the Aleksandrov-Fenchel inequality an
improved version, from which some useful consequences can be drawn. We
assume that n + 1 convex bodies ATq, Ki,. . ., K„e %" are given, and
we now abbreviate the (n - 2)-tuple {K->„ . . . , K^ by &. For i, j = 0, 1,
2 we write
U^j := V(A:„ Kj, m = y{Ki, K;, K,,..., K„). F.4.8)
For A], A2 = 0, the Aleksandrov-Fenchel inequality gives
0^ V{Ki + Aj/Co, K2 + hKo,m^
-ViKi + A./Co, Kt + A./Co, &)V(K2 + h^o, K2 + hKo, 6)
= (f/12 + Aif/02 + A2f/oi + KhVoof
-(f/„ + 2A,f/oi + X\Uoo){U22 + 2A2f/o2 + A|f/oo)
= absolute + linear terms
+ A?(f/g2 - f/oof/22) + Ai(f/oi - f/oof/u)
+ 2A,A2(f/i2f/oo - f/oif/02)
(the higher-degree terms cancel). Letting Ai^<», A2^<» we deduce
that the quadratic term is non-negative. Hence, its discriminant is
non-positive, which gives
(f/oof/12 - f/oif/02)' ^ (f/01 - f/oof/u)(f/o2 - f/oof/22)- F.4.9)
For a first important consequence, we assume that the inequality
Vl, ^ UooUu F.4.10)
holds with equality. Then F.4.9) gives
U00U12 - UoiUcG = 0
for all convex bodies K2- By E.1.18), this implies that the signed
measure
fi-.^ UooS(K„f^, •)- UoiS(Ko,€, •)
satisfies
L
s.-^>cdn = 0

336 Inequalities for mixed volumes
for all K e X". Using Lemma L7.9 we conclude that |/d/i = 0 for all
/ e C{S"~^) and hence that /i = 0. If we now assume that
f/oi > 0, F.4.11)
this implies that the measures S{Ko,&, •) and 5(Ari,6, ■) are
proportional (by F.4.11), none of them is zero). Vice versa, suppose that
SiKfiJi, •) = cxS{Ki,(i, ■) with a>0. Then integration of h{KQ, ■)
and h{Ki, •) respectively, gives f/oo = <xUq\ and f/oi = ^^^u and hence
equality in F.4.10). Thus we have proved:
Theorem 6.4.1. Let K, L, Ky,. . . , K„ e JC" he convex bodies and & =
{Ky, . . . , K„), and assume that
V{K, L,e^)>0. F.4.12)
Then equality holds in the inequality
V{K, L, &)^ ^ V{K, K, <^)V(L, L, <g) F.4.13)
if and only if
S{K,^, •) = aS{Lji, •) F.4.14)
with some a > 0.
Equality F.4.14) may be considered as a kind of generalized Euler-
Lagrange equation for the extremum problem connected with the
equality case in F.4.13).
To derive further consequences of F.4.9), we now assume that
C/o() > 0, f/o, > 0, C/o2 > 0 F.4.15)
and write F.4.9) in the form
(c/ooC/,2 - c/mf/02)' ^ vlxuUi - -~r^h - -~r^
\ t^Ol /V t^02
Taking square roots (the negative one on the left-hand side) and
applying the inequality 4fl& ^ (a -t- b)^, where a, b are the brackets on
the right-hand side, we obtain
2 ,, Tj + ^ = 0. F.4.16)
This improves the Aleksandrov-Fenchel inequality
U\2^UnV22. F.4.17)
If equality holds in F.4.16), then equality must hold in the arithmetic-
geometric mean inequality applied in the proof, hence
—^ = —^. F.4.18)
Uh UI2

6.4 Consequences and improvements 337
Together with F.4.16) this yields equality in F.4.17). Vice versa, if
F.4.17) holds with equality, the quadratic equation
Uiix^ - 2Ui2X + f/22 = 0
has only one real root, hence F.4.16) holds with equality.
We collect the results obtained so far, but change notation (ATj = K,
K2 = L, Ko = M).
Theorem 6.4.2. Let K, L, M, K^,. . . , K„e JC" he convex bodies and
(i = {K-i,, . .. , K„), and suppose that
V{K, M, B) > 0, V{L, M, &) > 0, F.4.19)
V{M, M, &) > 0. F.4.20)
Then
V{K, L, eJ ^ V{K, K, &)V{L, L, 6) F.4.21)
and
V{K, K, &) 2V{K, L, &) ^ V{L, L, e) ^
V{K, M, &f V{K, M, e)V{L, M, ^ V{L, M, (^^ ~ '
F.4.22)
The following assertions are equivalent:
(a) equality in F.4.21);
(b) equality in F.4.22).
The particular advantage of inequality F.4.22) lies in the fact that the
convex body M is at one's disposal. Observe that (b) for one convex
body M implies (a), which in turn implies (b) for all convex bodies M.
For the derivation of inequality F.4.22) and the equivalence of (a)
and (b) we had to assume non-degeneracy in the form F.4.19), F.4.20).
If F.4.19) holds, but not necessarily F.4.20), the inequality F.4.22) is
still true, by approximation. The implication (b) => (a), however, is no
longer valid. We shall now investigate equality in F.4.22) in the case
V(M, M,€) = 0. This will play a role in Section 6.7, where A/ is a
segment.
Let K, L, M, B be as in Theorem 6.4.2, but without assuming
F.4.20). We write AT = C, and L = C2, choose C3 e Xq to be arbitrary
and define
W^l := V(C„ q, &)
q, := V(C„ M, &)
for /, / = 1, 2, 3,
3
Kct := S^mC with x„i > 0
(=1

338 Inequalities for mixed volumes
for a = 1, 2, and
3
3
p„:= V{K^, M,&) = 2^«/9.-
1=1
By F.4.22), applied to K\, Ki instead of K, L, we have
Uu 2C/,2 U22
P\ P1P2 P2
F.4.23)
F.4.24)
We choose real numbers fXj and positive numbers v, and put x^ '■= v,,
xii '■= tVj + fii, where t is so large that X2i >0 (i = 1,2,3). Inserting
F.4.23) into F.4.24), we obtain
(the terms containing t cancel). Choosing V] = 1, /Ui = 0 and letting
V2,V3^0, an inequality results that is quadratic in fi2 and fi^. If now
equality holds in F.4.22), the coefficient of fil is zero. Since the
inequality holds for arbitrary real numbers /i2. /W3> the coefficient of 1X2H3
must vanish, too, which gives
li 93
Denoting this number by fc, we see that k is independent of C3 and that
W'i3'72 - W^239i - ^'73 = 0.
By E.1.18), this can be written in the form
j^.„.,/2(C3, ■)d[q2S{K,(^, •) - <7,S(L,6, ■) - kS{M,&, •)] - 0.
Since this holds for all C3 e IC^, we deduce that
q2S{K, (i. •) - 9i5(L, U.-) = kS(M, &, ■). F.4.25)
Vice versa, suppose that F.4.25) is satisfied. Integrating the support
functions of K and L with this measure, we obtain
^wqi - Wnq\ = ^<7l. ^^1292 - ^12^1 = ^92.
hence
Wuql - 2Wnq,qi + ^2291 = 0-
which is F.4.22) with equality.
Condition F.4.25), which thus has been shown to be equivalent to
equality in F.4.22), can be written in the form

6.4 Consequences and improvements 339
S{q2K,&, ■)=S{q,L^ kM,eJ, ■) if A: ^ 0 ^^ ^
S{qxL,&, ■)= SiqiK + \k\M,&, ■) if A: ^ 0.
We turn to a different class of consequences of the Aleksandrov-
Fenchel inequality, which concern a generalization of the Brunn-
Minkowski theorem.
Theorem 6.4.3 (the general Brunn-Minkowski theorem). Let a number
me{\,...,n) and convex bodies K^, K^, K^+i, ■ ■■ , K„ e X" be
given; define K\ '■= A - k)Ko + A/Cj and
f{X) ■■= V{KAm], K„+u ..., K„yi'" F.4.27)
for 0 = A ^ 1. Then f is a concave function on [0,1].
Proof We have to show that /"(A) = 0. It suffices to prove this for
A = 0: If 0 < A < 1, we put K^ ■= A - t)Kx + tK^ and
h{T) := V{R,[m],K„^u--,K„y^'".
Then /(A + /i) = h(ix/il - A)), hence /"(A) ^ 0 follows from h"{0) ^ 0.
Now
/"(O) = {m- l)vg'">-'[V(o)VB) - V^i)] ^ 0
with V(,) given by F.4.4). We have assumed that V@) > 0; the general
case is then obtained by approximation. ■
For m = n. Theorem 6.4.3 reduces to the Brunn-Minkowski
theorem, and in that case we know that for n-dimensional convex
bodies ATq, A^i the function / is linear only if K^ and Kx are
homothetic. An analogous assertion is not true for the general Brunn-
Minkowski theorem, but some conditions equivalent to linearity of/ can
be formulated and are of interest.
In analogy to F.4.4), that is,
V(,) := V(Ko[m- /],*:,[/],€),
we write
5(,) := S{Kf,[m - 1 - /], A:,[/], g, •) F.4.28)
for / = 0, . . . , m - 1; here g = {K„+i, . . . , K„).
Theorem 6.4.4. Under the assumptions of Theorem 6.4.3 and
v^o) > 0, v'^^) > 0,
the following conditions are equivalent:
(a) the function f is linear;
(b) Vl) = V(,-,)V(,+i) /or / = 1, . . . , m - 1;

340 Inequalities for mixed volumes
(c) V(% = V('SrV(„);
(d) the measures 5@) and 5(m-i) ore proportional.
Proof. Suppose that / is linear; hence
V{K,[m], m = [A - A)V('^)'" + AVj^^]'
for 0 ^ A S 1. By E.1.27), this gives
=0
hence
From
K'J'jd - xr-'x^v,,) = ll'J'jd - x)--rv%-^^""v'l;::^.
V(o = V%-'^l'"V'l:::^ for / = O, . . . , m.
T/m _ T/m-i+lT/i-1
i/"" — y/m-i-l'.ri+l
l^(/ + l)- '^@) V(„)
we deduce (b).
Since the function / is concave, it satisfies /'(O) = /(l) -/(O), with
equality if and only if / is linear. As in the proof of F.2.2), this yields
Vw S V'('S)-V(„), F.4.29)
with equality if and only if / is linear. Thus (a) and (c) are equivalent.
If (b) holds, then Theorem 6.4.1 yields that 5(,_j) and 5(,) are
proportional for / = 1, . . . , m - 1. This implies (d).
Suppose that (d) holds. After a dilatation applied to ATq we may
assume that 5@) = 5(m-i)- Integrating h{Ki, •) with respect to this
measure, we deduce V(\) = V(m) and hence, by F.4.29),
thus V(m) = V(o). Interchanging the roles of K^ and K^, we obtain
V(()) ^ V(„) and thus V^^) = ^@)- This implies equality in F.4.29), thus
(c) holds. ■
By Theorem 6.4.1, equality in the Aleksandrov-Fenchel inequality
F.4.13) implies the equality F.4.14) between two mixed area measures.
In the special case where one of the bodies involved in the (n - 2)-tuple
(:i is a ball, the latter equality in tum implies equality in a set of
lower-dimensional Aleksandrov-Fenchel inequalities for projections.
This opens the way to proofs employing induction with respect to the
dimension. We state the crucial step as a lemma.
As in Section 5.3, we use the abbreviation K" for the projection
K\HuQ, and we denote the (n - l)-dimensional mixed volume by v. If

Notes for Section 6.4 341
6 = (A:,, . . ., Km), we write &" := (K"i, . , ., /C^), and the unit ball
5" in E" is denoted in brief by B.
Lemma6.4.S. Let n ^ 3, let K, L, K4, . . . , K„e X" he convex
bodies and & = (Ki, ... , K„), and suppose that v{K", B",f^") > 0,
i;(L", 5",e")>0. If
S{K, B,e, ■) = S{L, B, &, ■), F.4.30)
then
v(K", L", ey"f = viK", K\ 6")i;(L", L\ €") F.4.31)
forallueS"-K
Proof. By inequality F.4.22), applied in //„o for any u e 5""', we have
i;(A:", a:", 6") 2i;(A:", L", 6") viL", L", (§") ^
i;(A:", 5", e"J i;(A:", B", ^")v{L", B", &") v(L", B", &f
By formula E.3.32), assumption F.4.30) implies
i;(A:", 5",(g") = i;(L", 5", 6"),
hence
i;(A:", a:", (g") - 2y(A:", -L", <&") + i;(L", L", 6") ^ 0. F.4.32)
By E.3.33), this yields
y{K, K, B, &) - 2V{K, L, B, &) + V{L, L, B, &) ^ 0. F.4.33)
Again using F.4.30), from E.1.18) we obtain
V{K, K, B, 6) = V{K, L, B, (i) = V{L, L, B, 6).
Thus F.4.33) holds with equality, which implies equality in F.4.32). By
Theorem 6.4.2, this implies equality F.4.31). ■
Notes for Section 6.4
I. Log-linear inequalities for quermassintegrals. Inequality F.4.6) says that the
quermassintegrals Wq, ..., W„ of a convex body in X" satisfy
iox0^i< j < k% n. Gritzmann A988) investigated systematically all
inequalities of this type. Let 0 g 1 < / < /t g n - 1, a, /3, y 6 R, a^ + /3^ + y^ + 0
and c > 0. Then Gritzmann showed that there is an inequality of the type
W';(,K)W^(,K)Wl{K) g c, F.4.34)
valid for all convex bodies K ^JC!^, if and only if
a ^ 0, a + P + Y ^ 0, a{n - i) + fi{n - j) + y{n - k) = 0.
Furthermore, in each case the constant c can be chosen such that equality
holds for the ball, and all the valid inequalities of type F.4.34) are generated
by the n — 1 special Aleksandrov-Fenchel inequalities
Wi(K)^ S W,_,(A:)W,^,(A:) for i = l, . . . , n - l.

342 Inequalities for mixed volumes
From this result, Gritzmann also derived a characterization of all log-linear
inequalities for the diameter D and two quermassintegrals. Examples are
Bieberbach's inequality F.2.6), Kubota's A925a) inequality
S
(r-
and the inequality
S"-' > K„.iDinV„r-\
which was proved in an entirely different way by Gritzmann, Wills & Wrase
A987).
2. Theorem 6.4.1 appears in Aleksandrov A937b) and Fenchel & lessen A938).
AleksandroVs proof is different. As mentioned before, if the question for the
equality case in F.4.13) is considered as a minimum problem with side
condition, then equality F.4.14) can be considered as a generalized Euler-
Lagrange equation for this problem. Correspondingly, Aleksandrov employed
a variational argument. The method used above goes back to Favard A933a)
and Fenchel A936b).
It appears that the algebraic manipulations leading to inequalities F.4.9),
F.4.16) and finally to Theorem 6.4.2 were, in special cases, first applied by
Favard A930) and slightly extended by Matsumura A932); see also Bonnesen
& Fenchel [8], Section 51. Full use of this method was made by Favard
A933a).
The proof of the equivalence of the equality in F.4.22) (without the
assumption F.4.20)) and the equation F.4.25), as given above, extends an
argument due to Favard A933a).
3. The general Brunn-Minkowski theorem. Theorem 6.4.3 is a corollary of the
Aleksandrov-Fenchel inequalities and consequently appeared in print right
after their proof; see Fenchel A936b) and Aleksandrov A937b). Theorem
6.4.4 is essentially taken from the latter paper.
Stability estimates for some special cases of the general Brunn-Minkowski
theorem were obtained by Diskant A975,1982). We sketch a method for
obtaining more general results.
With the assumptions and notations of Theorem 6.4.3 and of F.4.4), put
r^ = (A:„+,, ..., A:„)and
F,„(/Co, Ku (<, A) := v(/c4m],e) - [A - X)v%^ + Av;/,";]-.
Then Fm(Ko, /Ci,^, A) s 0 for A e [0,1], by Theorem 6.4.3. Now we assume
that
for some A e @,1) and some e e [0, Bq], where go > 0 is a given number. In
the sum
F^/Co. Ku (<, A) = i('")(l - A)"-'A'[V(,) - VjoT"'""^}'^)]
each summand is non-negative, hence
A)"-'A'[V(,, - VfJ']-')/"■ Vf-)] § e
(TV-
for I = 1, ..., w - 1. From the cases i = 1 and i = w - 1 we obtain
where the constant c depends on n, r, R, A, £,). if the bodies Ko, Kx, K„+\,
..., K„ have inradius at least r>0 and circumradius at most R. Under
different assumptions on the bodies K^+i, - .., K„, one can now obtain

6.5 Generalized parallel bodies 343
estimates for the distance between suitable homothetic copies of Ko, Kx in
terms of a function of e (involving constants that depend on n, r, R, A, Cp)-
For example, if K^+i, ■ ■ ■, A^m are balls, one can proceed precisely as in the
proof of Theorem 7.2.6.
The proof of Lemma 6.4.5, which in the form given above appears in
Schneider A985a), extends an argument due to Favard A933a).
Integral geometric proofs. For the special case i= n -2 (and hence
j = n - I) o( inequality F.4.7), that is,
two integral-geometric proofs, due to Santalo and to Blaschke are known for
n = 3. The n-dimensional extensions can be found in Gericke A937).
Non-convex sets- Ohmann A952b, 1954, 1956b) used an extension of Kubota's
integral recursion (formula E.3.27) for / = k) to define quermassintegrals for
non-convex sets, and he was able to extend some of the inequalities F.4.7) to
this general situation.
6.5. Generalized parallel bodies
Some of the deeper investigations of inequalities for mixed volumes and
of the equality cases make essential use of the method of inner parallel
bodies. In this section we study a more general concept and collect some
of the necessary tools and auxiliary results. In particular, we give the
proof of Theorem 6.2.3 and a result (Lemma 6.5.4) that will be needed
in the next section.
Let us assume that a closed subset QCS"~^ of the unit sphere, not
lying in a closed hemisphere, and a positive continuous function
/:5"~' -^ R are given. (Only the values of f on Q will be needed, but
without loss of generality we may assume that / is defined on all of
S"~^.) The closed convex set
K := f] H:,f(,) F.5.1)
is bounded, since Q positively spans E", and it has oas an interior point
since the restriction of f to Q has a positive lower bound. We say that
the convex body K is determined by {Q, f) if F.5.1) holds, and we say
that K is determined by Q \f K is determined by (i2,/) for some
positive continuous function /.
Let K, L e Xq. In Section 3.1, the system {Kp} of parallel bodies of
K relative to L was defined by
" ' [K --pL for-r{K,L)^
for -r(K, L) ^p>0,
where r{K, L) is the inradius of K relative to L. By C.1.16) and
Theorem 1.7.5(b),
^P ~ II '^«,/i(K.«)+pA(£„«)

344 Inequalities for mixed volumes
for -r{K, L) ^ p< 00. Hence, if we arrange by suitable translations that
o e intr(A:, L)L C a:, then for p> -r{K,L) the body Kp is
determined by E""', h,( + phi). Thus the operation F.5.1) is a very general
version of the formation of parallel bodies.
In the following, the set Q is fixed. The functions used to determine a
convex body, together with Q, are always continuous and positive. We
shall need some results on the volume of the bodies determined by Q.
Lemma 6.5.1. If K is determined by (Q, /), then
S„-i{K, S"-\Q) = 0 F.5.2)
and
V„iK) = ^ \jiu)dS„-,iK, u). F.5.3)
Proof. Let x ebd K. Then there exists a vector u e Q such that
X e Huj(u)' since otherwise {x, u) < f(u) for all ueQ and hence
{x, u) + £ < f{u) for all ueQ with some f>0, which would imply
X e int K.
Let i; e S"~\Q and x e K A H(K, v). Then x lies in the support
plane H{K,v) and, by the preceding remark, also in a different support
plane. Thus ;t is a singular point of K. From D.2.24) and Theorem
2.2.4 the assertion F.5.2) follows.
If weQ is such that h{K, w)-^ f{w), then any point xe
K n H{K, w) lies in H{K, w) and in some distinct support plane
Huj(u), hence x is singular. Thus
5„_i(A:, {w e Q\h{K, w) + /(w)}) = 0. F.5.4)
Now we conclude from
V„iK) = - \hiK, u)d5„_iGC, u)
together with F.5.2) and F.5.4) that F.5.3) holds. ■
Lemma 6.5.2. If Kj is determined by {Q,fj) for ; = 0, 1, 2, ... and if
ifj)j€K converges uniformly to fo, then (Ki)j^!>^ converges to Kq.
Proof. It is easy to see that each interior point of Kq is an interior point
of Kj for almost all ;. The assertion then follows with the aid of
Theorem 1.8.7. ■
We write V{f) for the volume of the convex body determined by
(i2,/). If g is a continuous function on S"~^ (not necessarily positive).

6.5 Generalized parallel bodies 345
then f + tg is positive, and hence V{f + tg) is defined, for all
sufficiently small \t\.
Lemma 6.5.3. Let K be determined by {Q, f). If g:5"~'-*IR is a
continuous function, then
^ Vif + tg)
1=0
= j^g{u)dS„-,{K, u).
Proof. For sufficiently small |f|, let K, be the convex body determined
by {Q,f+tg), and define Vi{t) ■= V{K,[j], K[n - j]). In particular,
V(f)=V„{K)=Vo{t)=-Vo.
From h(Kt, u) ^ f(u) + tg{u) forueQ and F.5.2) we have
/„[/(«) + tg{u)]dS„-r{K, u) ^ l„_M^„ u)dS„-,(K, u).
Together with F.5.3) this gives
f^tgiu)dS„-i{K, u)^ nV^it) - nVo.
Considering first positive t only, we deduce
g(u)dS„-,(K, u)^n limsup -^ -. F.5.5)
Again by F.5.3),
/„[/(«) + tgiu)]dS„-,{K„ u) = nV„it).
From h(K, u) = f(u) for u e Q and F.5.2) we have
fjiu)dS„-t(K., u) ^ {^.,MA^. u)dS„-i(K„ u).
Subtraction gives
j^^tg{u)dS„-r{K., u) ^ nV„{t) - nV„_,{t).
From K, —> K tor t —* 0, the weak continuity of 5„-i, and F.5.2) (for K
and K,) we obtain
g{u)dS„-i{K, u)^ n liminf -^'^ ^^-^^. F.5.6)
J^ i->o+ t
Using Minkowski's inequality F.2.2), we obtain
[v,@ - Vo] S[v,@/Vo]* = [^,@" - vsvvr'
k=0
^ [vr'v„{t) - vsyvs-' = v„it) - Vo.

346 Inequalities for mixed volumes
Since K,^ K for t-^0,we deduce
n limsup £ limsup . F.5.7)
f-.0+ t i-'0+ t
In an analogous way, one finds
n liminf ^ liminf . F.5.8)
(^0+ t ,_0+ t
Applying successively F.5.5), F.5.6), F.5.8), liminf ^ limsup and
F.5.7) we conclude that in all these inequalities the equality sign is
valid, hence
lim
Vnit) - Vo
= J^^g(u)d5„_,(A:, u).
f-«0+ t
If in this relation we replace f by -f and g by -g, we obtain
Vnit) - Vn
Both relations together prove the lemma.
lin, _JiW^ 1 ^ („)d5„-,(/C, u).
We are now in a position to present the proof of Theorem 6.2.3,
which was postponed.
Proof of Theorem 6.2.3. By assumption, K, L e .Tq ^re n-dimensional
convex bodies, and QC 5""' is a closed subset such that
K = nw-(A:, u).
Then Q is not contained in a closed hemisphere, since K is bounded,
and with the terminology introduced above we see that K is determined
by Q. The body L is defined by
L := f]H-{L,u).
The relative inradius r(K, L) of K relative to L will be abbreviated by
r. Without loss of generality, we may assume that o e int rL C AT.
If ;t e rL, then {x, u) = h{rL, u) ^ h{K, u) for u e O and hence
X e K. Thus rlc K and r{K, L) = r.
For -r<A^O, let K^ be the convex body determined by
{Q, fiK + Xhi). By Lemma 6.5.3,
^^ = J^/2(L,u)d5„_,(A:,, u), F.5.9)
and by F.5.2),

6.5 Generalized parallel bodies 347
j^h{L, u)dS„-,{K,, u) = l„.M^, u)d5„-,(A:,, u)
= nV,{K„ L). F.5.10)
We assert that
lim V„{K,) = 0. F.5.11)
A—•—r
For the proof, let A .= {u e Q\h{K, u) = h{rL, u)}. Then o e conv>l,
since otherwise one could construct a homothet of L contained in K and
larger than rL. There is a finite subset A'<Z A such that oeconv>l'
and hence a finite subset Q' <Z Q such that
is a poly tope containing no homothet of L larger than rL. If Px is
determined by (Q', h^ + ?.hL), then AT;. C P^ for -r< A^O, and it is
easy to see that V„(Pa)^ 0 for A^ -r. This proves F.5.11).
From F.5.9), F.5.10), F.5.11) we now obtain
V„{K) = nj_V,{Kx,L)dL F.5.12)
By the general Brunn-Minkowski theorem, 6.4.3,
Vi(K,. + \k\L, L)i/<"-i) S V,{Kx, lyl^"-^^ + |A|V„(L)i/<"-i).
From a:;. + |A|L C A: we have V^{Kx + |A|L, L) ^ V,(A:, L) and hence
Vi{K},, lyl^"-^^ g V,(A:, Lfl^"-^) + AV„(L)'/<"-'>.
Inserting this into F.5.12), we obtain
f" -
V„{K) ^ n j_^[V,iK, L)iA"-i) + AV„(L)i/<"-'>]''-'dA.
Since Vi(A:, L) = Vi(K, L) by F.5.2), we arrive at
v„{K)v„{Lyi("-'^
^ Vi(K, £)"<'■-'> - [Vt(K, £)'/<"-'> - rV„(L)'/<"-'>']",
which is the assertion of Theorem 6.2.3. ■
As we have seen, and shall further see in the proof of Theorem
6.6.18, differentiability assertions with respect to the parameter of a
system of inner parallel bodies play an essential role. Another result of
that type is needed in the proof of Theorem 6.6.7 and will now be
proved.
Let A, C e XS be convex bodies with interior points. We say that A
is adapted to C if to each point ;if e bd C there is a point y ebdA such
hat the normal cones at these points satisfy N{C, x) C N(A, y). For

348 Inequalities for mixed volumes
example, it follows from Theorem 2.2.1 that A is adapted to C if >1 is a
summand of C, but this is only a very special case.
In Section 3.1, the Minkowski difference of the two convex bodies C
and A was defined by
C - A = f]{C - a) = {x e l"\A + X C C}.
aeA
For 0 ^ T S r(C, A) (the inradius of C relative to A) we define
C, := (C ~ tA) + tA
and
/2„(t) := h{Cr, u) for u e S""'.
Lemma 6.5.4. If A is adapted to C, then h'u{0) = 0 for each u e S"~^.
Proof. Let u e S"~^ and 0< t< r{C, A) be given. We can choose
z eE" such that tA + z <^ C and {z, u) is maximal. Let
C/r •■= {t> e 5''-'|/2(C, u) = h(TA + z, v)}.
We assert that
ueposUr- F.5.13)
Assume this is false. Then u and the closed convex cone posU^, which
is pointed since t< r{C, A), can be separated strongly by a hyperplane.
Hence, there is a vector w e E" such that (w, u) > 0 and (w, u) < 0 for
each i; e t/r\{o}. On the closed hemisphere S"~^ n Ht.o, the
continuous function h{C, ■) - h(TA + z, •) is positive and hence attains a
positive minimum f. Then
h{C, v) - h{TA + z, v) ^ f(w, v)
for all i; e S"~' and hence r^ + z + f w C C Since {z + £w, u) >
{z,u), this contradicts the choice of z. Hence, the assumption was
false, and F.5.13) holds.
By F.5.13) and Caratheodory's theorem, we can choose (not
necessarily distinct) unit vectors vi, . . . , v„ e U^ and numbers Ai,. . . , A„ ^ 0
such that
n
1=1
For ;if e C we have
{x, Vi) ^ h{C, i>i) = h{TA + z, Vi),
hence
{x, u) g ZkMrA + z, Vi).
i=l

6.5 Generalized parallel bodies 349
Thus
n
h(C, u)^ 2^MrA + z,Vi).
(=1
By C.1.12) we have h^{T)^h^{0). Further, tA + z C C, hence
z e C ~~ tA and thus tA + z C C^, from which we infer that
^i,X^) - ^'ui'^) = h{C, u) - h{xA + z, u)
n
= "Z^MtA + z, Vi) - h{xA + z, u)
1=1
• n
= X XA,/2(>1, Vi) - h{A, u) .
We deduce that
K{^) - hM) "
0 ^ -Jii-^ "±J- ^ h(A, u) ~ JiXMA, Vi). F.5.14)
T 1 = 1
Now we let r tend to zero. The vectors u, and numbers A, chosen
above depend on r. For each re @, r{C, A)) we make a definite choice
of unit vectors Wi(r), . . . , j;„(t) and corresponding numbers Ai(r), . . . ,
A„(t). We assert that
n
lim 'Z^i{T)h{A, Vi(T)) = h{A, u). F.5.15)
r^O i=l
Suppose this were false. Then there exists a number a>0 and a
sequence (Tj)j^m such that ty ^ 0 for /-♦ <» and
n
h(A, u) - 'Z^iiTj)h(A, Vi{T^)) ^ -a F.5.16)
1=1
for / e N. For each Tj we choose a vector Zj such that TjA + Zj <^ C and
{Zj, u) is maximal. After selecting subsequences and changing the
notation we may assume that
lim Zj = z, lim i;,(Ty) = u, for i = 1, . . . , n,
and also that the compact sets U^ converge to a set U. Since pos U is
pointed, there is a unit vector ft and a number ^ > 0 such that
{h,v) = ^foTveUr and all sufficiently large ;'. From
n
{b,u)^ 2A,.(r;)(fo, v,iTj))
1=1
it follows that A,(r^) e @, (ft, u)/fi] for large /. Hence, we may also
assume that
limA,(Ty) = A, for I = 1, . . . , n.
\ssuming further that o € >1, we have Z/ e TjA + Zj, and we deduce that

350 Inequalities for mixed volumes
z e FiC u). u e N{C, z),
i>i e N(C, z) for J = 1, ... n,
n
u = 2^A,i;,-.
1=1
Since A is adapted to C, there is a point y e bd>l for which N{C, z) C
A'(>1, >■). Thus u e N{A, y), u, e NC/l, y) for j = 1,. . . , n, and
n n
h{A, u) - y^XMA, V.) = (y, u) - EA,()', (^> = 0.
.=1 (=1
But F.5.16) yields
n
h{A, u) - ^Xih{A, Vi) < 0.
This contradiction proves F.5.15), which together with F.5.14)
completes the proof. ■
Notes for Section 6.5
Lemmas 6.5.1-6.5.3 and their proofs are taken from Aleksandrov A938a).
Lemma 6.5.4. with essentially the same proof, appears in Schneider A990a).
The proof of Theorem 6.2.3 given here is due to Diskant A973b).
Inner parallel bodies. In the geometry of convex bodies, there are many more
applications of inner parallel bodies. Here we give a list of references
containing such applications: Bol A941, 1942, 1943a, b), Chakerian A972,1973),
Chakerian & Sangwine-Yager A979), Czipszer A962), Dinghas A943a), Diskant
A973b.c), Farv A983), Hadwiger [17], [18], Matheron A978a,b), Oshio
A955.1958,1962), Rcnyi A946), Sangwine-Yager A980.1983, 1988a, b),
Schneider A989a, 1990a)'and B. v. Sz.-Nagy A940). The application by F^ry
A986) contains a serious error (there are counterexamples to the higher-
order differentiability properties stated in Theorem 4).
H-convex sets. The notion of a convex body determined by Q can be
generalized: Let H C S"'^ be an arbitrary subset. Any halfspace W,7.a with
u e H and a e R is called W-convcx, and the intersection of any family of
W-convex halfspaces is called an //-convex set. A thorough study, mainly
from the combinatorial viewpoint, of the family of //-convex sets, for given
H, was made in the book by Boltjanski & Soltan [6].
Generalized outer parallel bodies and Wulff shape. For K e H^ and a given
continuous non-negative function g on 5" ', let F,K be the convex body
determined by E"', h^ + tg) for t ^ 0, thus
F,K = n W-,
hiK.u)*l!iUi)-
The map F: ICC) x l<^, x)^ Xfi defined in this way was studied by Willson
A980). In particular, he proved the semigroup property F,F,^r = F,+,/C and
described applications to the Wulff shape in the theory of the growth of
physical crystals. For an application of the Brunn-Minkowski theory to the
Wulff shape, see also Dinghas A944).

6.6 Equality cases and stability 351
6.6. Equality cases and stability
The Aleksandrov-Fenchel inequality
v(a:„ a:^, ATj,..., a:„J
^ V{K„ Ku K,,..., K„)V(K2, K2, K,, . . . , K„)
holds with equality if K^ and Kj are homothetic. For special choices of
Ki,. . . , K„ this is the only case of equality (certainly for n = 2, by
Theorem 6.2.1). In the following, we shall first study these special cases,
and the corresponding stability results and their applications. The
second part of the present section is devoted to the (still incomplete)
investigation of the cases of equality in general.
The classical result in this connection states that equality implies
homothety of K^ and K2 if K^, . . . , Ar„ are balls. We give two proofs
for this result and start with a lemma.
Lemma6.6.1. Let n ^ 3; let K, L e Ji" be convex bodies such that their
projections K" and L" are translates, for each u e S"~^. Then K and L
are translates.
Proof. For u e S" 'we define
G{u) ■■= {z e E"\K" = {L + z)"};
then G{u) is a line of direction u. Let u, v e S"~^ be linearly
independent. Let z e G{u), thus K" = (L + z)". Since (L + z)" is a translate of
L^ there is a vector y e E" with K" = {L + z + y)". For all w e 5""'
with w±u and w±v we have h(K, w) = h{L + z, w), because
K" = {L + z)" and w 1 u, and h{K, w) = h{L + z + y, w), because
K" = (L + z + y)" and w 1 v; hence {y,w) = 0. Since this holds for
all w with w ± u, i; we infer that y = ku + 1.1V with X, /ieU. The vector
X ■= z + y - IJV satisfies x e G{u), further K" = (L + ;if)'' and hence
X e G{i>). Thus we have proved that G{u) n G{v) ^ 0 for linearly
independent u, v.
Now let Ui,U2 e S"~^ be linearly independent; then G(ui) n
G{u2) = {t} for some t. If i; e 5""' is linearly independent of Ui and
U2, the line G(i;), which has to meet G(ui) and G(u2), necessarily
passes through t. Hence, K" = {L + t)" and thus h{K, w) =
h{L + t, w) for all w ± (;. The set of all w for which this holds is dense
in 5""', hence a: = L + f. ■
In the following, the unit ball B" is simply denoted by B.

352 Inequalities for mixed volumes
Theorem 6.6.2. If K, L e ^o ^f"^ convex bodies for which equality holds
in
ViK, L,B,...,Bf^ V{K, K, B,..., B)V{L, L, B, ..., B),
F.6.1)
then K and L are homothetic.
Proof. We use induction with respect to the dimension. For n -2, the
result is contained in Theorem 6.2.1. Suppose that nS3 and the
assertion is true in smaller dimensions. If F.6.1) holds with equality,
then by Theorem 6.4.1 the measures S{K/B, •) and S{L,li, ■) are
proportional, where H stands for the (n - 2)-tuple E, . . . , 5). Without
loss of generality, we may assume that
S{K,"B, ■) = S{L/B, •). F.6.2)
Then Lemma 6.4.5 yields
v{K\ L", Oi"[n - 3]f = v{K", K\ ''B"[n - 3])i;(L", L\ ^"[n - 3))
for u e S"~^. By the induction hypothesis, K" and L" are homothetic.
From F.6.2) and E.3.32) we get i;(A:", B") = v{L\ B"), hence K" and
L" are, in fact, translates. By Lemma 6.6.1, K and L are translates. ■
The second proof we shall give for Theorem 6.6.2 yields more,
namely a stability version. It depends upon the following analytic
inequality, which is closely related to a result known as Wirtinger's
lemma (or the Poincar6 inequality).
Lemma 6.6.3. Let f he a real function of class C^ on S"~^ satisfying
f/da = 0 F.6.3)
and
' f{u)uda{u) = o, F.6.4)
/.
where a denotes spherical Lebesgue measure and the integrations extend
overS"'^ Then
j f(f + -^ 4s/) da + ^4y S/' da ^ 0. F.6.5)
Proof. We use spherical harmonics (see the Appendix for details). Let
^m '■= ^mf (defined by A.3) for m = 0, 1, 2, ... , then Xo = 0 and
ATi = 0 by F.6.3) and F.6.4); hence the Fourier series of / with respect
to the system of spherical harmonics is
00
m = 2

6.6 Equality cases and stability 353
By Green's formula on the sphere 5""' and by (A.2),
Ds/, X„,) = (/, AsX„) = -m{m + n- 2)(/, X^),
hence
^s/ ' S '"('" + n - 2)X„.
m=2
The Parseval relation (A .5) gives
(fda= i ixido,
ffAsfda= -j;,m{m + n - 2) f Xida.
This yields
1 °° f
= 2 [2« - "i("i + n - 2)] xida ^ 0. ■
1-1 m=2 •'
If the preceding result is applied to differences of support functions,
one obtains estimates for the deviation of convex bodies; however, these
are not immediately in terms of the Hausdorff metric 6 but in terms of
the L2 metric 62- This is defined by
d2iK,L)' ■■= f^Jh^- hj^l'da. F.6.6)
Clearly 62= v (o„ S. Estimates in the opposite direction are given in
the following two lemmas, which we state without proof.
Lemma6.6.4. For K, L e X",
62(K, LJ S a„D(K U Ly-(K, L)"+'
with
o)„B{i, n - \)
a„ =
4^)'
where B(-, •) is the beta integral.
For a convex body K, we introduce its Steiner ball B(K) as the ball
that has the same Steiner point and mean width as K, thus
B(K) := B(s(K), |fe(A:)). F.6.7)

354 Inequalities for mixed volumes
Lemma 6.6.5. If K e X" and B{K) is the Steiner ball of K, then
I ^ \(n-l)/2
with B„ = ynCWn/ZATn.i), w/jcre
3 16(c + 2)<" - 2>/2
(n^ - l)(n + 3) l7r^n(n + 2J" (c + 1)
n-2
Lemma 6.6.4 was proved (in a more general form) by Vitale A985a)
and Lemma 6.6.5 by Groemer & Schneider A991). For the special case
of the deviation from a suitable ball. Lemma 6.6.5 gives a better
estimate, since the exponent (n + 3)/2 is smaller than the exponent
n + 1 of 6{K, L) in Lemma 6.6.4. (It is, of course, this exponent that is
essential for the strength of the estimate, and not the factor.)
When stability estimates are obtained in terms of the Li metric, we
shall refrain from translating them into Hausdorff metric estimates, but
the reader should keep in mind that this is always possible as a result of
Lemmas 6.6.4 and 6.6.5.
To deduce such a stability estimate from Lemma 6.6.3, we associate,
with each convex body K ^IC" oi dimension at least one, its
normalized homothetic copy K defined by
- . K - s{K)
'^■" hiK) ■
Let K, L e IC" be convex bodies of dimension at least one, and put
V,y := V{K[i], L[j], B[n - i - ;]).
Then equality F.6.1) reads
Vii - V20V02 ^ 0.
The following theorem contains a strengthened version of this
inequality.
Theorem 6.6.6. For convex bodies K, L e X" of dimension ^1,
Vli - V20V02 ^ „^„\\^ h{K)'Vo2^{K, L)'. F.6.8)
Proof First we assume that K and L are of class C+. The function
f-=h{K,-)-h{L,) satisfies F.6.3) and F.6.4), by (A.6) and
(A.7). Hence, Lemma 6.6.3 yields

6.6 Equality cases and stability 355
By formula E.3.17),
= V(K, K, B[n - 2]) - 2V{K, L, B[n - 2]) + V(L, L, B[n - 2])
V20 2V„ , V02
+
b{Kf h{K)h{L) b{L)^'
Applying the identity
h^ - ac = -c{a - 2/) + c) + (ft - c)^
with
„ _ i>{L) b{K)
we arrive at the inequality
Vu - V20V02 ^ ^l^\\) HK)'Vo262(K, If + (vn - ^ V02) ,
of which F.6.8) is a weaker version. By approximation, this inequality is
now extended to general convex bodies K and L. ■
The special case of F.6.8) where L is a ball can be written in a more
convenient form, using the Steiner ball of K. Since
d2{K, Bf = h{K)-^62{K, B(K))\
as a corollary of Theorem 6.6.6 we obtain, for the quermassintegrals
W„_j, W„-2 of the convex body K e X", the inequality
-^ wti - w„^2 s ~7-~r: ^(^. B(K))'. F.6.9)
From this, we can deduce a stability result for the more general
inequality F.4.7) between two quermassintegrals of convex bodies. Let
KeJCo and write Wk = Wk{Ky, let 0^ i<}< n. Using the special
Aleksandrov-Fenchel inequality Wl ^ W^/t-i W^/t+i, we obtain
Wj-'-'W^ \W,Wj^2J \Wj^iWj^3l \W„.2W„
and
— ^ 1
W,W'jl\ '
the latter from F4.6). This gives

356 Inequalities for mixed volumes
hence
w"-'
w
and thus
r''\w„-2i
Using the identity
.k _ rk k-l
^= -^a'-'-^h^ =:s,{a,h)
a - b r=o
with k = j - i, a = wI_i/k„, b = W„-2 and then F.6.9), we arrive at
ic'^'W]-- - Wr' S Yn.iAK)d2(K, B{K)f F.6.10)
with
By approximation, inequality F.6.10) can be extended to convex bodies
K of dimension ^2.
The case / = 0, ; = 1 of inequality F.6.10) is a strengthened version
of the isoperimetric inequality, namely
(A)- . (M"-' a -^l±l^ Yf. ^(^. swf. F.6.„,
\a)„) \kJ n(n - l)icr' W^n-2
Up to now, the considerations of this section centred around the fact
that equality in the Aleksandrov-Fenchel inequality
V(A:, L, C„ . . . , C„_2)^
^ V{K, K,Ci,..-, C„_2)V(L, L, Ci, . .. , C„_2), F.6.12)
under the special assumption Cj = ... = C„_2 = B" (unit ball), holds
only if a: and L are homothetic. Using this result, we shall now show
that the same conclusion can be drawn if we assume only that Ci, . . . ,
C„_2 are smooth. We first prove a more general result, which states that
equality in F.6.12) is preserved under replacement of the (n - 2)-tuple
(Ci,. . . , C„-2) by any other (n - 2)-tuple (Ai,. . . , A„.2) of convex
bodies such that Aj is adapted to C,, in the sense defined in Section 6.5.
Theorem6.6.7. Let C], . . . , C„_2 e Xq be n-dimensional convex bodies,
je; <& = (Ci, . . . , C„-2) (ind let K, L e JC" be convex bodies satisfying
V(K, L, (gJ = ViK, K, f^)V(L, L, &). F.6.13)
// c4 = {Ai,. . . , A„_{) where /I, e .l^S is adapted to Ci for
/ = 1, . . . , n - 2, then also
V{K, L, c^J = V{K, K, <4)V{L, L, c4). F.6.14)

+ -v-'-'^'--^ go F.6.17)
6.6 Equality cases and stability 357
Proof. Let n ^ 3, without loss of generality. If dim AT = 0 or dim L = 0,
then K and L are homothetic. If, say, dim AT = 1, then V{K, K,(i) = 0,
hence V{K, L,€)-=0. Since Cj, . . • , C„_2 are n-dimensional, this is
only possible if either dim L = 0 or L is a segment parallel to K. Thus
K and L are homothetic, and F.6.14) holds. We may, therefore,
assume in the following that dim K^2 and dim L = 2.
We write
Cy=- C, A^= A and (C2, . . . , C„-2) = &';
thus
V{K, L, C, €'f = V(K, K, C, (d')V{L, L, C, &). F.6.15)
By Theorem 6.4.1 we may assume, after a dilatation of K, that
S{K, C, &, •) = 5(L, C, 6', •). F.6.16)
Let B e ^" be a convex body with dim B = 1- By F.4.22), with (§
replaced by (B.6') and A/ replaced by C, we have
v(a:, a:, q, &) 2V(a:, l, e. ^')
V{K, Q, C, &f V{K, Q, C, &)V{L, Q, C, &)
V{L,L,Q,&)
V{L, Q, C, &f
By F.6.16) and E.1.18),
V{K, Q, C, &) = V{L, Q, C, 6'), F.6.18)
hence
V{K, K, Q, e') - 2V{K, L, Q, &) + V{L, L, Q, &) ^ 0. F.6.19)
Again by F.6.16) and E.1.18),
V{K, K, C, e') - 2V(K, L, C, &) + V{L, L, C, &) = 0. F.6.20)
We define
0(Q) := V{K, K, Q, &) - 2V{K, L, Q, &) + V{L, L, Q, &)
for QeX" and
g{x) := 0(C,)
for 0<T<r{C,A), with Cj := {C - tA)-k- xA as in Section 6.5;
further
fix) := 0(C - xA).
Then
fix) ^ 0 F.6.21)
and
<P{A) g 0 F.6.22)

358 Inequalities for mixed volumes
by F.6.19). By the linearity of the mixed volume,
g{T) = fit) + T<P{A). F.6.23)
We assert that
g'{0) = 0. F.6.24)
For the proof, let
(fir) := V(C„ i),
where ^ is an arbitrary (n - l)-tuple of convex bodies. Then, by
E.1.18) and with h^{T) := h{C„ u),
<fir) - (fjO) ^ 1_ r h,{T) - h,{0)
T n jS"-' t
We show that the integrand remains bounded for r^O. Choose a
positive homothet XA + z contained in C. For given u e 5"~', choose
X e F{C, u). For 0 < t< A, let tA + Zt be obtained from kA + z by
dilatation with centre x. Then tA + Zt^ C and hence h{Cj,u)^
h(TA + Zz, ")■ Writing
h(C, u) - h(TA + Zr, u)=: a,
h{C, u) - h{kA + z, u)=: /3,
we have a//3 = r/A and hence
0 ^ h^@) - h^(T) = h(C, u) - h{Cr, u) § a = /3t/A ^ yr
where the constant y is independent of u. This proves that the integrand
remains bounded, and we can apply the bounded convergence theorem
together with Lemma 6.5.4 to deduce that cp'{0) = 0. Since & was
arbitrary, this proves F.6.24).
From F.6.23) and F.6.24) we get /'(O) + ^A) = 0, hence /'(O) ^ 0
by F.6.22). On the other hand, /(O) = 0 by F.6.20) and /(t)^0 by
F.6.21). This yields /'(O) = 0 and thus ^{A) = 0, hence
V{K, K, A, &) - 2V{K, L, A, &) + V(L, L, A, &) = 0.
This, together with F.6.18) for Q = A, shows that there is equality in
F.6.17) for Q = A.By Theorem 6.4.2, this implies
V{K, L, A, &f = V{K, K, A, ^')V{L, L, A, &')
or, returning to the former notation,
V{K, L,A^, C2, ..., C„_2J
= V{K, K, Ai, C2, . . . , C„-2)V(L, L, Au C2, . . . , C„_2).
This equality has been deduced under the assumption that F.6.13) holds
and that A\ is adapted to d. In a similar way, each of the bodies C2,

6.6 Equality cases and stability 359
. . . , C„_2 can be replaced by another one adapted to it. This proves the
theorem. ■
Now we can prove
Theorem 6.6.8. If equality holds in
V{K,L, C„...,C„_2J
^ V{K, K,Cu.-., C„_2)V(L, L, Ci, .. . , C„_2), F.6.25)
where Ci,...,C„_2 are smooth convex bodies, then K and L are
homothetic.
Proof. If C, is smooth, then the unit ball B" is adapted to C,; hence by
Theorem 6.6.7 equality in F.6.25) implies equality in F.6.1). By
Theorem 6.6.6, K and L are homothetic. ■
We are now in a position to complete the results on the general
Brunn-Minkowski theorem that are collected in Theorem 6.4.4, if we
introduce smoothness conditions.
Theorem6.6.9. Let a number m e {2,. . . , n} and an (n - m)-tuple
€ = {Km+\, • • • . Kn) of smooth convex bodies be given and let Kq,
Ki e J^" be convex bodies of dimension ^m. Then the conditions
(a)-(d) of Theorem 6.4.4 hold only if K^ and K\ are homothetic.
Proof. We use the notation introduced before Theorem 6.4.4. Assume
that the equivalent conditions (a)-(d) of that theorem are satisfied.
Since (b) holds, that is,
V\k) = V(/t-i)V(/t+i) for A: = 1, . . . , m - 1, F.6.26)
then by Theorem 6.4.1 the measures 5^^) and S^^k-\) are proportional.
The proportionality factor is obtained by integrating h{Ko, •), and we
obtain
^ (k-l)S(k) = y (k)S(k-\)-
Integrating the support function of the unit ball B" with this measure,
we obtain
V(,_,)V(A:o[m - 1 - A:], K,[k], 5",^-)
= V(,)V(A:o[m - kl Kilk - 1], B",6^).
If we define
Vik) ■■= V{Ko[m - 1~ k], Ki[k], B\&),

360 Inequalities for mixed volumes
this reads
V(*-i)V(/t) = V(A)V'(/t-i) for fc = 1, . . . , m - 1,
hence
V{k)V^k+i) = V{k+i)y{k) tor k = 0, . . . , m - 2.
Together with F.6.26), this yields
Vlk) = V^k-i)V^k+i) for k = I, . . ., m - 2.
Thus we are again in the situation described by F.6.26), but with m
replaced by m - 1 and ^ replaced by {B",&). Repeating the procedure,
we end up with the equality
V{Ko.K„B"[m -21,6J
= V(A:o, /Co, B"[m - 2], ^)V{K„ A:„ B"[m - 2], 6).
By Theorem 6.6.8, this implies that Ko and Ky are homothetic. ■
Theorem 6.6.8 generalizes Theorem 6.6.2, in which the bodies Cj,
. . . , C„_2 were assumed to be balls. For the latter result. Theorem
6.6.6 gives a strengthened version in the form of a stability result. We
shall now investigate the possibilities of improving Theorem 6.6.8 in a
similar way. For convex bodies K, L, Ci,. . . , C„_2 e J(" we introduce
the deficit A by
A(K, L,Ct,-., C„_2)
:= V{K, L,Ci,..., C„_2J - V{K, AT, C„ . . . , C„_2)
X V{L, L,Ci,..., C„-2),
so that the Aleksandrov-Fenchel inequality can be written as
AiK, L, Ci, . . . , C„_2) S 0. F.6.27)
Inequality F.6.8) tells us that
A(K, L,B", ..., B") ^ " ^^ a{K, L) 62{K, L)^ F.6.28)
(if dim K, L^l), where
a{K, L)
:= mAx{h{KfV{L, L, B"[n - 2]), h(LfV(K, K, B"[n - 2])}.
For general convex bodies C\,. . . , C„_2 it is possible (as we shall see
later in this section) that equality holds in F.6.27) without K and L
being homothetic. This implies that Theorem 6.6.8 cannot be improved
to a stability version of the type F.6.28) that would hold for all bodies
Ci,...,C„_2 in a dense subclass of X". However, one can show
stability for F.6.27) under the assumption that the convex bodies Ci,
..., C„-2 are p-smooth, for some fixed p>0. The convex body

6.6 Equality cases and stability 361
K e J(q is called p-smooth if pB" is a summand of K, equivalently (by
Theorem 3.2.2), if to each boundary point x of K there is a ball
pB" + t of radius p such that x e pB" + tC K.
For given numbers r, R>0, we denote by X"(r, R) the set of all
convex bodies in X" that contain some ball of radius r and are
contained in some ball of radius R.
Theorem6.6.10. Let p, r, R>0 and an integer p e {I,. . ., n - 2} be
given. If K, L, Ci,. . . , Cp e X"(r, R) and if Ci,. . . , Cp are p-smooth,
then
A{K, L, Ci, . . . , Cp, B , . . . , B")
^ YP*^^''-^>A{K, L,B",...,B")^\ F.6.29)
where the constant y depends only on n, p, r, R.
Together with F.6.28), the inequality F.6.29), with p = n ~2, yields
a stability estimate for the Aleksandrov-Fenchel inequality F.6.27),
provided that Ci, . . . , C„_2 are /^-smooth. It is inevitable that the
factor yA on the right-hand side of an inequality of the type F.6.29)
tends to zero if p tends to zero and hence Theorem 6.6.8 is not a
corollary of Theorem 6.6.10.
The gist of the proof of Theorem 6.6.10 is the following lemma.
Lemma6.6.11. Let n ^ 3. Let K, L, M e X"{r, R), let K4, ■ ■ ■ ,
K„ e JC" be arbitrary convex bodies and set {Ki,. . . , K„) = <^. // the
convex body M' is a summand of M, then
A{K, L, M', &) ^ aVA(K, L, A/, (g) F.6.30)
with
R^V{B"[3],e).
-m-{^.
The lemma shows, in particular, that A(K, L, M,&) = 0 implies
A{K, L, M',&) = 0 if M' is a summand of M. Since a summand M'
of M is adapted to M, this is simply a special case of Theorem 6.6.7.
This special case, however, admits a stability version in the form of
Lemma 6.6.11.
Proof of Lemma 6.6.n. For convex bodies A, B, C, D e X", we use
the abbreviations
ViA, B, C) := V{A, B, C, K^,..., K„),
A(A, B, C) := A(A, B, C, K^, . . . , Kn),

362
Inequalities for mixed volumes
F{A, B, C; D)
VjA, A, C)
2V{A, B, C)
V{B, B, C)
V(A,D,C)^ ViA, D, C)V(B, D, C) V(B, D, Cf
provided that the denominators are positive.
First we assume that the bodies K4, . . . , K„ appearing in the lemma
have interior points. Given Ko, K^, K2, K3 e Xq, we use the notation
Uij := V(Ki, Kj, K3) for /, ;■ e @, 1, 2}.
A{Ku K2, Ki)= U\2- UnUn,
F '■- F{K\, K2, Ki, Kq) —
Ui
2U
12
U
22
f/01
V01U02
Then A^O, and by F.4.16) we have F ^ 0. By F.4.9),
f/02'
(f/oof/,2 - f/oif/02)' ^ (f/oi - f/oof/ll)(t/o2 - U00U22).
Rearranging the terms, we deduce that
A s —— F.
U,
F.6.31)
00
From the identity b^ - ac = -c{a - 2b + c) + (b - c)^ with
a =
U,
UoiUfa'
c =
t/22
we obtain
A _ i^ p ^ / t/12 _ U21Y (f. f. -xo^
Uiul2 UI2' \UorUo2 UI2I' ^'-'-'^
and in particular
A ^ Ul,UnF.
Inequality F.6.32) together with F^O yields
F.6.33)
U
12
U22
U01U02
f/02
Multiplying respectively by 1/02/1/12 and by L/oyUcfi/U22, we obtain
U
22
f/,
02
u
12
f/oi
f/02
and
U
12
u,
01
u
22
f/02
f/0lf/l2
VT
t/
F.6.34)
F.6.35)
22
For convex bodies A, B e JC^ we define
V(L, M, B)
a{B) :=
v(a:, m, B)

6.6 Equality cases and stability 363
and
fi{A, B) := V{K, K, A)a{B) - 2V{K, L, A) + V(L, L, A)a(B)-K
Applying inequality F.6.34) with Ki = K, K2 = L, Kj = M, we obtain
\rv(i^ ..■(i^^\^ nKo,L,M) Va{K,L,M)
l^^"-^ - ^^''^^l - ViK„ K, M) ViK, L, M) ' (^■^■'^^
and inequality F.6.35) yields
V{L, L, M)
Applying F.6.33) to Ko = L, Ki= K, K2 = L and K^ = M, we obtain
«"■ ^> =' - l^lxw- <^"«
Now, for A e XS,
\p(A, A) - p(A, L)\
^ ViK K, A)\a{A) - (x{L)\ + V{L, L, A)\aiAy' - a(Ly'\,
hence F.6.36) and F.6.37) yield
\fi{A, A) - piA, L)\ ^ (fiA)V^{K, L, M) F.6.39)
with
"^""^ ■= V(K, L, M)V(K, M, A) ^ V(L, L, M) ' ^^■^'^^
By assumption, M' is a summand of M, thus M = M' + M" with a
convex body A/"e r^". We assume first that M' and A/" have interior
points. The inequality F^O, applied to ATq = M, Ki = K, K2 = L and
A:3 = A/" yields fi{M", M") ^ 0. Thus /3(A/", L) ^ /3(A/", L) -
/3(A/", A/") and F.6.39) gives
fi{M", L) ^ 9<A/")V4(A:, L, M). F.6.41)
Since /3(A/', L) = /3(A/, L) - /3(A/", L) by the Unearity of the mixed
volume, inequalities F.6.38) and F.6.41) yield
^(^'' ^) = - vii^' f' i!!! - 9<A/")V4(/C, L, A/). F.6.42)
K(K, L, A/)
Writing /3(A/',A/') = /3(A/', L) + [/3(A/',A/')-/3(A/', L)], we deduce
from F.6.42) and F.6.39) that
/3(A/', M') § -V' F.6.43)
with
A{K, L, M)
n> ■■= 7(^^^ + [<P(M') + <P(M")]V^{K, L, M).
Dividing F.6.43) by V{K, M, M')V{L, M, M'), we obtain

364 Inequalities for mixed volumes
F{K, L. M'; M) ^
V{K, M, M')V{L, M, M')
Inequality F.6.31) with AT, = K, K2 = L, Kj = M' and Kq = M yields
Both inequalities together show that
^ V{M, M, M')
y^-;^ + [cdM') + <KA/")]V4(/c, l, a/)}.
F.6.44)
This estimate can be simplified if we now use the assumption that
K, L, M e X"{r, R). Then K is contained in some translate of {R/r)M,
hence the monotoneity of the mixed volume gives
V{K, M, M') ^ R
V{M, M, M')~ r'
Since M = M' + M", we may assume that M' C M, hence
V(L, M. M') ^ R^V(B", B", B").
From A{K, L, M) ^ V{K, L, Mf v/e obtain
Since M' is contained in M and, thus, in a translate of (R/r)K, we
have
<fiM')^{j^ +1,
and the same inequality holds for (p{M"). Now F.6.44) leads to the
inequality F.6.30). By approximation, we can finally extend this
inequality to the case where K4,. . . , K„ and M', M" are not necessarily
of dimension n. ■
Proof of Theorem 6.6.10. Under the assumptions of Theorem 6.6.10,
the ball M'= pB" is a summand of Cj. Hence, Lemma 6.6.11 with
M = C, and
& = iK4, ■■■, K„) = (C2, . . . , Cp, B"[n - p-2])
yields
AiK, L, pB", &) ^ a\/A{K, L, Ci,e),
hence

6.6 Equality cases and stability 365
A{K,L,Ct,...,Cp, B"[n - p - 2])
S a-^p*A{K, L, C2, . . . , Cp, B"[n - p - l])^.
Repeating this procedure with C2, ■ . , Cp, we arrive at the inequality
F.6.29). ■
In the second part of this section, we shall now investigate the
problem of equality in the Aleksandrov-Fenchel inequality
V(K, L, ef ^ V{K, K, e)V{L, L, &) F.6.45)
in the case where & = (Ci,. . . , C„_2) and the bodies Ci, . . . , C„_2 are
not necessarily smooth. In that case, equality in F.6.45) may hold even
if K and L are not homothetic. An example is given by Corollary 6.6.17
below. Before treating such examples and special cases, we formulate a
general conjecture.
Conjecture6.6.12. If K, L e J(" and Ci, . . . , C„_2 e Xq, then equality
in F.6.45) holds if and only if, after applying a suitable homothety to
K or L, h(K, u) = h{L, u) for all (Ci,. . . , C„.2-, B)-extreme unit
vectors u.
The notion of (Ci,. . . , C„_i)-extreme vectors was defined in Section
2.2. In particular, u is (Ci, . . . , C„^2< 5)-extreme if and only if there
exist (n - l)-dimensional linear subspaces £1, . . . , E„-2 of E" such
that
T(Ci, u) G Ej for / = 1, . . . , n - 2
and dim Ej n . . . n E„-2 = 2. Here T{Ci, u) is the touching cone of C,
at u.
Without the assumption in Conjecture 6.6.12 that Ci,. . . , C„-2 are
n-dimensional, more cases of equality in F.6.45) are possible. This was
pointed out by Ewald A988). For example, let « = 3, let 5,, 52 C E^ be
segments of independent directions and let M d X^ be an arbitrary
convex body. Then put K ■= M -^ Si, L ■= M + S2 and C := 5i + 52.
Since V(A/, 5,, 5,) = 0, one computes V{K, K, C) = V{L, L, C) =
V{K, L, C) and thus V{K, L, Cf = V{K, K, C)V{L, L, C). The
condition of Conjecture 6.6.12, however, is not satisfied.
Of course, if Ci,. . . , C„_2 are smooth, then each vector u e 5""' is
(Ci,. . . , C„_2, 5)-extreme, hence Theorem 6.6.8 would follow from
the truth of Conjecture 6.6.12. In fact. Theorem 6.6.7 would also
follow.
The set of (ig, 5)-extreme unit vectors appears in Conjecture 6.6.12
since it is probably closely related to the mixed area measure
5(<§, B, •), according to another conjecture (of which a special case is
\'erified by Theorem 4.6.3):

366 Inequalities for mixed volumes
Conjecture6.6.13. Let AT],... , K„-i e J(" he convex bodies. The
support of their mixed area measure, suppS(Ki,..., K„-i, •), is the
closure of the set of (Ki, ... , K„^{)-extreme unit vectors.
Before discussing further the relation between the two conjectures,
we state a result on supports of mixed area measures from which further
consequences can be drawn.
Lemma6.6.14. Let K, L, C^, . . . , C„_2 e X" and 6 = (Ci,. . . , C„_2).
If L is smooth and strictly convex, then
suppS{K,e, •) C supp5(L,<g, •)■
Proof. Suppose the assertion were false. Then we can choose a vector
Uo e supp5(Ar,6, • )\supp5(L,6', •) and an open neighbourhood a of
Uq in 5""' such that S(L,€, a) ~ 0. Let
L' := {x e L\{x,Uo)^hiL,U(,)- e),
where e > 0 is so small that each outer unit normal vector to L at a
point of cl(bd L\L') belongs to a; this is possible since L is smooth.
For u e S"~^\a, the unique boundary point of L at which u is an outer
normal vector is also a boundary point of L'. We deduce that
S{L,e^, /3) = S{L',Q, /3) for every Borel set /3 C S"-\c^. This yields
0 ^ V{B, L, fri) - V{B, L\ m = - S{L, &, a) - - S{L', 6, a)
n n
and hence S{L',e,a) = 0. Thus S{L,&, •) = S(L',&, •) and,
therefore, V{K, L,€) = V{K, L',B). On the other hand, we can choose an
open neighbourhood /3 of Uq such that h{L, u) > h{L', u) for u e fi.
Since uq ^ supp S{K,&, ■), we have S{K,&, fi) > 0, which implies
V{K, L, g) = - L,/!(L, u)dS{K, fd, u)
> - L,/!(L', u)dS{K, e, u) = V{K, L', m-
This contradiction completes the proof. ■
Instead of the condition appearing in Conjecture 6.6.12, let us now
assume that the following is satisfied;
h(K, u) = h{L, u) for each u e supp5F, B, ■).
Then for any convex body M e X" we have

6.6 Equality cases and stability 367
- j^iM, u)d[S{K, g, u) - S{L, &, u)\
= V{M, K,Q) - V{M, L, 6)
= 0,
because supp5(Af,<g, •) C supp5E, ^, •) by Lemma 6.6.14. Since M
was arbitrary, we deduce that S{K,&, •) = S(L,§i, •) and thus equality
holds in F.6.45).
One may conjecture that the converse of this observation is also true:
Conjecture6.6.15. If K, L e JC", Ci,. . . , C„_2 e .To ond equality holds
in F.6.45), then, after applying a suitable homothety to K or L,
h{K, u) = h{L, u) for each u e supp5((g, B, •).
The truth of Conjectures 6.6.13 and 6.6.15 would imply the truth of
Conjecture 6.6.12.
The problem of equality in the Aleksandrov-Fenchel inequality is
related to another unsolved problem. This is the equality case for
E.1.23), which expresses the monotoneity of mixed volumes. We shall
first prove a special result concerning this problem. It is useful in
treating a special case of Conjecture 6.6.12, and its corollary gives a first
example of non-trivial equality in F.6.45). The result deals with the
special mixed volumes V,(A:, L) = V{K[n - i], L[i]y, for L = B these
are the quermassintegrals Wi(K). If K, L are convex bodies such that
LC K, then the monotoneity E.1.23) implies
Vo{K, L) S V,{K, L) S . . . g V„{K, L). F.6.46)
The following theorem characterizes the pairs K, L for which
Vp_i(Ar, L) = Vp{K, L) for some p. For n = 3, /? = 2 it goes back to
Minkowski A911); the general case is essentially due to Favard A933a).
Theorem6.6.16. Let K, L e Xo he convex bodies satisfying LG K; let
p e {I, . . . , n}. Then
Vp.iiK, L) = Vp{K, L) F.6.47)
holds if and only if K is an (n - p)-tangential body of L; in this case,
Vo{K, L) = V,iK, L)= ... = Vp{K, L).
Proof. Assume that F.6.47) holds. First let p = \. As remarked before
E.1.23), V(s{K, L) = V,(A:, L) implies
hffK, u) = h{L, u) for u e supp5„_i(A:, •).

368 Inequalities for mixed volumes
By Theorem 4.6.3, supp5„_j(Ar, •) is the closure of the set of extreme
unit normal vectors of K, hence each extreme supporting hyperplane of
K supports L. By definition, K is an (n - l)-tangential body of L.
Now let p^\. For n = 2, only /? = 1 is possible, hence the assertion
is true. We assume that n > 2 and that the assertion (i.e., sufficiency of
F.6.47)) has been proved in dimension n - 1. We may also assume
p^2. From Vp^2{K, L)^Vp_i{K, L)= Vp{K, L) and the Aleksan-
drov-Fenchel inequality we have
Vp.,{K, Lf = Vp-2(A:, L)Vp{K, L).
By Theorem 6.4.1 (where F.6.47) implies a = 1) and E.3.32) this yields
v(K"[n - p + 1], L"[p ~ 2]) = v(K''[n - p], L"[p - 1]) F.6.48)
for each u e S"~^.
Let H{K, v) be a (/? - l)-extreme support plane of K. Choose a unit
vector u orthogonal to v and, if dim T(Ar, u) § 2, in the linear hull of
the touching cone T{K, v). Then H' := H{K, v) n //„,o is a (/? - 2)-
extreme support plane of K" (relative to //„o). Since L" A K", we
deduce from F.6.48) and the induction hypothesis that K" is an
[(n - I) - (p - l)]-tangential body of L". Hence, the (p - 2)-extreme
support plane H' of K" supports L". This implies that H(K,v)
supports L. Thus K is an (n - /7)-tangential body of L. This completes
the induction.
Vice versa, suppose that K is an (n - /7)-tangential body of L for
some p e {I, . . . , n}. Then
h{K, u) = h{L, u) for u e extnp_i K,
where extny AT denotes the set of ;-extreme unit normal vectors of K.
For 0^7^/7-1, we deduce from Lemma 6.6.14 and Theorem 4.6.3
that
supp5(A:[n - ;■ - 1], L[j], •) C supp5(A:[n - ; - 1], B[j], ■)
= clextny K C clextnp_i K
and hence that h{K, u) = h{L, u) for u in the support of
S{K[n - j - \], L[i], •). Thus
\^„_^K,u)dS{K[n- i- \],L[j],u)
= \^^_^h{L,u)AS{K[n- i- \],L[j],u),
for / = 0, ...,/?- 1, which by E.1.18) gives
Vo(A:, L) = V,{K, L) = ... = Vp{K, L), F.6.49)
as asserted. ■

6.6 Equality cases and stability 369
Corollary 6.6.17. Let K, L e Xn. If i e {1, . . , n - 1} and K is homo-
thetic to an {n - i -\)-tangential body of L, then
V,{K, Lf = V,_,(A:, L)V,+,(A:, L). F.6.50)
Proof. We may assume that rL is a maximal homothetic copy of L
contained in K. By assumption, K is an (n - / -l)-tangential body of a
homothet of L and necessarily of rL. From F.4.49), with L replaced by
rL, we obtain
V^{K, L) = rV^{K, L) = ... = r'^'V,^,{K, L) F.6.51)
and thus F.6.50). ■
The next theorem establishes a converse to Corollary 6.6.17, for
/ = 1. It thus settles a special but important case of the general equality
problem for the Aleksandrov-Fenchel inequality. It was conjectured
(for n = 3) by Minkowski A903) and proved by Bol A943b). His proof
(which we follow) is a nice application of the powerful method of inner
parallel bodies.
Theorem 6.6.18. Let L e X^ be an n-dimensional convex body and
K e X" an arbitrary convex body. In the inequality
V{K, ...,K,Ly^ V„{K)V{K, ..., K,L,L) F.6.52)
equality holds if and only if either dim K < n - I or K is homothetic to
an (n - 2)-tangential body of L.
Proof If dim K < n - I, both sides of F.6.52) are zero. If K is
homothetic to an (n - 2)-tangential body of L, then equality in F.6.52)
is a special case of Corollary 6.6.17. We assume now that equality holds
in F.6.52) and that dim K = n-1. Since dim K = n - I would imply
V„(K) = 0 and V{K, . . . , K, L)>0, this case cannot occur.
We use the system of parallel bodies Kx of K relative to L, as
defined in Section 3.1. In particular, K) = K-—XL for -r^A^O,
where r = r{K, L) denotes the inradius of K relative to L. Let
V,{X):=Vi{K,,L)=V{K>[n-i],L[i\) and fi{X) := V,{X)'l("-'\ By
the concavity of the system of parallel bodies (Lemma 3.1.9), the
monotoneity of mixed volumes and the general Brunn-Minkowski
theorem 6.4.3, the function /) is concave on [-r, »). Therefore, -/) has
the properties collected in Theorem 1.5.2, some of which will be used in
the following.

370
We have
Inequalities for mixed volumes
dVo(A)
dA
= nV,{k),
F.6.53)
which is a very special case of Lemma 6.5.3. For the left derivative of
V, we write
dV,(A)\
= : (n - 1)V|(A) F.6.54)
dA /I
for A > -r, and we define
A{X) := V,(AJ - Vo(A)V|(A).
Then
since
V|(A) s V2(A)
F.6.55)
n-l
Vi(A:, + orL, L) = 2J" . ^jar'V,+i(A)
for o-^O; furthermore Kx + aLd Kx+a by C.1.12) and hence
V,(/Ca+ aL, L)^ V,(A+ or) for or^0 with equality for or = 0, and this
gives
(n - 1)V|(A) =
1<v=0
— V,(A + a)
aa
da
Aa
a=0
V,(A + or)
a=0
(n - 1)V2(A).
The fact that /o is concave and hence satisfies [/o(A)]i' = 0 yields
A{k) ^ 0, F.6.56)
by F.6.53) and F.6.54).
Proposition. The function A is non-decreasing on (-r, <»).
For the proof, let -r<Ai<A2 and assume, on the contrary, that
A(X2)<M^i)- Then Zi(A,)>0 by F.6.56). Put 6 := inf {A|A > A,,
MX) < AiXi)}. We write /, := /, thus
Vi(A) = /(A)"-'. F.6.57)
Since / is concave and increasing,
/((A) ^ /;(A) ^0 for A > -r F.6.58)
and
/;(ar,) ^/Kara) for-r < otj < 0^2. F.6.59)
By F.6.57) and F.6.54),
K|(A) = /(A)''-Vi(A). F.6.60)

6.6 Equality cases and stability 371
If A, < A < 6, then F.6.60), F.6.58) and F.6.59) yield
zi(A,) ^ AW = V,(AJ - Vo(A)Vf(A)
= V',(AJ - Vo(A)/(A)''-Vi(A)
^ Vi(AJ - Vo(A)/(A)''-V;(A)
^ V,(AJ - Vo(A)/(A)''-VK6).
With A ^ 6 we obtain
0 < Zi(A,) S ViFJ - V,{b)f{bY-^fm = Mb). F.6.61)
Now we consider the function F defined by
r(A) := v,(AJ - VoWfixy-'nib)
for A > -r. By F.6.58) we have
A(b) ^ r{b), F.6.62)
and by F.6.59),
r(A) ^ Zi(A) for A > 6. F.6.63)
The function F has a right derivative at b, for which we obtain, using
F.6.63) and F.6.59),
r;F) = Bn - 2)nbf"-'n{b) - nV,{b)f{by-^f',{b)
-Vo{b){n - 2)f(br-'f',(b)^
= (n- 2)V,(b)-'f(br-'f',{b)F(b) > 0
since f'r(b)>0 (because /((A) > 0 for all A>-r by F.6.60)) and
F(b) > 0 by F.6.62) and F.6.61). Hence, there is a number d>0 such
that
r(A) ^ F(b) ioT b^ X^ b + d. F.6.64)
For b < X^ b + 6 we now deduce from F.6.63), F.6.64), F.6.62) and
F.6.61) that
A(X) g r(A) ^ rF) a ziF) ^ zi(Ai),
which contradicts the definition of b. This contradiction proves the
proposition.
We have assumed equality in F.6.52); thus V,@)^ - Vo@)V2@) = 0.
By F.6.55) and F.6.56) this implies A@) = 0 and
Vm = V2@). F.6.65)
The proposition yields 4(A) = 0 for A€(-r,0). According to the
deduction of F.6.56), this is only possible if /o is a linear function on
(-r,0). For A-^ -r, Vo(A) converges to 0 (see F.5.11)), hence
V„(K,) = (l + -j)" Vn(K). F.6.66)
K contains a translate of rL, and after a translation we may assume that

372 Inequalities for mixed volumes
rLdK. Then (-A)L + [1 + (A/r)]A: C/C for -r^A^O and hence
[1 + {k/r)]K C Kx\ thus
^' <
r
1 + - V„{K) ^ V„{K,).
Since equality holds here, each /C^ is homothetic to K. By Lemma
3.L10, K is homothetic to a tangential body of L.
From F.6.65) we have
/ A\""^ 1 / A\""^ 1
V,(A) = 1 + - - V„{K), V|(A) = 1 + - -, V„{K).
\ -I r ' \ r j r^
For A = 0, together with V,@) = V,(A:, L), F.6.66) and F.6.65) this
yields
V^,{K, rL) = V,{K, rL) = V2{K, rL).
Since rLC K, Theorem 6.6.16 now shows that K is an {n - 2)-tangen-
tial body of rL. This completes the proof. ■
A converse to another special case of Corollary 6.6.17 is contained in
the following theorem. It concerns the case where L is a ball.
Unfortunately, for a technical reason we have to assume central symmetry for K,
which is very probably unnecessary.
Theorem 6.6.19. If K e X" is centrally symmetric and i e {I, . . . ,
n - l), then the equality
W,(Kf = W,_i(A:)W,+,(/C) F.6.67)
holds if and only if either dim K < n - i or K is an (n - i -I)-tangential
body of a ball.
Proof. Since Wj(K) > 0 if and only if dimK ^ n - i, we may assume
that dim K^ n- i + I. That F.6.67) holds for (n - i - l)-tangential
bodies of balls is a special case of Corollary 6.6.17.
We prove by induction with respect to n that F.6.67) and the
assumption dim K^ n - i + I imply that K is an (n - / - l)-tangential
body of a ball (/ = 1, . . . , n - 1). For n = 2 the assertion is true by
Theorem 6.2.1. Assume that n ^ 3, that the assertion is true in
dimensions less than n and that F.6.67) holds for some /€ {1, . . . ,
n - 1}, where dim K ^ n - i + I. If / = 1, K is an (n - 2)-tangential
body of a ball by Theorem 6.6.18. Let /^2. From F.6.67) and
Theorem 6.4.1 it follows that, after applying a suitable dilatation to K,
we may assume that
5„_,(/C, •) = 5„_,-_i(/C, ■). F.6.68)

6.6 Equality cases and stability 373
By Lemma 6.4.5, this implies
w,_,(a:"J = w,_2(a:")w,(a:")
for M € 5"~\ where we have written
W/.iK") := v{K"[n - \ - k], B''[k]).
By the induction hypothesis, K" is an (n - /-l)-tangential body of a
homothet of B". Let r^ be the radius of this homothet. Equations
F.6.51) in dimension n - 1 give
rriw,_i(/C") = rLw,(A:").
From F.6.68) and E.3.32) we have w,_,(A:") = w,(A:"), thus r„ = L
Let Br be the maximal ball in K with centre at the symmetry centre
of K\ here r denotes the radius of Br- K and Br have a common
supporting hyperplane //, and by symmetry a second one parallel to it,
say //'. Let m be a unit vector parallel to H. The intersection H fl //„(>
is a common support plane (in //u,o) of K" and B" and hence an
extreme support plane of K". The same holds true for H' fl //^.o- Since
both (n - 2)-planes must support some ball in //„ o of radius 1, it
follows that r = \. Hence, we may assume that Br = B. Now let
M6 5""' be arbitrary. The projection K" is an (n - /-l)-tangential
body of a unit ball and necessarily of B".
The argument is completed in a similar way to the proof of Theorem
6.6.16. Let H(K, v) be an /-extreme support plane of K. We choose a
unit vector u orthogonal to v and, if dim T(K, v) ^ 2, in the linear hull
of T(K, v). //' := //(/C, y) n //„ 0 is then an (/ - l)-extreme support
plane of K". Since K" is an (n - / - l)-tangential body of B", we
deduce that H' supports B" and hence H{K,v) supports B. Thus K is
an (n - i - l)-tangential body of B. ■
The general conjecture 6.6.12, with distinct bodies C|, . . . , C„_2, has
so far only been verified under strong restrictions for these bodies. One
example is given by the following result, which extends the equality
assertion of Theorem 6.3.2.
Theorem6.6.20. Conjecture 6.6.12 is true if Cy, . . . , C„_2 are strongly
isomorphic simple polytopes (and K, L are arbitrary convex bodies).
Proof. Let Pi,. .. , P„€ Xq be strongly isomorphic simple polytopes.
We have to apply some results from Sections 5.1, 5.2 and 6.3 and
therefore use the notation introduced there. In particular, <4 is the
fl-type of Pi, . . • , P„ and P e <^. is an arbitrary member. Further, u^,
..., Ufj, Fj, Fjj, Vij, ©ij, J, py^ := F(Pr,Ui) etc. are defined as in
Section 5.1. We abbreviate (Pj, . . . , P„)by f^.

374 Inequalities for mixed volumes
Let K, L e X" be convex bodies. After applying a dilatation to K or
L we may assume that
ViK, K,f^) = V(L, L,f^). F.6.69)
We may also assume that V{K, L,(k)> 0, since otherwise K and L are
parallel segments (possibly degenerate), and the assertion is trivial. By
Theorem 6.4.1, equality in the inequality
V(K, L, e^y ^ ViK, K, 6)V(L, L, &) F.6.70)
holds if and only if
S^KJi, ■) = 5(L,6, ■). F.6.71)
We must, therefore, examine the measure Si^K, P-^, . . . , P^^ ■). First,
S{K, P,,..., P„, («,}) = viF{K, «,), fP>, . . . , F\"^) F.6.72)
for i = I,. . . , N, by the definition of the mixed area measure. For
(i,j)eJ (recall that this means that Fjj ■= F{P, Uj) f) F(P, Uj) is an
(n - 2)-face of P), we denote by a;, the spherical image of Fjj and by w^,y
the orthogonal projection onto the two-dimensional linear subspace L;,
orthogonal to Fjj. Observe that Ojj and 7r,y depend only on (/,;) and the
fl-type c •/.
Let CO e 3E""') be a Borel set. If wC relinta,y, then
SiK, Pi,..., P„,co)
= iA"-^KFf, ..., F\;'^)siniiK, @ n Lij), F.6.73)
where s denotes the area measure in Ljj. If w fl Ojj = 0 for all (/, /) e J,
then
SiK, Pi, P„,co) = 0. F.6.74)
The last two equalities are proved by computing 5„_i(A/C + XiPi+ ...
+ X„P„, co) for A, A3, . . . , A„ > 0, using D.2.24) and Fubini's theorem,
and then applying E.1.17).
Now we assume that equality in F.6.70) and hence F.6.71) hold.
Suppose that (/,;) e J and that (o C relint a,y is a Borel set. Then
F.6.73) yields 5Gr,y/C, (o) = sGTijL, (o). By the local uniqueness theorem
4.3.2, the convex arcs in ttjjK and iTijL that are the reverse spherical
images of relint a,y must be translates of each other. Hence, there exists
a vector //y € E", orthogonal to F,-,, such that
h{K, u) - h(L, u) = </,y, m) for u e Ojj F.6.75)
(first for u e relint a/,, then by continuity for u e a,y). Define
h(K, Ui) - h{L, Ui) = </;,, M,) =: or;,
h(F{K, Ui), Vij) - h(F{L, Uj), Vij) =: A,y
for / € {1, . . . , A^} and (/,;) e J. From F.6.71), F.6.72) and E.1.18)
we deduce that, for each fixed i e {\, . . . , N),

Notes for Section 6.6 375
2 ^v^''-'\F'!f\ .... F^,")) = 0. F.6.76)
Taking directional derivatives in F.6.75), we infer from Theorem 1.7.2
that A,y = {tii,Vjj). Since /,y is a linear combination of m, and vij, we
deduce that /;, = ar,M,- + A;yy,y. As /y = /,, = ar,My + Ay;y,,, multiplication
by My yields
A,y = aryCosec6>,y - a';Cot6>,y F.6.77)
for (/, /) € y. Writing Z := (a,, . , . , o-jy), we see from E.2.6) that
A,Z = (A,i, .. ., A,w)
with A,y := 0 if (/,/) $ J. With <p, as defined in the proof of Theorem
6.3.2, we have
= -^ 1 A,^,(''-^)(Fl/), . . . , F\p),
n - 1 (,,y)ey
hence <p,(Z,/'3) = 0 by F.6.76). This means that Z is an eigenvector
with eigenvalue 0 of the bilinear form 4> used in the proof of Theorem
6.3.2; hence from that proof (Proposition 3) it follows that Z is the
support vector of a point. Thus there exists a vector ^ € E" such that
a-, = (^, M,) for / = 1, . . . , n. After applying a suitable translation to K,
we may assume that ^ = o. Then A,y = 0 by F.6.11) and hence /,y = o;
thus F.6.75) now reads
h{K, u) = h(L, u) for u e a,y, (/, /) € J. F.6.78)
The (Pj,.. . , P„, B)-extreme unit normal vectors are precisely the
vectors in U(,,y)£y<7,y.
Vice versa, from F.6.78) we deduce F.6.71) and hence equaUty in
F.6.70). This completes the proof. ■
Another case where the truth of Conjecture 6.6.12 could be
established is the following result (Schneider 1988a), which we formulate here
without proof.
Theorem6.6.21. If K, L e X^ are centrally symmetric and Cj,. . . ,
C„_2 6 X") are zonoids, then Conjecture 6.6.12 holds for these bodies.
Notes for Section 6.6
1. Lemma 6.6.1 is due to Suss A932a); proofs also appear in Aleksandrov
A937b) and LeichtweiB [23], pp.'141-3. Corresponding stability results were
proved by Groemer A987a); see also Groemer A991b).
2. The first proof given above for TTieorem 6.6.2 is essentially due to Favard
A933a) (compare also LeichtweiB 1981). The second proof, using Lemma

376 Inequalities for mixed volumes
6.6.3, extends a method of Kubota A925b). Theorem 6.6.6 was proved by
Schneider A989c) and independently by Goodey & Groemer A990). The
stability estimate F.6.10) was derived in Groemer & Schneider A991).
3. Theorems 6.6.7 and 6.6.8 and their proofs are taken from Schneider A990a).
The proof of Theorem 6.6.9 (given Theorem 6.6.8) is modelled on the
procedure of Fenchel A936b). Theorem 6.6.10 appears in Schneider A990b).
It is a quantitative improvement of a result obtained earlier in Schneider
A985a).
4. Equality cases in the Aleksandrov-Fenchel inequality. Conjectures 6.6.12,
6.6.13 and 6.6.15 and Lemma 6.6.14 were formulated in Schneider A985a),
where credit for 6.6.15 and a special case of 6.6.14 is given to A. Loritz.
Conjecture 6.6.12 states that the equality
V(K, L, e^Y = V{K, K, e^)V(L, L, ^),
where K,Le X", (^ = (C,,..., C„-i) and Ci,..., C„_2 e .TJ holds if and
only if Suitable homothets of K and L have the same (<!', B)-extreme
supporting hyperplanes. When this conjecture was formulated, the following
classical cases in favour of it were known.
(a) C< consists of balls. This is the case mentioned in Note 2, first proved by
Kubota A925b), with a different proof given by Favard A933a).
(b) <& consists of strongly isomorphic simple polytopes, and K, L are
polytopes having the same system of normal vectors to the facets as the
polytopes of ^. This comes out in Aleksandrov's first proof for the
Aleksandrov-Fenchel inequality, as presented in Section 6.3. Theorem
6.6.20 extends this result to arbitrary convex bodies K, L.
(c) <^ consists of bodies of class C+, and K, L have support functions of
class C-. This case is a byproduct of Aleksandrov's A938b) second
proof and was also noted by Favard A938).
(d) C, = .. . = C„-2 = K. This is the case treated in Theorem 6.6.18,
which is due to Bol A943b). Corollary 6.6.17 (for n = 3) had already
been discovered by Minkowski A903), §7. A special case of Theorem
6.6.18 was treated by Knothe A949) in a different way.
5. Tangential bodies. Theorem 6.6.16, as mentioned, is due to Favard A933a),
but we found it necessary to give a different proof for the necessity of
condition F.6.47).
\i K e TC!^ '\% a convex body with inradius r, then its quermassintegrals
satisfy
If equality holds here, then Theorem 6.6.16 implies that Jf is a cap body of a
ball. Sangwine-Yager A991) obtained an explicit estimate showing that K
must be close to some cap body of a ball if Wn-i^K) - rW„_i(/C) is small.
Theorem 6.6.19 is taken from Schneider A985a).
6. The proof of Theorem 6.6.20 was given in Schneider A991 + ). The second
part of the proof, in which it is shown that the vectors t,, associated with the
\n - 2)-faces f/y of P are all equal to a fixed vector, is similar to (but
different from) a method of Aleksandrov [2], Chapter XI, §3, which shows
the infinitesimal rigidity of convex polytopes in E' with the aid of mixed
volumes. A connection between rigidity of polytopes and mixed volumes was
first exploited by Weyl A917); his proof is sketched in Efimov A957), §100.
6.7. Linear inequalities
Under certain circumstances, the quadratic Aleksandrov-Fenchel
inequality

6.7 Linear inequalities 377
V(K, L, ey ^ V(K, K, e)V(L, L, &), F.7.1)
where K, L, K3, . . . , K„ e X" and & = (K^,. . . , K„), admits the
linear improvement
2V(K, L, (&) ^ V{K, K, 6) + V(L, L, 6). F.7.2)
A sufficient condition for the validity of F.7.2) is that there exist a
convex body M fulfilling the conditions
V{K, M, e) = V(L, M, (i) > 0. F.7.3)
This follows immediately from F.4.22). For example, M could be a
segment. If m is a unit vector parallel to it, then, by E.3.31), condition
F.7.3) is equivalent to
v(K\€") = v(L\B") >0.
A special case of the latter inequality occurs when K" = L", that is, K
and L have the same circumscribed cylinder of direction u. In this case,
the Brunn-Minkowski theorem for K and L also holds, in an improved
version: The volume of A - ■d)K + i?L, and not only its nth root, is a
concave function of i? for 0 ^ r?^ 1. This is clear from Fubini's theorem
and the fact that each line G of direction u satisfies
G n [A - ff)K + &L] D A - i?)(G n K) + diG n L).
Less immediate is the fact that the general Brunn-Minkowski theorem
(Theorem 6.4.3) also admits an improved version of this type. In the
following we describe a theory of linear inequalities for mixed volumes
of bodies inscribed in a cylinder; this unifies some special results in the
literature obtained by other methods.
We assume that a unit vector u e 5""' and an (n - l)-dimensional
convex body C in the hyperplane //„ 0 orthogonal to u are given. Then
we define
Xc := [K e X"\K'' = C}
and call such a set of convex bodies a canal class. The (n - l)-dimen-
sional mixed volume of convex bodies in //„ 0 is denoted by v. If i/ is a
segment of length one parallel to u, then
uiKl . . . , K^.i) = nV(Ku ..., /C„_„ [/) F.7.4)
by E.3.31). The mixed area measure in //„(, is denoted by s and,
without loss of generality, considered as a measure on S"~^ that is
concentrated on the great subsphere
Su '■= S" n Hu,o-
Then we have
s(Ki..., a:"„_2, •) = («- iM(a:„ ..., /c„_2, u, •) F.7.5)
on 5"''. This follows from E.1.17) and the equality

378 Inequalities for mixed volumes
5„_,(/C + At/, •) = S„-i{K, •) + hiK", ...,K\-),
which is clear from D.2.24), if one then replaces K by Ai/C| + . . .
+ A„_2/C„_2 and compares the coefficients.
Theorem 6.7.1. Let d be an (n - 2)-tuple of convex bodies in X"
satisfying
y(C, <&") > 0. F.7.6)
For K, L in the canal class Xc the inequality
V(K, K, e) - 2V(K, L, &) + V(L, L,(k)^0 {6.1.1)
is true. Here equality holds if and only if
S(K,€, ■) = S(L + aU,^^, ■) F.7.8)
or
S{L,e^, ■) = S{K + aUJ^, ■) F.7.9)
with some number a ^ 0.
Proof. By F.7.4) and F.7.6), each convex body A e Xc satisfies
nV{A, U, ti) = t;(>l", 6") = t;(C, ^^") > 0.
In particular, V{K, f/,<f^) = V(L, i/,^) >0, hence F.4.22) with M = V
yields inequality F.1.1) (observe that F.4.20) is not necessary for the
validity of F.4.22), as remarked after the formulation of
Theorem 6.4.2). As shown in Section 6.4, equality in F.4.22) is
equivalent to F.4.26), with a suitable number k. The numbers ^i, qi
occurring there are, in our present case, given by qi = q2 = v{C,(k")/n.
Hence, the first equation of F.4.26) is equivalent to F.7.8) with
suitable a ^ 0, and the second equation of F.4.26) is equivalent to
F.7.9). ■
Next we show that for canal classes the general Brunn-Minkowski
theorem 6.4.3 holds in a stronger form, and there is also an analogue of
Theorem 6.4.4. Let a number m€{2,...,n}, an (n - m)-tuple
(i* = (Km+i,. . . , K„) of convex bodies /C„n, . . . , K„ e X", and two
bodies Kq, K^ 6 Xc be given. We use the notation of Section 6.4, in
particular
V(,) := V{Ko[m - /], A:i[/], e^) for / = 0, . . . , m,
5(,) := S{Ko[m - 1 - /], Kx\i], (&, •) for / = 0, . . . , m - 1,
and in addition we define a measure y on S"~^ by
y -.^s^Cim -2],&\ ■).

6.7 Linear inequalities 379
By F.1.1), the sequence (V@), . . . , V'(„)) is concave, hence F.4.1)
tells us that, for 0 ^ / < / < A: ^ m,
{k - /)V(,) + (i - A)V„) + (/ - /)V(,) ^ 0. F.7.10)
This is a stronger version of F.4.5). The following result is the general
Brunn-Minkowski theorem for canal classes.
Theorem6.7.2. Let a number m 6 B,. . . , n} and convex bodies K^,
Ki 6 Xc, ^m+i. ■ . ■ , K„ e X" be given and define (^ := {K„+i, . . . ,
K„), K-^ := A - A)A:o + A/Ci and
/(A) := V{K,[m], (&)
for 0 ^ A ^ 1. Then f is a concave function on [0,1].
Under the assumption
v{C[m - 1], Q") > 0
the following assertions are equivalent:
(a) the function f is linear;
(b) the sequence (V(o), • ■ • , ^(m)) " linear, that is,
V(o) - V'(i) = V(,) - VB) = ... = v^„_^ - v^„y,
(c) (m-l)V(o)-mV(i) + V(„) = 0;
(d) S^k-l) - 5(/t) = ay for k = I, . . . , m - I with a constant a;
(e) 5@) = 5(„_i)On5''-'\5r^-
Proof. We have
/"(O) = m(m - l)[V(o) - 2V(,) + V^2^],
hence /"@)^0 by F.7.7). As in the proof of Theorem 6.4.3, we
deduce that / is concave.
The concavity of / yields /'(O) ^ /(I) - /(O), thus
(m - l)V(o) - mV(,) + V(„) ^ 0, F.7.11)
with equality if and only if / is linear. Hence (a) and (c) are equivalent.
Inequality F.7.11) is also a special case of F.7.10) and hence equality
holds here (see the remark after F.4.1)) if and only if equality holds in
V(/t_,) - 2V(/t) + V^k+l) ^ 0 for A: = 1, . . . , m - 1. Thus (b) and (c) are
equivalent.
Suppose that (b) holds. By Theorem 6.7.1, this implies
S(k-i) - 5(/t) = fl/ty F.7.12)
for k = 1,. . . , m - 1, with a constant C/t possibly depending on k.
Integrating the support function of /C, with the measure given by
F.7.12) we obtain
V(/t) - V(,+,) = -^4-^ aMC[m - 1], e").

380 Inequalities for mixed volumes
For k^ m -2 we integrate the support function of Ko with the
measure given by F.7.12), but with k replaced by /c + 1, and obtain
V(/t) - V(,+i) = -^^ fl/t+iKC[m - 1], &").
Thus fli = ... = flm-u and (d) holds. Trivially, (d) implies (e), since y
is concentrated on s"~^.
Suppose that (e) holds. Then
l„.,^„.JiHKo, o) - h{Ku i;)]dE@) - 5(^-,))(i^) = 0.
Since h{K(„ v) = /i(/C,, v) for v 6 5""^, we also have
[AKK^, v) - h(K„ v)]d(S^o) - 5(„-i))(i^) = 0.
Thus
V@) - V(,) - V(„_,) + V(„) = 0.
Adding F.7.11) and the inequality
(m - 1)V(„) - mV(^_i) + V(o) ^ 0,
which also follows from F.7.10), we see that equality holds in F.7.11),
which is condition (c). This completes the proof of Theorem 6.7.2. ■
There is also a counterpart to Theorem 6.6.9. We say that the convex
bodies Kq, /Cj e Xc are equivalent by telescoping if K(,= Ki + XU or
Ki = Kq + \U with A^ 0, where U is a unit segment parallel to u.
Theorem 6.7.3. If the assumptions of Theorem 6.7.2 are satisfied and if
K„+i, . . . , K„ are smooth, then the conditions (a) to (e) of Theorem
6.7.2 hold if and only if Kq and Ki are equivalent by telescoping.
Proof. It is clear that condition (a) is satisfied if Kq and K^ are
equivalent by telescoping. To prove the other direction, we use
induction over m. Assume that conditions (a)-(e) of Theorem 6.7.2 are
fulfilled. First let m = 2. Then condition (d) can be written in the form
S{Ko, &, ■)- S{Ku e, ■) = a{n - l)S{U, &, •).
If a ^ 0, this is equivalent to
S{Ko,e, •) = 5(A:i + a(n - l)i/,6, •)■
By Theorems 6.4.1 and 6.6.8, the bodies Kq and Ki + a(n - 1I17 are
homothetic, and since Kq, /C, e JCc, this implies that Kq and /Cj are
equivalent by telescoping. If a < 0, the argument is similar.

Notes for Section 6.7 381
Now assume that m ^ 3 and that the assertion of the theorem is true
for m - 1. By condition (d),
5(/t_i) - 5(/t) = ay for k e {I, . . . , m - 1}.
Integrating the support function of the unit ball B, we obtain
V(Ko[m - k], K,[k - 1], B, &) - V(A:o[m - k - \], K,[k], B, (&)
= " ~ ^ av(C[m - 2], B", <&"),
which is independent of k. Thus, writing
i/(,) := V(A:o[m-l-fc], A:,[fc], fi,e^).
we have
^@) - V'(i) = V(i) - VB) - ... - v^„_2) - y(m-\)-
By the induction hypothesis, this implies that Ko and K^ are equivalent
by telescoping. ■
Notes for Section 6.7
1. As remarked above, for Ko, Kj € Xc the improved version of the Brunn-
Minkowski theorem is easily obtained; it says that V'„((l - d^Ko + ^K^) is a
concave function of & on [0,1]. As in the proof of Theorem 6.2.1, this
implies the inequalities
(n- 1I/@, - nVd,+ !/(„, go, F.7.13)
V^o)-2V^,^+V^2)^0 F.7.14)
for K,,, := V(Ko[n - i], Kj[i]); see Bonnesen & Fenchel [8], p. 94. If instead
of K^= K'{ one assumes only that
V„.,(K"o) = V„.,(K'{), F.7.15)
then, as described in Bonnesen & Fenchel [8], p. 95, a volume-preserving
symmetrization procedure transforms K0 = (I - d)Ko + &Ki into Kg such
that(K'o)" ={K'i)" and
V„(K'o) g A - »)V„(K') + W„(K\).
It follows that F.7.13) holds also under the assumption F.7.15). However,
V„(K'ff) need not be a concave function of &, so that F.7.14) cannot be
deduced if merely F.7.15) is assumed, contrary to an assertion made in
Bonnesen & Fenchel [8], p. 95. This error was pointed out by Diskant A984),
who constructed a counterexample.
2. The results and proofs of this section are taken from a paper (Schneider
1989b) that partially extends ideas of Favard A933a). The above
investigation, together with the remark that F.7.3) implies F.7.2), unifies and
generalizes several special results on linear improvements of inequalities for
mixed volumes scattered in the literature. These are obtained by different
methods. Some particular results are not covered by the above. We refer the
reader to Bonnesen & Fenchel [8], p. 99, Gcppert A937), Bol A939),
Hadwiger [18], F.4.4) and p. 279, and Dinghas A949a).
3. An isoperimetric problem of Hadwiger. As mentioned in Section 6.2, Note 7,
Hadwiger A968b) proved that the volume and surface area of a half-body
satisfy

382 Inequalities for mixed volumes
25" s n"(K„ + 2K„.t)V"„-\ F.7.16)
and that equality holds if K is the union of a half-ball of radius p, say, and a
right cylinder of height p attached to it. Hadwiger did not prove that equality
holds only in this case. The following proof of F.7.16) proceeds in a different
way and settles the equality case. Moreover, it could be generalized to
quermassintegrals other than the surface area.
As Hadwiger did, we consider, more generally, the class .1^5 of convex
bodies K with the property that K has a supporting hyperplane H such that
V„.,{K n H) S V„.,(K n H')
for each hyperplane H' parallel to H. Let K e Xs be given and let H be as
above. By Schwarz symmetrization with respect to a line G orthogonal to H
we obtain from K a convex body Ko with V„(Ko) = V„{K) and S(Kq)S
S(K); here strict inequality holds unless K \s a body of revolution with axis
parallel to G. Moreover, Koe Xg; hence Ko belongs to the canal class Xc,
where C = H C\ Ka is an (n - l)-dimensional ball, of radius, say, p. By Kj
we denote the half-ball of radius p with basis C and such that K^C H~,
where H~ is the halfspace bounded by H that contains Ko-
By F.7.10) and Theorems 6.7.2 and 6.7.3, with m = n and empty 6, we
have
(n - 1I/@, - "V'(„ + V^„y g 0,
with equality if and only if Ko and K^ are equivalent by telescoping. Now
V(o, = V„(Ko),
V,„,= V„(K,) = \k„p\
V^,,= - [ ^h(Ku u)AS„.,(Ko, u)
= -fAS(Ko) - K„-,p'-'].
n
The latter equality is easily seen if one chooses the origin at the centre of C.
With V„(Ko) = V„(K) =: V„ and S(Ko) ^S(K)=: S we obtain
(n - 1)V„ - p5 + p"(\k„ + K„.,) g 0.
Writing the inequality
(n - \)a - npb + p"c ^ 0 F.7.17)
in the form
, ^ n - 1 1
ph = a + — p"c,
n n
taking logarithms and using the concavity of log, we obtain b" Sfl""'c, with
equality if and only if it holds in F.7.17) and a = p"c. Hence, inequality
F.7.16) follows. If equality holds in F.7.16), then S = S(Ko), Ko and K^ are
equivalent by telescoping, and V„ = P"(\k„ + k„_,). The second condition
implies that Ko is, up to a translation, the union of K^ and a circular cylinder
attached to C, and the third condition implies that the cylinder has height p.
Finally, S = S(Ko) yields that K must be a translate of Ko-
6.8. Analogous notions and inequalities
In this section we describe, without proofs, a number of inequalities that
are in analogy, and often related, to the inequalities for mixed volumes.

6.8 Analogous notions and inequalities 383
either in their general form or in their specialized versions for quermass-
integrals. The results to be considered are of a rather different type, the
unifying viewpoint being simply that they are reminiscent, often
strikingly, of the inequalities for mixed volumes.
First we quote Aleksandrov's inequalities for mixed discriminants.
(For a definition of the mixed discriminant D{Ai,. . . A„) see Section
2.5, in particular B.5.32).)
Theorem6.8.1 (Aleksandrov). Let Ax, . . . , A„ be real symmetric n x n
matrices, where A2, ■ ■ . , A„ are positive definite. Then
D(Ai,A2,Aj, ...,A„y
m D(Au Au Aj, ..., A„)D(A2, A^, A^, . . . , A„). F.8.1)
Equality holds if and only if A^ = XA2 with a real number A.
Aleksandrov A938b) proved these inequalities and used them in his
second proof of the Aleksandrov-Fenchel inequalities for mixed
volumes.
The major part of this section will be concerned with the description
of various analogues of notions and inequalities from the classical
Brunn-Minkowski theory. There are counterparts to Minkowski
addition, mixed volumes, quermassintegrals and to the inequalities
connecting these. Often the analogies are surprising, although many of the
inequalities go in the opposite direction to that of their analogues, and
are easy consequences of familiar analytic inequalities (those of Holder,
Minkowski). But there are also some unsolved problems and interesting
connections with the affine geometry of convex bodies (see Section 7.4).
Most of the functionals of convex bodies considered subsequently will
not be translation invariant and will require the origin to be an interior
point of the body. We introduce, therefore, the class JC^ of convex
bodies that have the origin o as an interior point. A centrally symmetric
convex body with centre at the origin will be called centred. Two convex
bodies K, L e ^00 are said to be equivalent by dilatation ii K = XL with
some A > 0.
Some of the notions defined below can be applied to
lower-dimensional convex bodies and star bodies, but we do not treat these here.
First, we consider extensions or analogues of Minkowski addition. Let
p^l be given. For K,LeXoo and A,/i^O (A-i-/i>0) there is a
convex body A- /C +p //• L for which
h(X-K +pfi- L, •)" = XhiK, •)" + lihiL, •)". F.8.2)
Such p-sums were introduced by Firey A962). He proved the extended

384 Inequalities for mixed volumes
Brunn-Minkowski inequality
Wi(K +p £,)''/<"-'> ^ W,(/:)''/<"-'' + W,(L)''/<"-" F.8.3)
for / € {0, 1, . . . , n - 1} and p ^ 1. For p = 1 and / = 0, this is the
ordinary Brunn-Minkowski inequality. For p>l, equality holds in
F.8.3) if and only if K and L are equivalent by dilatation.
Lutwak A991+a) studied p-sums more systematically. He defined
mixed p-quermassintegrals by
W„i(K, L) ■■= hm
p '^ f-.o+ e
(/ = 0, . . . , n - 1), showed that
w^,,(a:, l) = -^L/'(i., uyh(,K, «)'-''d5„_,_,(A:, «)
and deduced F.8.3) in a new way. He was also able to prove an
analogue of Minkowski's existence theorem (Theorem 7.1.2) and a
corresponding uniqueness result.
If in definition F.8.2) one uses gauge functions instead of support
functions, then another combination of convex bodies is obtained. It was
introduced by Firey A961a). Let p ^ 1 be given. For K, L e ^oo and A,
fi^O (\ + fi>0), there is a convex body \* K +p fi* L) ior which
g(X*K+,^*L, ■)" = Xg(K, •)" + fig{L, -y. F.8.4)
Theorem 1.7.6 (together with Theorem 1.6.1) shows that
x*K +pn*L = (A« a:* +ph* l*)*.
Firey proved that
V„(K +, Lyl" ^ V^iK)-"'" + V„(L)-''/", F.8.5)
with equality if and only if K and L are equivalent by dilatation.
The special case p = 1 of F.8.4) was studied further by Firey
A961b), who showed that
Wi{K +1 £,)-'/<"-" ^ W,(A:)"^/<""'> + W,(L)-'/<"-'> F.8.6)
for / = 0, 1, . . . , n - 1, with the same cases of equality as in F.8.5).
Next, we consider some analogues of mixed volumes and quermass-
integrals. The known inequalities satisfied by these, for which analogues
are sought, include the following:
m
V(/C„ . . . , K^r ^ nv(^„ • ■ ■ . a:,, K„^x, ■■..K„) F.8.7)
1=1
for m = 2, . . . , n, which includes the Aleksandrov-Fenchel inequalities
F.3.1) and is deduced from these by induction (Aleksandrov A937b),
Busemann [12], p. 50); the cyclic inequality F.4.6), that is,
W,(A:)*-' ^ WiiKy-'Wk^K)'-', F.8.8)

6.8 Analogous notions and inequalities 385
valid for 0 ^ / < / < /c ^ n; in particular
K'„Wi(K)"-' g k'„Wi(K)"-' F.8.9)
for 0 ^ / < ;" < n; and the generalized Brunn-Minkowski inequality
WiiK + L)'/<"-" g W,(A:)'/<"-" + W,(i)^/<"~'> F.8.10)
for / € {0,. . . , n - 1}, a special case of Theorem 6.4.3.
Many of the formal properties of the mixed volume (multilinearity,
translation invariance, continuity and monotoneity) are shared by the
mixed width integral introduced by Lutwak A977). It is defined by
A(K„ ...,K„) := -^ l„.M^i, «) ■ ■ ■ b(K„, u)da{u)
for Ki, ..., K„eX", where b(K,u) denotes the width of K in
direction m € 5"'' and a is spherical Lebesgue measure. It follows from
H61der's inequality that
m
A(K„ ..., K„)'" S YlMK,, ..., Ki, K„^i, ...,K„)
1=1
for 1 < m ^ n. Lutwak also proved that
A(Ku...,K„r ^V(K^)--V(K„),
with equality if and only ii Ki, . . ., K„ are balls.
By specialization, one obtains the width integrals
B,(K) := A(K,..., K, B", . . . ,B")
n - i i
of Lutwak A975b). They satisfy
BjiK)'-- g B^(K)''->B,(Ky-'
for 0 ^ / < ;" < k = n, with equality if and only if K is of constant width;
they also satisfy
Bi(K + Lfl^"-') S B,(A:)'/<"-') + B,(L)^/<"-'>
for / € {0, . . . , n - 1}, and other inequalities.
In a similar way as for the mixed width integral, but using the radial
function p{K, •), one defines the dual mixed volume
V{Ku ...,K„) -^l^.A^^u ") ■ ■ ■ PiK„, u)da(u)F.8.11)
for Ki,. . . , K„ e Zoo. It was introduced by Lutwak A975a). By
specialization, one defines
Vi{K, L) := V{K,..., K, L, . . . , L)
n - i i

386 Inequalities for mixed volumes
and the dual quermassintegrals
W^(,K) := V,(,K, B") = - L,p(A:, w)""' da(«) F.8.12)
ft J^
for i = 0, . . . , n. The inequalities
m
V{K„ ..., K„r ^ n^(^'. ■■ ■ ' Ki, K„^„ ...,K„) F.8.13)
1=1
(l< m = n) and
Wj(K)''-' ^ Wi(K)''-'Wi,(K)'-' F.8.14)
@^i<j<k = n) are valid. Equality holds in F.8.13) if and only if
Ki, . . . , K„ are equivalent by dilatation, and in F.8.14) if and only if
K is a centred ball.
For the dual quermassintegrals W,, a representation (or inductive
definition) in analogy to Kubota's integral recursion is possible. This was
pointed out by Lutwak A975d). Recall that a special case of Kubota's
formula E.3.27) can be written in the form
W,(K) = —^ f W;_,(A:|//„,o)da(«) F.8.15)
for / = 1, . .. , n, where W/_i is computed in the (n - l)-dimensional
subspace Hu.o- In anology to F.8.15), the dual quermassintegrals satisfy
W
iiK) = —i— L..tv;.,(A: n //„.o)da(«), F.8.16)
where W,'_, is the (/- l)th dual quermassintegral in the (n - l)-space
fiu.o- Together with Wo(K) = V„(K), this can be used for a recursive
definition of the dual quermassintegrals. It is also true that
W„_,(K) = -^ f V,(K n p£,)dv(p) F.8.17)
(Lutwak A979a)) for /c = 1, . . . , n - 1, in analogy to
W„_k{K) = ^ j^^^^VdK\pEk)dvip) F.8.18)
(another special case of E.3.27)). Here, as in the following, £/t denotes
a /c-dimensional linear subspace of E".
Hadwiger [18], p. 267, has defined harmonic quermassintegrals by
W,iK) := V„(K), W„(K) := k„ and
-1
lv,(K\pE,)-'dvip)
F.8.19)
for K € XS and k = I, . . . , n - 1, and has shown that
Wi(K + D'A"-') g WiiK)^^^"-'^ + Wi(Ly^^"-'K F.8.20)
Lutwak A988a) has proved the inequality

6.8 Analogous notions and inequalities 387
K'„Wi(K)"-' ^ <#y(A:)"-'. F.8.21)
for 0 ^ i < 7 < n and K e X^; equality holds if and only if /C is a ball.
A special case is
ic'/"V„(A:)<"-'>/" ^ Wi(K). F.8.22)
For / = n-l, this is the harmonic Urysohn inequality proved by
Lutwak A975a), and for / = 1 it is the harmonic isepiphanic inequality
of Lutwak A984).
As special cases of more general averaging processes treated in
Lutwak A975c, 1984), Lutwak A984) proposed to define affine
quermassintegrals *o(^). *i(^)» ■ •■ > *n(^) for K eJCf, by taking
*o(A:) := V„{K), <P„iK) := k„ and, ioxO<k<n,
F.8.23)
*„_,(A:) := -J
\ V,{K\pEk)-''A^ip)
JSO{n)
Then
0i{K) g Wi(K) ^ W,(K),
by Jensen's inequality. Lutwak showed that
0,(K + L)^/<"-" ^ *,(A:)^/<"-" + *,(!,)'/<"-". F.8.24)
In analogy to F.8.9), Lutwak A988a) conjectured that
K'„0^(Kr-i g K'„0i(Kr-' F.8.25)
for 0 ^ / < 7 < n and K e Xo- A special case would be the inequality
ic'^"V„(Ky"-''>^" ^ ^i(K). F.8.26)
The cases / = n - 1 and i = 1 are true; they follow, respectively, from
the Blaschke-Santal6 inequality and the Petty projection inequality (see
Section 7.4), as noted by Lutwak.
The dual affine quermassintegrals, also proposed by Lutwak, are
defined, for K € JC^, by letting <>o(K) := V„(K), 0„(K) := k„, and,
forO< A:< n,
*„_,(/C) := ^ [/ V,(K n pE,r dv(p)
From Jensen's inequality,
Wi(K) g *,(^)-
Grinberg A986, 1991) proved the inequality
0i(K) g ^f V„(A:)<''-'>/'' F.8.28)
for 0 < / < n - 1, with equality if and only if /C is a centred ellipsoid.
For / = 1, this is the Busemann intersection inequality (see Section 7.4).
Grinberg A991) also proved that the affine quermassintegrals and the
dual affine quermassintegrals are, as the names suggest, invariant under
F.8.27)

388 Inequalities for mixed volumes
volume-preserving linear transformations (and the affine
quermassintegrals, of course, under translations).
Notes for Section 6.8
1. Aleksandrov's inequalities for mixed discriminants. Aleksandrov's A938b)
proof of F.8.1) (which is sketched in Busemann [12]) exhibits some formal
analogies to his proof of the inequalities for mixed volumes of strongly
isomorphic polytopes. No really simple proof seems to be known (the one
given by Schneider A966b) is erroneous, and the more general result stated
there does not hold). New interest in the inequalities for mixed discriminants
arose when Egorychev used a special case in his proof of van der Waerden's
permanent conjecture; this led to simpler proofs of that special case. We refer
the reader to the references given in the survey article of Lagarias A982).
Extensions of the inequalities F.8.1) are treated by Khovanskii A984) and
PanovA985a,b).
2. Other types of addition for convex bodies. Besides the combinations of convex
bodies defined by F.8.2) and F.8.4), other analogues of Minkowski addition
have been introduced and applied. These are radial linear combination and
radial Blaschke addition (Lutwak 1988b), harmonic linear combination and
harmonic Blaschke addition (Lutwak 1990b, 1991). For ordinary Blaschke
addition, see Section 7.1, Note 5. The process defined by F.8.4) was called
harmonic p-combination, and was further studied, by Lutwak A991+b).
3. Inequalities for integral means. Inequalities for power means of the width of a
convex body were treated by Lutwak A975c), and similar inequalities for the
brightness by Lutwak A984). Lutwak A987a) also obtained various
inequalities for the power means of quermassintegrals of projections of convex
bodies.
4. Quermassintegrals of polar bodies. For 0 = i, ; < n and ( + / g n - 1, one has
for K € Xoo- Equality holds precisely for centred balls. For i + j = n - I, this
is due to Firey A973) and for / = n - 1 to Lutwak A975a), and the general
result was deduced from Firey's case, independently by Heil A976) and
Lutwak A976). Firey A963) proved the sharp inequalities V{K, K*) § ki for
n = 2 and ViK, K*, B^) g Kj for n = 3.
For the problem of the infimum of Wq{K)Wo(K*), see Section 7.4,
Note 13.
Heil A976) remarks that for n > 3 the minimum of Wi(K)Wi{K*) is not
attained by balls.
5. Mean dual affine quermassintegrals. In analogy to F.8.27), one may also
define
f llAn + l)
y,(K n £)'-' dti,iE)\
JA{n.k) J
K„
for 0<it<n and K e Xq. Then 4>„_t(/C) is invariant under unimodular
affine transformations of K. Among convex bodies of given positive volume,
precisely the ellipsoids attain the maximal value of *„_*■ This was proved by
Schneider A985b), Theorem 1, second part.

Selected applications
The Brunn-Minkowski theorem and the inequalities for mixed volumes
are useful in many geometric applications. Here we select only a few of
them. Minkowski's existence theorem, to be treated in the first section,
is, strictly speaking, not necessarily an application of the inequalities for
mixed volumes, but it is a classical result of the Brunn-Minkowski
theory. The corresponding uniqueness result, on the other hand, is a
consequence of the equality condition in Minkowski's inequality and,
thus, in the Brunn-Minkowski theorem. The Aleksandrov-Fenchel
inequalities are a powerful tool for obtaining extensions of Minkowski's
uniqueness result; these are the subject of Section 7.2.
The rest of the chapter will be devoted to some inequalities in the
proofs of which the Brunn-Minkowski theorem or the Minkowski
inequalities have been used (as well as other techniques such as
symmetrization). However, this is only a superficial aspect, and the
main reason for presenting these inequalities and the notions involved is
that they belong to an interesting and promising part of the geometry of
convex bodies; this is the theory of affine-invariant constructions and
functionals.
7.1. Minkowski's existence theorem
If cp is the area measure 5„_i(/C, •) of the n-dimensional convex body
K € JC^, then
|^..,.,Md()p(M) = 0, G.1.1)
which is a special case of E.1.28). From the geometric meaning of the
area measure it is clear that it cannot be concentrated on any great

390 Selected applications
subsphere of S"~^. Minkowski's existence theorem says that these trivial
necessary conditions are also sufficient in order that (p be the area
measure of a convex body K e Xo (by Theorem 7.2.1, this convex body
is unique up to a translation). This beautiful and useful result was
proved by Minkowski for special cases, in which, however, the essential
ideas were already contained; it was later extended independently by
Aleksandrov and by Fenchel and lessen, when the full notion of area
measure was available.
Although Minkowski's existence theorem is not an application of the
inequalities for mixed volumes, its proof is motivated by Minkowski's
inequality. In fact, if K e J^") with S„-i(K, •) = qf> exists, then any
convex body L e Xq satisfies
-^|^„.,/«(L, u)d<piu) = V(L, K,...,K)^ V„(LI/''V„(A:)'-V«,
with equality if and only if L is homothetic to K. Thus, up to
homothety the convex body K is characterized by the fact that it
minimizes the functional
L^l^_^h(L,u)d<piu)
under the side condition V„(L) = 1.
Following Minkowski A897, 1903), we first treat the special case of
polytopes.
Theorem 7.1.1 (Minkowski). Let Ui,. . . ,Uf^ e S"~^ be pairwise distinct
vectors linearly spanning E", and let fi, . . . , f/^ be positive real numbers
such that
N
IfiUi = o. G.1.2)
(=1
Then there is a polytope P e^o having (precisely) Mj,. . . ,mjv os its
normal vectors and satisfying
V„^,(F(P, ud) = /, G.1.3)
fori=l,...,N.
Proof. For an /V-tuple A = (ai,. . . , a^j) e U'"^ with non-negative
components o-j, . . . , ar,v we define
N
P(A):= n «»',.«,•
(=1
From lin {«!,..., Mjv} = E" and G.1.2) it follows that pos {«!,...,
Ufj} = E"; hence P(A) is a polytope, with o € P(A). Its normal vectors

7.1 Minowski's existence theorem 391
are among the vectors mi, . . . , u/^, and
h(P(A), Ui) ^ or, for / = 1, . . . , A^,
with equality if V„-i(F(P(A), «,)) > 0. Let
M := {A e R'^Iot; s O, V„(P(A)) ^ 1};
then M is closed, since V„(P{A)) depends continuously on A. The
linear function 0 defined by
1 '^
^^) ■= — ^Oiifi, A = (OTi, . . . , OTyv),
n ,=1
attains a minimum on M, because /, > 0 for all /. Let //""' be this
minimum and suppose that it is attained at A* = (a*, . . . , a%y, put
P(A*) =: P*. We assert that nP* solves the problem.
By G.1.2), <P{A) is not changed under a translation of P(A), hence
we may assume that P* has o as an interior point and, hence, that af,
. . . , a%>0. Write
V„-,{F(P*, «,)) := ft for / = 1, . . . , N.
We have V„(/'*) = 1, since otherwise XP* with suitable A<1 would
satisfy V„(XP*) ^ 1 and yield a value less than ii"~^ for the function <P.
From
V„(/'*) = -2/t(/'*, M,)/f
n ,=1
and h(P*, «,) = arf if /f + 0 it follows that
1 ^
-lc^fff=l. G.1.4)
n ,=1
Since ^>l*) = /i"~^ we have
1 '^
-l^ffi^f^"-'- G.1.5)
n ,=1
Now consider the hyperplanes in R'^ given by
f 1 '^
\ " (=1
//2:=|>l€(R'^|i-2ar/f=l|.
From G.1.4) and G.1.5) it follows that A*e HiCi Hi- Suppose that
A = (or,,. .. , ota,) € Hi, where ori,. . . , ar,v ^0. Then V„{P{A)) ^ 1,
since otherwise //""' would not be the minimum of * on M. For
0^i?Sl we have {\ --S^A* + ^A e Hi- Further, a point y g
A - i?)/'* + i?/'(>l) satisfies
(y, Ui) ^ A - i?)arf + i?ar, for i = 1, . . . , A^,

392 Selected applications
hence
A - ff)P* + &PiA) C Piil - &)A* + dA).
It follows that V„((l - d')P* + '»P(A)) ^ 1 for 0 ^ r?^ 1, which together
with V„{P*) = 1 gives
ViPiA), P*,..., P*) ^ 1. G.1.6)
Since a*, . .. , a% > 0, there is a neighbourhood f/ of >l* in R'^ such
that, for A e HyCi U, the polytope P(A) satisfies o-j, . . . , o-jy > 0 and
V„-,(F(P(A), u.)) > 0 if ft >0.
If this holds, then h(P(A), «;) = a,, hence
1 '^
V(P(A), p*,...,P*) = - 2aiff.
n ,=1
Inequality G.1.6) now shows that //i n U Q //J, where //i" is one of
the two closed halfspaces bounded by Hi- Since A*e HiH Hi, we
conclude that Hi = Hi and hence that
f, = ii"-'n= v„.,iF(,iip\u^)
for i = \,. . ., N. This completes the proof that fiP* is the desired
polytope. ■
The general case of Minkowski's existence theorem will now be
obtained by approximation.
Theorem 7.1.2. Let tp be a measure on 7i(S"~^) with the properties
l^_^ud<piu) = o G.1.7)
and (p(s) < <pE"~') for each great subsphere s of 5""'. Then there is a
convex body K e Xo for which 5„_i(/C, ■) = q>-
Proof. For each /c € N, we decompose the sphere S"~^ into finitely
many pairwise disjoint Borel sets of diameter at most l/k and with
spherically convex closure. We fix k and denote by Ay, . . . , A/^ those
sets of the decomposition on which q> is positive. For / = 1, . . . , A', let
then c, 4 o since Aj lies in an open hemisphere, and hence c, = /,«, with
M.eS""' and f>0. Observing that c,€convZ\,, one finds that
1 - Bfc2)-i g /. ^ 1. For (o € 7iE"-'), let
(Pki<o) ■•= S (fi^df-

7.1 Minowski's existence theorem 393
Then qp^ is a measure on '3(S"~^) satisfying
j^„.," d<p/t(M) = j^^_,Md(p(M) = o.
For g € C(S"~^) we have
/,„.,«(«)d<?>/t(«) - /^„.,g(«)d,p(«) = ij^iMud - g(u)]d^u)
and
\Mud - g(u)\ ^ \g(ud - g(u)\ + \\g\\/2k\
If u e Aj, then |m, - m| ^ l//c, since m, epos4,. From the uniform
continuity of g we deduce that jgdqpk -> jgdfp for /c -» «», hence
(pk -^ (p for /c -» 00.
There is a number a > 0 such that
j ^_^{u,v)-'d<p{u)^a for ye 5""'
(where the superscript plus denotes the positive part), since the integral
is a continuous function of v that is positive by the properties of q>. The
estimate above shows that
uniformly in v. Hence, there exists ko such that
I..
g„.,{u, v)''<i(Pk(u) > —
for fc ^ fco and all v € S"~^. Thus, for k ^ k^, the measure (pk is not
concentrated on any great subsphere.
By Theorem 7.1.1, for k^k^ there is a polytope Pk^^Po with
S„-i(Pk, •) = (Pk and, without loss of generality, oePk- Since
<Pk(S"~^) = <p(,S"~^), the surface areas of the polytopes Pk remain
bounded; hence, by the isoperimetric inequality, V„(Pk) = b with some
constant b. Let x e Pk and write x = \x\v with y € 5""'; then
/j(/'/t, m) ^ /j(conv{o, x}, m) = |x|(m, v)'^
for M € 5""' and hence
b ^ V„(Pk) ^ ^ l^Ju, u)-dcpk(u)^^j-,
thus PkC B(o,2nb/a). By the Blaschke selection theorem 1.8.6, there
is a subsequence of (Pk)ksk„ converging to some convex body K eX".
From the weak convergence of the sequence ((pk)kskc, to (p we conclude
that 5„_,(A:, ■) = (p. U

394 Selected applications
Minkowski's theorem makes it possible to define a new addition for
n-dimensional convex bodies: For convex bodies Ko, /C, e Xo, the sum
of their area measures satisfies the conditions of Theorem 7.1.2; hence
there exists a convex body M for which
5„_,(A/, ■) = S„_r(Ko, •) + 5„_,(A:„ •)■
By Theorem 7.2.1, the body M is unique up to a translation, and we
may assume that it has its area centroid at the origin. This body M is
called the Blaschke sum of Ko and Ki and is denoted by /Co# /Ci. For
Blaschke addition, there is a counterpart to the Brunn-Minkowski
theorem:
Theorem 7.1.3. Let Ko, Ki^Xq and A e @,1) and let K^x] be a convex
body for which
5„_,(a:[,], •) = (!- AM„_,(a:o, •) + A5„_i(a:„ •)■
Then
Vni.Ky.^Y"-'^!" S A _ K)V„{Kof''-'^l" + AV„(A:0<''-'>/"- G.1.8)
Equality in G.1.8) holds if and only if Ko and Kx are homothetic.
Proof Using Minkowski's inequality F.2.2) in the form
v„(a:[,,)-i/''v(a:(„, K[n - 1]) ^ v„(a:)(''-i)/",
we obtain
v„(a:,,,)<''-')/'' = v„(a:(„)-i/'' -^|^„.,/«(a:[,], «)d5„_,(/C[„, u)
= V„(A:„])-'/''(A - A)V(A:„], Ko[n - 1])
+ AV(A:,„, K,[n - 1])]
^ (I - A)v„(a:o)<''-'>/'' + Av„(a:,)<"-'>/",
together with the assertion on the equality sign. ■
Notes for Section 7.1
1. The polytopal case of Minkowski's existence theorem, Theorem 7.1.1, is due
(for n = 3) to Minkowski A897); see also Minkowski A903), §9. We have
essentially followed the presentation of the proof given in Bonnesen &
Fenchel [8], §60. Related, but slightly different versions of the proof have
been given by, e.g., Aleksandrov [2), Chapter VII, §2, and McMullen A973),
§7. Aleksandrov A939a) found a different proof, using his so-called 'mapping
lemma', an application of the Mnvariance of domain'. This proof appears also
in Aleksandrov [2|, Chapter VII, §1, and Pogorelov A975), §2. Similar
existence results for unbounded polyhedra can be found in Aleksandrov [2],
Chapter VII, §3.

Notes for Section 7.1 395
Minkowski A903), §10, extended his result to convex bodies more general
than polytopes, namely to convex bodies K for which, in present-day
terminology, the area measure 5„_i(/C, •) has a continuous density with
respect to spherical Lebesgue measure.
Once the notion of the area measure had been introduced, the general
version of Minkowski's result, Theorem 7.1.2, could be deduced from the
polytopal case by means of approximation, using the weak continuity of 5„_i.
This was done by Fenchel & Jessen A938) (above we followed their
approach; see also Fenchel A938)) and later by Aleksandrov A939b).
Independently of Fenchel and Jessen, Aleksandrov A938a), §3, gave a proof
that generalizes Minkowski's approach via an extremum problem, but in the
general case the necessary variation argument (which uses Lemma 6.5.3) is
less elementary. A similar attempt, but with an insufficient variation
argument, was made by Suss A931a). Related existence theorems for infinite
convex surfaces are stated without proof in Aleksandrov [2], p. 305-6. A
similar existence result for convex caps was obtained by Busemann A959).
2. Minkowski's problem in differential geometry. If /C is a convex body of class
C+, then its area measure 5„_i(/C, •) has a continuous density (with respect
to spherical Lebesgue measure), given by 5„-i, the product of the principal
radii of curvature. Its reciprocal value is the Gauss-Kronecker curvature, as a
function of the outer normal. Thus Minkowski's existence theorem is related
to the problem of the existence of a convex hypersurface with Gauss-
Kronecker curvature prescribed as a function of the normal vector.
Classical contributions to this problem are the papers by Lewy A938), Miranda
A939) and Nirenberg A953). If one wants to utilize the general existence
result 7.1.2 to solve this problem, one needs regularity results for convex
surfaces with sufficiently smooth curvature functions. For such results and
for more information on Minkowski's problem and related problems from the
viewpoint of differential geometry, we refer to the books of Pogorelov ([27],
1975), the survey article of Gluck A975), and in particular to Gluck A972)
and Cheng & Yau A976).
3. Algorithmic version. The essential idea of the proof of Theorem 7.1.1 is the
minimization of the linear functional 0 over the region MdW^, which is
convex by the Brunn-Minkowski theorem. Known iterative methods for such
optimization problems can therefore be used to derive algorithms for a
constructive solution of Minkowski's problem for polytopes. Such an
algorithm in E', of interest for computer vision problems, was described by Little
A983). The role of possible applications to robot vision is discussed in Horn
A986), Chapter 16.
4. Applications of Minkowski's existence theorem in stochastic geometry appear
in Schneider A982a, 1987a), Weil A988) and Wieacker A989).
An elegant application of Minkowski's theorem, Blaschke addition and
Minkowski's inequalities to a problem in discrete geometry is made in a paper
by Boroczky, B^r^ny, Makai jr. & Pach A986); another application is in F^ry
&Makaijr. A982).
5. Blaschke addition. The addition of convex bodies now called Blaschke
addition is mentioned briefly in Blaschke [5], p. 112. For the case of
polytopes it had occurred already in the work of Minkowski A897), p. 117.
Theorem 7.1.3 was proved by Kneser & Siiss A932); weaker forms had been
proved previously by Herglotz (unpublished) and Siiss A932a).
Blaschke sums were investigated and used by Firey & Griinbaum A964),
Firey A965a, 1967a, c), Grunbaum [15], Chapter 15.3, Schneider A967c,
1987a), Chakerian A971), Kutateladze & Rubinov A969), Kutateladze A973,
1976), Goikhman A974), Bronshtein A979), Goodey & Schneider A980),

396 Selected applications
McMullen A980) and Lutwak A986b, 1987b, 1988b, 1991). An addition of
Blaschke type is also used in Bronshtein A978c).
6. The existence problem for the area measure of order i. Necessary and
sufficient conditions for a measure <p on ^E""') to be the ith area measure
S,{K, •) of some convex body K ^IC", where 1 < (" < n - 1, are not known.
The necessary condition
is not sufficient, even if (p has an analytic density, as shown by Aleksandrov
A938a). There is a complete solution of the existence problem for the special
case of sufficiently smooth bodies of revolution due to Firey A970c) and in a
special case to N^denik A968). Some necessary conditions follow from results
of Firey A970a) and Weil A980a) and from Theorem 4.6.3. The latter,
together with a result of Larman & Rogers A970) (applied to the polar body)
implies that the support of an I'th area measure is arcwise connected if
(< n- 1 (Fedotov 1982).
A special case of the existence problem asks whether the sum of two ith
area measures is always an (th area measure, where i e {2, . . ., n - 2). This
question was posed, more or less explicitly, by Firey A967a, 1970b,c, 1975)
and Chakerian A971). Negative answers were given by Fedotov A979b, 1982,
and, in a weaker form, 1978b) and independently by Goodey & Schneider
A980). The latter authors showed that even suitably chosen parallelepipeds
with parallel edges provide counterexamples; here Theorem 4.6.4 had to be
used. They also proved: Let P, Po, Pi e ''Pq be convex polytopes, i e
{1, . . . , n - 1}, and suppose that
5,(P, •) = 5,(Po, ■) + Si(P„ ■).
If n denotes orthogonal projection onto any (i + l)-dimensional plane, then
nP is the Blaschke sum (relative to the carrying plane) of nPo and ttPj.
Since no general criterion for (th area measures is known if (' e {2,. . . ,
n -2), it is of interest to investigate qualitatively the set Si of all (th-order
area measures of convex bodies in .Xg. This was done by Weil A980a) and
Goodey A981). Weil showed that, for / < n - 1, Si is not dense in -//?, the set
of all finite measures on '3E""') with barycentre o, endowed with the weak
topology, but that Si - S, is dense in -^ - Jff- Goodey proved for i < n - 1
that cV, - Si is of first Baire category in the Banach space Jff - jH with the
total variation norm. Goodey also investigated weak limits of /th-order area
measures.
7. Mixed bodies. For convex bodies Jf i,.. . , Jf„_i e ICS, the mixed area
measure S(Ki,.. ., K„-\, •) satisfies the assumptions of Minkowski's
theorem; hence there is a convex body [Ki,. . . , K„-i], unique up to a
translation, for which
■^n-ld-S^I' ■ • ■ ' ^n-lj> ■ ) = S{Ki, . . . , K„-i, • ).
[Ki,.. . , K„-i] is called the mixed body of Ki, . . . , K„-i. Mixed bodies
were introduced, in special cases, by Firey A967a), and were extensively
studied by Lutwak A986b), who obtained a number of inequalities for these
bodies.
8. Existence problems for curvature measures. Existence problems for the
curvature measures C, have apparently only been considered for the case
1 = 0. For a convex body K e X" with o e int K, let f(u) := p{K, u)u for
u e 5"-', so that f(u) ebdK.
Theorem. Let k be a measure on 1i(S"'^). There exists a convex body
K e JC" with oexnXK for which Co{K, f{a))) = *c(w) for w e ^E"-') if and

7.2 Uniqueness theorems for area measures 397
only if the following two conditions are satisfied:
(a) *cE"-') = 555"-'E"-'),
(b) KE"~'\a)) > ^""'(w*) whenever wC 5""' is spherically convex and w*
denotes the set polar to w.
The body K is unique up to a dilatation.
Existence was proved by Aleksandrov A939a), first for polytopes and then
by approximation in general. The polytopal case is an existence result for
polytopes with vertices on given rays through o and preassigned curvatures at
these vertices. Uniqueness up to a dilatation was shown in Aleksandrov
A942a).
Analogous results for unbounded convex surfaces and orthogonal
projection onto a plane were obtained by Aleksandrov A942a) and [2], Chapter IX.
Oliker A986) treated, by means of a variational method, an existence
problem similar to Aleksandrov's theorem.
The theorems of Minkowski and Aleksandrov are existence results, which
a priori give no additional information on the convex bodies whose existence
they ensure. It is an interesting problem to find conditions on the given
measures from which conclusions on the shape of the solution bodies can be
drawn. For Aleksandrov's theorem, such a result was obtained by Treibergs
A990). For w C 5""' and given or, 0 < or < 7r/2, let w^ be the set of all points
of 5""' at spherical distance S a from w. Treibergs showed that if in
Aleksandrov's theorem stated above one assumes in addition that
*c(w) S ^"-'(wj forall we7:iE"-'),
then the body K satisfies
%n^{p{K,u)\u 6 5"-'}^
; ;—s c(a, n)
\ni{p{K,u)\ueS''-')
with a real constant c(a, n) depending only on a and n.
7.2. Uniqueness theorems for area measures
The theory of mixed volumes is a powerful tool for obtaining some
uniqueness theorems for convex bodies in the style of differential
geometry. For convex bodies of class C\ the assumptions of these
theorems can be formulated in terms of elementary symmetric functions
of principal curvatures or radii of curvature. In the general case they
correspondingly involve curvature measures or area measures. In the
present section, we prove some of these results and state others without
proof.
We start with the uniqueness assertion for the Minkowski problem.
Although this theorem will be improved and generalized by later
theorems, we give its formulation and proof separately, to show in a
basic example the close connection with results on mixed volumes.
Theorem 7.2.1. If K, L e JCq are convex bodies with
S„-:(K, •) = 5„_i(L, •), G.2.1)
then K and L are translates of each other.

398 Selected applications
Proof. We integrate the support function h{K, •), using G.2.1) and
E.1.18), and apply Minkowski's inequality F.2.2) to obtain
V„{KY = V{K, L,...,LY^ V„{K)V„{LY-\ G.2.2)
Thus V„{K) g V„(L), and analogously we obtain V„{L) ^ V„{K).
Hence, equality holds in G.2.2), which implies that K and L are
homothetic. Because of G.2.1), they must be translates. ■
Suppose that K and L are convex bodies of class C\. Then the
assumption G.2.1) is, by D.2.20), equivalent to the assumption that the
function 5„_i, the product of the principal radii of curvature as a
function of the outer unit normal vector, is the same for K and L.
Equivalently, the Gauss-Kronecker curvatures of the convex hypersur-
faces bd K and bd L coincide at points with the same outer normal
vector. This is the classical assumption for the uniqueness part of
Minkowski's problem in differential geometry.
Along with any uniqueness assertion comes the question for stability:
if the assumption enforcing uniqueness is only satisfied approximately,
can one ascertain approximate uniqueness? One would want to have
explicit estimates, implying the uniqueness assertion, and sharpening it
in a quantitative way. The following stability version of Theorem 7.2.1 is
due to Diskant A972). Recall that JC"{r, R) denotes the set of convex
bodies in E" with inradius at least r and circumradius at most R.
Theorem 7.2.2. Let 0<r<R. There exist numbers Eo>0 and y,
depending only on n, r, R, with the following property. If K, L e
DC"{r, R) are convex bodies satisfying
\S„.,(K, (o) - 5„-,(L, co)\ S e G.2.3)
for (oe^(S"~^) with some e e[0, eo], then
6{K, L') ^ ye'/" G.2.4)
for a suitable translate L' of L.
Since the exponent 1/n in G.2.4) is probably not optimal, it is of little
interest to look more closely at the size of the constant y. Similar
remarks refer to the stability assertions made below.
Part of the proof will be stated as a lemma, in a more general form
since this will be needed later. We assume that a number m e {2,. . . ,
n) and convex bodies K, L are given and we use the notation F.4.4)
with an (n - m)-tuple of balls, that is
V(,) := V(K[m - i], L[/], B"[n - m]).

7.2 Uniqueness theorems for area measures
399
Lemma7.2.3. If K, Le X"{r, R) and
\S„-,(K, (o) - 5„_,(L, (o)\ ^ e
for all (o e C3E""') with some e^O, then
2R
2R
G.2.5)
G.2.6)
G.2.7)
Proof By E.1.18),
V@) - V(„_,) = - L/t(A:, u)d[S„_,(K, u) - S„.y(L, u)].
n -^^
Using the Hahn-Jordan decomposition of the signed measure
S„.i{K, •) - S„-i(L, •), we deduce from G.2.5) the first inequality of
G.2.6), and the second one is obtained similarly. By F.4.5) with / = 0,
j = m - I, k = m we have V(^_i)^ Vl'm)'^(O) (and similarly the left-
hand inequality in G.2.7)). We conclude that
[V, VI'"
+
(m)
@)
1/m
@)
(m-1)
[V(m-X) - V@)] + [V(,)
(m)\
^(f-)^^
Proof of Theorem 7.2.2. We use inequality F.2.12), which states that
r{K, L) ^
(")]
i/i/C-i)
, G.2.8)
and Lemma 7.2.3 for m = n. In the following, c,,...,C6 denote
constants depending only on n, r, R. By G.2.6), V(„) - V(,) ^ Ci£,
hence
Jii
C)
lAn-l)
^ A - C2£)
l/(n-l) > 1 _
^ 1 - C2£
1/n
provided that e^Ei := min {1, l/ca}. By G.2.7),
Using x*" - y*" ^ px''-^(x - y) for X ^ y ^ 0 and p > 1 (Hardy, Little-
wood & P61ya 1934, p. 39), we obtain
Now G.2.8) yields
r(K, L)^l- CsE^I" =: or.

400 Selected applications
Since K and L may be interchanged, we also have r{L, K)^ a. By the
definition of r(L, K), there is a translate L' of L such that aK C L'.
Similarly, there is a translate K + t of K such that L' C a~^(K + /).
Thus a=l, where the trivial case or = 1 can be excluded. The bodies
aK and 0-"^^ + 0 are homothetic. Choosing the origin at their centre
of homothety, we have / = 0, thus aK C L' C a~^K, and hence o e K.
Now
K C or-'L' = L' + ^—-^ L' C L' + ^-^-^2RB"
a a
and similarly L' C K + [A - a)/a]2RB". This yields
a
if £^ £0 ■= niin {e, (Zcs)""}, which implies a^ 1/2. ■
The uniqueness assertion of Theorem 7.2.1 can be generalized
considerably, namely to certain mixed area measures.
Theorem 7.2.4. Let a number m e {2,. . . , n} and an (n - m)-tuple
6 = (/Cm+i, . . . , K„) of smooth convex bodies be given. If K, L e X"
are convex bodies of dimension = m satisfying
S(K[m - 1], 6, •) = S(L[m - 1], 6, •), G.2.9)
then K and L are translates of each other.
Proof The assumption G.2.9) says that condition (d) in Theorem 6.4.4
is satisfied. By Theorem 6.6.9, this implies that K and L are
homothetic. By G.2.9), they must be translates. ■
The special case K„+i = . .. = K„ = B" gives the Aleksandrov-
Fenchel-Jessen theorem:
Corollary 7.2.5. Let m e {2,. . . ,n). If K, L eJC" are convex bodies of
dimension ^ m satisfying
5„_,(A:, •) = 5„-,(L, •). G.2.10)
then K and L are translates of each other.
A special case is the differential-geometric version: If K and L are
convex bodies of class C+ such that, at points with the same outer
normal, the (m - l)th elementary symmetric functions of the principal
radii of curvature are the same, then K and L differ only by a
translation.

7.2 Uniqueness theorems for area measures 401
For the cases m = n and m = 2 of Corollary 7.2.5 we already know
stability versions (Theorems 7.2.2 and 4.3.5). We shall now prove a
stability result for the general case.
Theorem 7.2.6. Let m e {2,. . . , n) and positive numbers £o> '"> ^ be
given. If K, L e X"(r, R) are convex bodies satisfying
|5„_i(A:, (o) - S„_i{L, (o)\ ^ e G.2.11)
for all (o € ^E"~^) with some e e [0, £o]. tf^en
6iK,L')^Y£'' with q = ^ r-
'^ (n + 1J""-^
for a suitable translate L' of L, where the constant y depends only on n,
£o, r, R.
The proof makes use of Lemma 6.6.4, for which we did not
reproduce the proof, but the argument involving mixed volumes will be
given completely.
Proof. In the following Ci, . . . , C15 denote constants depending only on
n, £o> r, R. From Lemma 7.2.3 we have
Interchanging K and L we obtain
y (m-l) = ^ {„) •'@) + <^1^'
and multiplication leads to
V(i)V(„_,) ^ V@)V(„) + C2E
and thus
i^ ^ la + e,e. G.2.12)
Combined with the Aleksandrov-Fenchel inequalities, this yields
+ Cr,e.
YM.<:Yj11<: < JVzil <: Za
V(i) VB) F(„) - V(,)
For / € {1, . . . , m - 1} we deduce that
and thus
V?,)^ V(,_,)V(,>,) + C4£. G.2.13)
We use inequality F.4.9) with the choices Ko= K, Ki = L,

402 Selected applications
(/C3, . . . , K„) =■ (K, . . . , K, L, . . . , L, B" . . . , B")
m - \ - j /-I n - m
and the notation
V(i) := V{K[m - 1 - ;], L{j], B''[n - m + 1])
to obtain
^ [yl) - ^(;-i)^0>i)]
X [Ff,_,) - V(,_i)V(A:[m - 1 - /], L[i - 1], B"[n - m + 2])]
= c^e,
where G.2.13) has been used. This leads to
^ c,ei/2
for / = 1, . . . , m - 1, from which we deduce that
Thus we have shown that inequality G.2.12) implies the inequality
^J^HzlL^YM + cs^n, G.2.14)
This is the same as G.2.12), except that on the left-hand side one
argument L has been replaced by B" and on the right-hand side one
argument K has been replaced by B"; furthermore, the error c^e has
been replaced by c-]E^^. The derivation of G.2.14) from G.2.12) applied
only general inequalities for convex bodies and not the assumptions of
the theorem. Hence, we can repeat the argument and finally arrive at
^ ^^ + c«£<i/2r
V02 Vn
with Vij = V{K[i\, L[j], B"[n - i - /]), which implies
V?i - V20V02 ^ C9£<'/2)"-\
From Theorem 6.6.6 and Lemma 6.6.4 we conclude that
d{K, L) %. CioE* with q = 7,
V ' / "^ („ + lJ'"-2
and where K^\K- s(K)]/b{K).
Replacing L by a translate, we may assume that s(K) = s(L). Let

7.2 Uniqueness theorems for area measures 403
b(K)ML) := A; then d(K, XL) ^ a with a = c,o6(A:)£«. We have
AL C a: + orB" C A + Cua)K,
and from G.2.11) we obtain |V„_, o - V'o.m-il = ^/n, hence
A""-'Vo.„-, ^ A + CHar)""-'V„_i,, ^ A + Cnar)'"-'(Vo.„-i + e/n).
This yields A ^ 1 + Ciac* and therefore
K C XL + aB" C L + c^iS^B".
Similarly we obtain L C /C + che''B" and thus 6{K, L) ^ Cist*. ■
The theory of mixed volumes is flexible enough to permit also
applications to non-closed convex hypersurfaces. As an example, we
prove a version of the Aleksandrov-Fenchel-Jessen theorem for convex
hypersurfaces with boundary and spherical image in a hemisphere.
Let a unit vector u e 5""' and an (n - l)-dimensional convex body
CC Hu.o be given. By a convex hypersurface of the canal class Xc we
mean a point set F in E" that by orthogonal projection to //^o is
mapped bijectively onto C and is such that the set
K(F) := {x - Xu\x € F, A g 0}
is convex. Let Qu '■= {x e S"~^\{x,u) >0}. We define the yth-order
area measure of F by
Sj(F, (o) := Sj(K(F), (o) for w € ^(i3;t)-
Thus Sj(F, ■) is defined on the half-sphere Qu- Observe that we do not
demand that the hypersurface F be tangential to the right cylinder with
cross section C. Thus the relative interiors of two convex hypersurfaces
of the canal class Xc may well have different spherical images.
Theorem 7.2.7. Let Fo, Fi be convex hypersurfaces of the canal class
Xc satisfying
•^m-i(^o> •) = 5„_,(Fi, •)
for some m e {2,. . . , n). Then there is a translation parallel to u
carrying Fo into Fj.
Proof We choose XeR with Fo, F, CHJ^a and put /C, : =
K(Fi) n Hu,x for / = 0,1. Then K(, and /C, are convex bodies belonging
to the canal class Xc defined in Section 6.7. We have
5„_i(A:o, w) = S„.i(Ku w) for Borel sets (o C 5"-'\//„.o. This holds by
assumption if wC i2„. For u> C Q~ ■= {x e 5"~'|(x, m) <0} it is true
since F(Ko, u) = F(Ki,u) for u e Qu- Thus condition (e) in Theorem
6.7.2 (with K„+i = . . . = K„ = B") is satisfied. By Theorem 6.7.3, K^
and Ki are equivalent by telescoping. This proves the assertion. ■

404 Selected applications
We turn now to the extension of some classical differential-geometric
characterizations of balls. First, let K he a convex body of class C\ and
assume that, for some ; € {1,. . . , n - 1}, the y'th elementary symmetric
function of the principal radii of curvature is constant. Then K is a ball.
This well-known result can be generalized as follows. The assumption
Sj = a with a constant a gives, by integration using D.2.20), that
Si(K, (o) = aZ'"-\(o) for w € ^E""'). G.2.15)
Since X"-^((o) = Sj(B",(o) for we ^E""^), condition G.2.15) is
equivalent to Sj(K, •) = Sj{a^l'B", •)■ By the Aleksandrov-Fenchel-Jessen
theorem, this assumption implies that /C is a ball of radius a^^'. This
conclusion, namely that a convex body K e Xo satisfying G.2.15) with a
constant a must be a ball, holds for arbitrary K, and this fact
generalizes the classical differential-geometric result.
There is also a stability version of the latter uniqueness result (where
assumption and conclusion are stronger than in the corresponding case
of Theorem 7.2.6).
Theorem 7.2.8. There exist numbers £o '^'^^ 7. depending only on n, with
the following property. If K e X" is a convex body satisfying
1 - e^ '^ -gl + E
for each (o e Ti{S"~') with X"~^((o)> 0 and for some e e [0, £o]. then K
is contained in a ball of radius 1 + ye^/*""') and contains a ball of radius
1 _ y£l/(n-l).
This was proved by Diskant A971a). The proof uses symmetrization
and will not be given here, as it is not closely related to the theory of
mixed volumes.
The differential-geometric result mentioned above has a counterpart
where radii of curvature are replaced by curvatures. If /C is a convex
body of class C^ for which the function //,-, the /th (normalized)
elementary symmetric function of the principal curvatures, is constant
for some /€{l,...,n-l}, then /C is a ball. If we want to formulate
the condition
Hi = a G.2.16)
with a constant a- in a way that makes sense for arbitrary convex bodies,
we have (at least) two possibilities. By D.2.19), condition G.2.16) (for
K of class C^) is equivalent to
c„-x-i{K, p) = aX"-'@ n bd a:)

7.2 Uniqueness theorems for area measures 405
and therefore, in view of D.2.23), to
C„_i_,(A:, •) = aC„-,(K, •)• G.2.17)
But if K is of class Cl, then by B.5.28) the condition G.2.16) is
equivalent to
and this, by B.5.27), is equivalent to
5„_,_,(A:, •) = aS„-r(K, •)■ G.2.18)
Thus, aiming at an extension of the differential-geometric result
concerning the condition G.2.16) to general convex bodies, we may take
either G.2.17) or G.2.18) as an assumption. As it turns out, the results
are different. Let us first consider G.2.18). This condition is not only
fulfilled by balls. In fact, suppose that K is an (n - 1 - /)-tangential
body of a ball, say of radius one. Then it follows from Theorem 6.6.16
that Wo(K) = Wi(K) = ... = W,+i(K), and by Theorem 6.4.1 this
implies that
S„.r(K, ■) = S„-2(K, •) = ■■■ = S„-i-i(K, •).
The converse is also true:
Theorem 7.2.9. Let ; e {0,. . . , n - 2}. // K e JC^ is a convex body
satisfying
Sj(K, ■) = aS„_i{K, •) G.2.19)
with some constant a, then K is a j-tangential body of a ball of radius
^-l/(n-/-l)_
Proof After replacing /C by a suitable homothetic copy, we may
assume that ar= 1. From G.2.19) and E.1.18), integrating the support
function of either B" or K, we obtain
W„_, = Wi and W„_,_i = Wo G.2.20)
for the quermassintegrals W^ = W,iK) = V(K[n - i], B"[i]). By the
Aleksandrov-Fenchel inequalities,
^ ^ —^ ^ . . . ^ 2-^. G.2.21)
By G.2.20), equality holds here throughout; in particular wl= WoW2-
By Theorem 6.6.18, K is an (n - 2)-tangential body of a ball. If this
ball has radius r, then Theorem 6.6.16, together with a suitable
dilatation, implies Wo = rWy. Since equality holds in G.2.21), we have
Wo= r"~'W„-j, and together with G.2.20) this yields r = 1. Thus
Wo = Wi = . . . = W„.j. By Theorem 6.6.16, K is a /-tangential body of
B". ■

406 Selected applications
If, however, condition G.2.17) is accepted as a substitute for G.2.16),
then a proper generalization of the classical differential-geometric result
is obtained:
Theorem 7.2.10. Let ; € {0 n-2). If K e Xq is a convex body
satisfying
q(K, ■) = aC„-,(K, ■) G.2.22)
with some constant a, then K is a ball.
For the proof, we sketch only the first steps. We may assume that
0-= 1. Let coC S"~' be a closed set; then
Si(K, (o) ^ q(K, x{K, w)) = C„_,(A:, t{K, w))
= C„-x{K, t(K, (o) n reg/C) = q(K, x{K, (o) n reg/C)
^ 5,(A:, w),
where we have used successively Lemma 4.2.3, the assumption G.2.22)
with 0'= 1, D.2.23) and Theorem 2.2.4, the assumption G.2.22), and
Lemma 4.2.3. Since the equality sign must hold, we deduce that
Sj(K, CO) = C„_i(A:, t{K, CO)) = X"-\t{K, co)) = S„_,{K, co)
by D.2.24). The equality Sj{K, co) = S„-i{K, co) for all closed sets co
permits us to conclude that Sj(K, •) = S„-i(K, •). Now Theorem 7.2.9
shows that /C is a /-tangential body of a ball. The rest of the proof
consists in showing that among /-tangential bodies of balls only balls
satisfy G.2.22). We omit this part of the proof, since it is not
particularly relevant to the theory of mixed volumes. We mention,
however, that Theorem 2.5.5 and the result of Section 4.2, Note 5, are
essential tools. The proof can be found in Schneider A979a).
For the case ;' = 0, and only for this case, a stability version of
Theorem 7.2.10 is known. This is due to Diskant A968), and we quote
it without proof.
Theorem 7.2.11. Let K eXo,0<E< 1/2, and suppose that
- c„_,(/c,P)-'^'
for each set /3€^(E") with C„_,(/C,/3) > 0. Then K lies in the
ye- neighbourhood of a unit ball, where the constant y depends only on
n.
Finally, the uniqueness part of Aleksandrov's theorem formulated in
Section 7.1, Note 8, can be extended to curvature measures of different
orders.

Notes for Section 7.2 407
Theorem 7.2.12. Let j e {0,. . . , n - 1}. Let K,LeJCo be convex
bodies with o as an interior point, and let the radial projection
/: bd a: -» bd L be defined by f{x) = X(x)x ebdL with A(x) > 0 for
xebdK. If
for each Borel set pchd K, then K and L differ only by a dilatation
with centre o if j = 0, and they are identical if j > 0.
The case ; = 0 is due to Aleksandrov A942a), and the case ;' > 0 was
proved by Schneider A978a).
Notes for Section 7.2
1. Minkowski's uniqueness theorem. The method of proof for the uniqueness
theorem 7.2.1 is the original one found by Minkowski A897, 1903).
The first stability result of the type of Theorem 7.2.2 was proved by
Volkov A963). (A sketch of his proof is given in Pogorelov [27], Chapter
VII, §10.) His result was slightly weaker than that of Diskant A972), since it
had an exponent of e equal to l/(n + 2) instead of Diskant's 1/n. Later
Diskant A979) showed that the exponent can be improved to l/(n - 1) if one
makes the stronger assumption that K and L have continuous radii of
curvature and that \s„-i{K, u) -s„.i(L, u)\ < e for all u e 5""'. The optimal
exponent is unknown. A short alternative proof of a weaker form of Theorem
7.2.2 was given by Bourgain & Lindenstrauss A988b).
A paper worthy of recommendation that has historical interest is Stoker
A950), which treats Minkowski's and Christoffel's uniqueness theorems and
the infinitesimal rigidity for closed convex surfaces in E' from a differential-
geometric viewpoint, showing clearly the interrelations between these
problems and how the methods for their proofs are related to the classical
geometry of convex bodies.
2. The Aleksandrov-Fenchel-Jessen theorem. Corollary 7.2.5, the classical Alek-
sandrov-Fenchel-Jessen theorem, was proved independently by Aleksandrov
A937b) and by Fenchel & lessen A938) (see also Busemann [12], p. 70,
LeichtweiB [23], p. 319, and LeichtweiB A981)). Theorem 7.2.4 for the
special case where the bodies of 6 are of class C\ was obtained by
Aleksandrov A938b); the general version for smooth bodies as given here is
new. For the special case of the Aleksandrov-Fenchel-Jessen theorem where
the bodies K and L are of class C+, a differential-geometric proof was given
by Chern A959).
A version of the Aleksandrov-Fenchel-Jessen theorem for convex hyper-
surfaces with boundaries was proved by Busemann A959). His assumption of
identical boundaries is rather strong; it seems more natural to assume that the
support functions coincide at the boundary of the spherical image. For
differential-geometric uniqueness results of this type, see Aleksandrov
A956/8) and Oliker A979a, b). The uniqueness theorem 7.2.7 for general
convex hypersurfaces with boundaries is due to Schneider A989b).
The stability theorem 7.2.6 and its proof are taken from Schneider A989c).
A weaker theorem of this type had previously been proved by Diskant A985)
(for the special case m- 1 = n -2 also by Diskant 1976). His assumptions
are stronger in several respects: He assumed K and L to be smooth and to
satisfy an inequality of the form |5„-i(/C, •)- 5„_i(Z,, • )| S e^/"-'(to) for

408 Selected applications
w € ^E""'), and he had to impose the assumption that one of the bodies has
a projection on some hyperplane that is a ball. Moreover, his exponent for e
is smaller than the one given in Theorem 7.2.6.
Under stronger regularity assumptions, Oliker A982), Theorem 4, proved
a stability theorem involving 5„_i for convex hypersurifaces with boundaries
and with common spherical image contained in an open hemisphere. Earlier,
more specialized results of this type were obtained by Volkov & Oliker
A970) and Oliker A973).
One may ask whether the Aleksandrov-Fenchel-Jessen theorem can be
generalized by imposing more general relations between area measures, for
example: If K, L € Xq are convex bodies satisfying
"iajS,(K, ■) = "ia,Sj{L, •).
;=i ;=i
where the constants ori,. .. , a„-i are non-negative and not all zero, must K
be a translate of LI An affirmative answer for n = 3 is contained in
Schneider A976).
3. Stability results via symmetrization. Extremal properties of balls and thus
uniqueness theorems characterizing balls can often by proved by means of
symmetrization procedures. Sometimes symmetrization is also useful for
obtaining stability properties of balls. In this way, Diskant A971a) obtained
Theorem 7.2.8, and the proof of Diskant A968) for Theorem 7.2.11 also
made use of symmetrizations. Earlier applications of such methods to similar
results can be found in Fet A963) and Diskant A964, 1965). Symmetrization
is also used in Schneider A983) to obtain an estimate for convex hypersur-
faces with unique projection to a hyperplane and satisfying an inequality
Co = aC„-i with or > 0.
4. Characterizations of balls and stability. Theorem 7.2.9, which characterizes
tangential bodies of balls, appears in Schneider A978c), and Theorem 7.2.10
was proved in Schneider A979a). For the latter result, Kohlmann A988)
developed a new method of proof employing generalized Minkowskian
integral formulae and permitting extensions to spaces of constant curvature.
In the literature, there are several investigations, differential-geometric or
convex-geometric, of the stability of the sphere within a class of convex
hypersurfaces having some curvature restriction, such as almost constant
Gauss-Kronecker curvature or mean curvature, or almost umbilical points.
Besides the references in Note 3, we mention Diskant A971b), Koutroufiotis
A971), Moore A973), Treibergs A985), Schneider A990b), Pogorelov A967),
ReSetnjak A968a), Guggenheimer A969) and Vodop'yanov A970).
5. Uniqueness results involving curvature measures. Aleksandrov's A942a) proof
of the case / = 0 of Theorem 7.2.12 proceeds by direct geometric reasoning
(see also Busemann [12], p. 30); the proof for / > 0 by Schneider A978a) has
to use in addition the integral-geometric formula of Theorem 4.5.5. Using the
latter method, the following more general version can be obtained. Let K, L,
f be as in Theorem 7.2.12, and suppose that
n-l n-l
^a.C.(K,fi)= ^a,Q(L,fm)
1=0 1=0
for each Borel set /ScbdJf, where aro,-.-,a„~i are non-negative real
constants with an + ... + a„-i > 0. Then K = L.
In Schneider A978a), the following result was also deduced. Let K e X"
be a convex body with o e int K. Suppose that CjiK, H^.o) = CjiK, H~o) for
some ; e {0,..., n - 1} and all u e 5""'. Then K is centrally symmetric with
respect to o.

7.3 The difference-body inequality 409
The following characterization of the ball was proved for n = 3 by Blind
A977) and for n § 3 by Schneider {1977e). Let K e XS be a smooth convex
body with the property that every Jordan domain in the boundary of K that
halves the curvature measure Co also halves the area C„-i; then K is a ball.
6. Characterizations involving inequalities. Aleksandrov A961) proved the
following uniqueness theorem, which involves only inequalities. Suppose that K,
L e X-S are convex bodies for which
S,{K, ■) S 5,(L, ■) and V,MK) g Ky^^L)
for some j € {I,. . . , n - l). Then K and L are translates of each other.
It would be very interesting to know whether a convex body K e 1^1
satisfying the inequality
S2{K, ■) - 2cSi{K, ■) + c^So(K, •) = 0 G.2.23)
with some constant c is necessarily the sum of a ball and a segment (possibly
degenerate). If true, this would imply that a regular closed convex surface of
class C\ in E' whose principal curvatures ki, ki satisfy (Ac, - c)(k2 - c) S 0
must be a ball. This is only proved for analytic surfaces and surfaces of
revolution. Under the strong additional assumption that K admits some
circular projection, the conjecture concerning G.2.23) is easy to prove; see
Schneider A975b).
7. Aleksandrov's theorem on projections. The following theorem is due to
Aleksandrov A937b) (see also Chakerian 1967b).
Theorem. Let i e {0,. .. , n - 2} and let K, L € JC" be centrally symmetric
with respect to o and of dimension g n - /. If
w,{K\Hu.o) = Wi(L\H„fi) for all u e 5"-',
where w, denotes the (th quermassintegral in the (n - l)-dimensional sub-
space Hu,o, then K = L.
For the proof, one has to observe that the assumption implies
5„_i_,(/C, ■) = 5„_i_,(^. ■). by formula E.3.32) and the uniqueness part of
Theorem 3.5.3. The Aleksandrov-Fenchel-Jessen theorem (Corollary 7.2.5)
then gives the assertion.
Stability results corresponding to the case i = 0 are due to Bourgain
A987), Bourgain & Lindenstrauss A988a,b) and Campi A988), and for the
case ( = n-2 to Campi A986) and Goodey & Groemer A990). Further
related references are given in Section 7.4, Note 1.
7.3. The difference-body inequality
In the final sections, we collect a number of inequalities for convex
bodies, the proofs of which make use of the Brunn-Minkowski
inequality or of inequalities for mixed volumes. The first of these inequalities
concerns the difference body.
For a convex body K 6 X", the difference body DK is defined by
DK := K - K = {x e l"\K D (K + x) =^ 0}.
Theorem 7.3.1. Let KeJCo- Then
2''V„{K) ^ V„(D/C) ^ (^„")v„(/C). G.3.1)

410 Selected applications
Equality holds on the left precisely if K is centrally symmetric and on the
right precisely if K is a simplex.
We note that the left-hand inequality of G.3.1), together with the
characterization of the equality case, is an immediate consequence of
the Brunn-Minkowski theorem. For the right-hand inequality, which is
due to Rogers & Shephard A957), we first prove a lemma, following
Chakerian A967a).
Lemma 7.3.2. Let B e Xo, xo e B, and let f: B-*R be a non-negative
concave function on B. If h: R-*R is strictly increasing, then
l^h(f{x))dx s nV„(B)|^/t(rf(xo))(l - O'-'dr.
Equality holds if and only if, for each y ebd B, the function f is linear
on [xo, y] and f(y) = 0.
Proof. We may assume that xq = o. For x e B\{o}, let x e[o, y]C B
such that [o,^] is maximal, and put g(x) ■= f(o)(l - \x\/\y\) and,
further, g(o) := /(o). Then f = g, since / is concave and non-negative.
It follows that
l^h(f(x))<ix ^ j^h{g(x))<ix
p(B.u)
/t(/(o)(l - r/p(B, u)))r"-'ArAX''-\u)
= /^„.,P(B, uYlh{tf{o)){\ - tY-'AtAX''-\u)
= nV„(B)|^/t(rf(o))(l-r)''-'dr.
Equality holds if and only if / = g. ■
Proof of Theorem 7.3.L Let K e Xo and define D(K,x) := K D {K + x)
for X 6 DK. Let x^, x^ € D/C, A 6 [0,1] and
a 6 A - A)D(A:, X,) + kD{K, xi).
Then a = A - A)a, + Aa2 with aj6D(/C,x,), hence aieK and a, =
bi + Xj with 6; €/C (i = 1,2). It follows that aeK and ae
K + (I - X)xi + Xx2, thus a € D(/C, A - A)xi + Axa). We have proved
that
A - A)D(A:, xi) + XD(K, X2) C D(K, A - A)xi + Ax2).

7.3 The difference-body inequality 411
From the Brunn-Minkowski theorem we conclude that the function
defined by
f(x) := v„(K n(K + x)y^\ X 6 da:,
is concave. Now Lemma 7.3.2, with B = DK, xo = o, h(^) = ^" and
the function /, together with the identity
Lk^"^^ n (a: + x))dx = jJ^iK(y)iK(y - x)dydx
= V„(K)\
where 1 denotes the characteristic function, yields the right-hand
inequality of G.3.1).
Equality holds if and only if / is linear on each segment joining o to a
boundary point of D/C, and this is true if and only if K r\ (K + x) is
homothetic to K, for each x eDK. Simplices have this property, and it
remains to show that they are characterized by it.
Assume, therefore, that KC\(K + x) is homothetic to K for each
xeDK. Let peexpK. Let u e S"~^ be such that the ray
{p + Am|A^O} meets K in a non-degenerate segment [p,?]. Let
A 6 @,1) and Kx-= K r][K + A(p - q)]; then Kx = hxK with a homo-
thety hx- Since peexpK, there is a hyperplane H such that
K D H = {p}. The point hxp must be an exposed point of Kx, lying in
the supporting hyperplane hxH. It follows that hxP = p and hence that
p is the centre of homothety of hx-
Now we use the cones P(K, p) and S{K, p) = cl P{K, p) defined in
Section 2.2. Suppose that
X 6 [P{K, p) + p] n [P{K, q) + q].
Then there are points y,z€K such that y - p = a(x — p) and
z - q = P(x - q) with a-, /3 > 0. For A < 1 and sufficiently close to 1, the
segments [p, y] and [q, z] + A(p - q) will intersect in a point xx- Since
Xx € Kx and, by similarity, xx = hxx, we deduce that x e K. It follows
that
[P{K, p) + p] n [P{K, q) + q]C K
and hence
K' := [S{K, p) + p] n [S(K, q) + q]C K.
Since K C K' holds trivially, we conclude that K = K'. This implies
that each two-dimensional halfplane bounded by aff{p,^} intersects K
in a (possibly degenerate) triangle with side [p, q]. This holds for any
point q for which [p, q] is the (non-degenerate) intersection of K with a
line. It is now easy to see that all possible points q make up a facet of
K and that K is the convex hull of p and this facet. Since p was an

412 Selected applications
arbitrary exposed point of K, induction with respect to the dimension
now shows that K is a simplex. ■
Notes for Section 7.3
1. Special cases of the difference body inequality
V„{DK)s{^^)v„(K) G.3.2)
are treated in Section 53 of Bonnesen & Fenchel [8], where references to
earlier work are given, and in the thesis of Godbersen A938). The general
case was settled by Rogers & Shephard A957). Chakerian A967a) slightly
simplified and extended their argument; we have followed his approach.
Lemma 7.3.2 appears in his paper, which also includes some generalizations
of the difference body inequality. In Rogers & Shephard A958a, b) one finds
a variant of their proof and similar and related inequalities,
2. Intersections of translates. The characterization of simplices by equality in
G,3.2) depends on the following result: A convex body K with the property
that each nonempty intersection K (^{K + x) is homothetic to K is
necessarily a simplex. The proof given above is a modified version of a proof by
Martini A990). The original proof of Rogers and Shephard was rather long.
A shorter proof for polytopes was given by Eggleston, Griinbaum & Klee
A964).
The Rogers-Shephard characterization of simplices can be considered as a
special case of a rather general type of problem concerning characterizations
of specific sets by intersection properties of their translates: Let Jfl be a
family of subsets of E", and let a be a binary relation on Jff. The problem is
to determine the subfamily Jff(a) C ./// defined by
jn(a) := {K e jn\{K + x) n (K + y) e Jft, x, y e E"
^ (K + X) n {K + y) a K).
One may also restrict the translation vectors x, y to smaller sets.
Contributions to this problem, of various kinds, are due to Schneider A967a, 1969b),
Gruber A969, 1970a,b, 1971), McMullen, Schneider & Shephard A974) and
Fourneau A985) (the latter includes references to earlier work).
3. A conjectured strengthening of the difference body inequality. Godbersen
A938) and independently Makai jr. A974) conjectured that
ViKli], - Kin - i]) S i^l)v„(K) G.3.3)
for (■ e {1,. . . , n - 1}, with equality only for simplices. For / = 1 and
(■ = n - 1, this follows from the fact that - K C nK if K has centroid o (see
Bonnesen & Fenchel [8], p. 53), but the general case is unknown. If G.3.3) is
true, it implies the inequality G.3.2), by
V„(DK) = V„(K - K) = t(l)v(K[il -K[n - i])

7.4 Affinely associated bodies 413
It would also imply the inequality
[ V'„((l - k)K + X(-K))AX g -^ VM),
which was established in a different way by Rogers & Shephard A958a).
A generalization of the difference body. For K e IC'i and a number p eN,
define
DpK := {(xu ...,x,)e (E")p\K n (K + x,) n . . . n {K + x^) j^ 0},
so that DiK is the difference body of K. It was shown by Schneider A970b)
that the np-dimensional volume of DpK satisfies
with equality if and only if /C is a simplex.
Write dp{K) := V„,(D,K)V.,{K)-P. For n = 2,
MK) = \p(p + 1N,(K) + 1 - p2.
hence 6^ attains its minimum precisely for centrally symmetric convex bodies.
This is not true for n S 3 and p^2. For these cases, one may conjecture that
the minimum of dp is attained by ellipsoids.
More generally than in Schneider A970b), one can consider, for convex
bodies Kq, Ki, . . . , K^e X^, the translative integral-geometric quantity
D(Ko,K,,..., Kp)
:= J... J^/^o n (/;:,+ X,) n ... n {Kp + Xp))<ix, ... dx^.
Then an obvious extension of the argument used in Schneider A970b)
(essentially an application of Lemma 7.3.2) yields the inequality
r.^;^ K ,,,^lpn + n\yn{Ko)V„{KO-.-V„iKp)
D(Ko,Ku...,Kp) = \ ^ Jv„(KonK,n...nKp)-
Here each Jf, may be replaced by any of its translates K',. The best estimate
is obtained if V„{Ko n K'l n . . . n K'p) is maximal.
7.4. Affinely associated bodies
The mixed volume V(Ki,. . . , K„) remains unchanged if the same
volume-preserving affine transformation of E" is applied to each of the
convex bodies Ki,. . . , K„. The general theory of mixed volumes thus
belongs to the affine geometry of convex bodies. The Euclidean
character of most of the related material treated in former sections is
due to the fact that the mixed volume was specialized by taking balls for
some of its arguments. However, several of the former results are of a
purely affine nature, although metric tools, like support functions and
mixed area measures, may have been used in their proofs.
In the present section, we collect a few more results and problems
from the affine geometry of convex bodies. In particular, we consider
some constructions leading to interesting affine-invariant functionals.
More precisely, with a given convex body K e Xo or K € .Xoo we

414 Selected applications
associate, in several ways, a convex body TK 6 Xqo such that, for each
ae SL(n) (the special linear group of E"), either TaK = aTK for all
K or TaK = a~'TK for all K. In the latter case, where a~' denotes
the inverse of the adjoint linear map of a, the affine character is slightly
obscured. This is a result of the fact that we have identified E" with its
dual space, using a fixed scalar product. The proper setting for the
constructions under consideration would be the n-dimensional real
vector space with a volume form, together with its dual space and the
induced volume form. Then TK, in the second case, should be defined
as an object in the dual space, with the transformation rule
TaK = (ar*)~'T/C, where a* denotes the map dual to a. It is, however,
sometimes convenient to use metric tools in definitions or proofs.
In the treatment of inequalities for the associated bodies, the Brunn-
Minkowski theory is often useful, but symmetrization techniques also
play an essential role. We give no proofs in this section, but rather refer
to the original literature. Much more related material can be found in
the survey article by Lutwak [53]. The affine geometry of convex bodies
could fill a book in its own right; however, some central problems are
still open.
First we recall equation E.3.34). For K € X^, the projection body
riK oi K was defined by
h(nK,u)= v„_,(a:|//„,o)
= \lj{u,v)\dS„-y(K,v) G.4.1)
for M 6 E". Thus UK is a zonoid with centre at the origin. Since K is
n-dimensional, the origin is an interior point of TIK, hence its polar
body (UK)* is defined. One abbreviates (UK)* by U*K and calls this
the polar projection body of K.
It is not immediately obvious from the definition, but not difficult to
show (see Petty 1967 or Bourgain & Lindenstrauss 1988b) that
UaK = a-'UK for a € SL(n).
Since also
this implies that
(aK)* = a-'K* for or e SL(n),
U*aK = aU*K for a e SL(n),
and, further, that the second projection body U^K := UUK satisfies
U^aK = aU^K for a e SL(n).
If K is an ellipsoid then both U*K and U^K are homothetic to K.
Weil A971) determined the polytopes for which K and U^ K are

7.4 Affinely associated bodies 415
homothetic; these are the direct sums of centrally symmetric polygons or
segments. Martini A991) proved that an n-polytope P in E", n ^3, is a
simplex if and only if its polar projection body n*P is homothetic to its
difference body DP.
By suitably combining volumes of associated bodies as considered
here with volumes of the original bodies, one can obtain affine-invariant
functionals. They are continuous and hence attain their extreme values
(see, e.g., Macbeath 1951). It is of considerable interest to determine
these and the bodies characterized by the corresponding extremum
properties. For centrally symmetric convex bodies K e Xo it has been
conjectured that
k"„.iI^„-" ^ V„(nK)V„{Ky-" ^ 2". G.4.2)
In the left-hand inequality, the restriction to centrally symmetric bodies
is not essential. In fact, suppose that K e Xo is a convex body for which
V„(nK)V„(Ky-" is minimal. For L € .Tg we have
V(nK, L, . . . , L) = V(nL, K, ..., K),
by E.1.18), G.4.1) and Fubini's theorem. With L := UK, Minkowski's
inequality F.2.2) yields
v„(UK)v„(Ky-'' ^ v„(n^K)v„(nKy-".
By the assumed minimum property of K, the equality sign must hold
here, hence Tl^K and K are homothetic. In particular, K must be a
zonoid. Thus, the left-hand inequality of G.4.2) need only be proved for
zonoids. It was conjectured by Petty A972) that this inequality holds
and that equality characterizes ellipsoids. The right-hand inequality of
G.4.2), with equality precisely for direct sums of planar centrally
symmetric convex bodies, was conjectured by Schneider A982a). Both
conjectures are open.
We remark that an integral representation for the volume of the
projection body is obtained from G.4.1) and Theorem 5.3.1, namely
V„(nK) =
"^Js"-' ■ ■■ L-.'^''^"'' ■ ■■' "n)d5„-i(/C, Ml) ■■■ dS„-i(K, u„).
G.4.3)
Concerning the volume of the polar projection body, which is given by
V„(U*K) = -^ J^„.,V„_i(A:|H„,o)-''da(«), G.4.4)
it is known that
n-" (^^" j ^ V„(n*K)V„(Kr-' ^ (K„/K„_,r G.4.5)

416 Selected applications
for K € JCo- The right-hand inequality, together with the equality
condition characterizing ellipsoids, was proved by Petty A972). This
inequality is now known as the Petty projection inequality. The left-hand
inequality in G.4.5) had been conjectured by Ball A990+a) and was
proved by Zhang A991); here equality holds if and only if /C is a
simplex.
For the definition of two other types of associated bodies, it is
convenient to enlarge the domain of convex bodies slightly. A star body
in E" is a nonempty compact set K satisfying [o,x]C K for all x e K
and such that the radial function p(K, ■), defined by
p(K, u) := max {A ^ 0|Am € K} for u 6 S"-\
is positive and continuous. The set of star bodies in E" is denoted
by ^S.
For KeSo, there is a unique star body IK whose radial function
satisfies
p(lK, u) = V„_i(A: n //„.o) for u 6 5"-'
(where V„_| is (n - l)-dimensional Lebesgue measure). It is called the
intersection body of K. From the result of Busemann mentioned in
Section 6.1, Note 7, it follows that IK \s convex if K is convex and
centrally symmetric with respect to the origin.
The intersection body plays an essential role in Busemann's A949b,
1950) theory of area in Minkowski spaces. Let B e J^o be a convex
body that is centrally symmetric with respect to the origin. In the
Minkowski space with unit ball B (i.e., E" with the norm defined by the
gauge function of B), Busemann introduced a notion of Minkowski
surface area in such a way that, in particular, the surface area of a
convex body /C € ZJ is proportional to the mixed volume
V(K, ... , K,l*B),
where l*B ■= (IB)*, the polar of the intersection body of B. It follows
from Minkowski's inequality that K solves the isoperimetric problem,
that is, has maximal volume among all convex bodies with given
Minkowski surface area, if and only if K is homothetic to the convex
body I*B. This body is, therefore, called the isoperimetrix.
A different, and in some sense 'dual', notion of surface area in the
Minkowski space associated with B was proposed by Holmes &
Thompson A979). In their theory, the surface area ol K e JC^ turns out to be
proportional to the mixed volume
v(a:, ..., K,nB*),
so that the isoperimetrix is now the zonoid Y\B*. In both theories,
interesting problems remain open. For example, which centrally sym-

7.4 Affinely associated bodies 417
metric convex bodies B yield the minimum or maximum for the
affine-invariant functional V(B,. . . ,B, l*B) or for V{B,. . . ,
B, UB*), the 'self-surface area* of the unit ball in the respective theory?
And for which B is the isoperimetrix homothetic to B? More explicitly:
If IB is homothetic to B*, must B be an ellipsoid? If IIB is homothetic
to B*, must B be an ellipsoid? These questions were posed by
Busemann & Petty A956) and Holmes & Thompson A979),
respectively. Further discussion and related problems can be found in the
survey of Lutwak A989+b).
Returning to the intersection body, we remark that its volume is
given by
V„(IK) = - iv„-i(K n //„.o)" da(«). G.4.6)
Busemann A953) proved that
V„(lK)V„(Ky-'' ^ kUk"-" G.4.7)
for K € .^ob. with equality for n ^ 3 if and only if K is an ellipsoid
with centre at the origin. G.4.7) is called the Busemann intersection
inequality.
For the proof of G.4.7), Busemann first established an integral-
geometric identity of the Blaschke-Petkantschin type, namely
v„(a:,) • ■ ■ v„{K„_,) =
2 J5"-' [JK,n//„„
■/,
^ ^„ i>„_i(xi, . . . , x„_i)dx, ■■ ■ dx„_,
da(M),
G.4.8)
where dx,- denotes the (n - l)-dimensional volume element at x, in the
hyperplane //„ o- He then applied Steiner symmetrization to the right-
hand side to obtain the inequality
v„(a:,) • ■. v„(a:„_,) § 1^ f n v„_,(a:, n H„.o)''A''-"da(«).
G.4.9)
Here for n ^ 3 the equality sign holds if and only if the sets Kj are
homothetic centred ellipsoids. Inequality G.4.7) is a special case.
Busemann assumed that Ki,. . . , K„-i are convex bodies, but this
assumption can be relaxed.
Petty A961a) reinterpreted G.4.8) and G.4.9) by introducing the
so-called centroid bodies, which have since proved quite useful for
obtaining other affine inequalities. The centroid body TK of a star body
K eS^\s defined by
h(rK,x):=,^^^\^\{x,y)\Ay forxeE",
where Ay denotes the volume element at y. Thus

418 Selected applications
''^^^' ^^ = (n + l)V„(K) U^'' "^l'^^' "^'"' '*^^"^'
and r/C is a zonoid with centre o. One has
For a: = aTK for a e GL(n)
(see Lutwak A990b), Section 7, for a systematic study of transformation
rules for associated bodies). If K e X^ is centrally symmetric with
respect to o, then bdFK is the set of all centroids of the intersections
K n //uo, " € 5"~*. For this case (and n = 3), centroid bodies appear
(though not under this name) as early as the paper of Blaschke A917),
§5, where they are attributed to Dupin. Blaschke also posed the
problem solved by inequality G.4.12).
Petty established the formula
v(rA:,,...,rA:„) =
2" 1
Ik " Jk ^''(^>* • ■ • '^n)'*^! ' ■' ^^'"
V„(rK) ^ S^ StT Vn{K) G.4.12)
n\ V„{K,) ■ ■ ■ V„(A:„)
G.4.10)
which also follows from E.3.35). In particular,
V„(TK) = ^ f • ■ • f D„{xi, .... x„)dx, • • • dx„.
Together with Busemann's A953) inequality
/^ ■ • \^J^n{xu . . . . x„)dx, ■ . . dx„ ^ ^ ^ V„{Ky^' G.4.11)
this gives the Busemann-Petty centroid inequality
.n — \
in + 1)! <
(or
^"(^^) = ^" (G?Tk ^)' (^•^■^^)
if the identity «!*■„*■„_, =2"n^"~' is used). Here, equality holds if and
only if /C is a centred ellipsoid.
Now G.4.10), the inequality
ViTK,, ..., rK„r ^ V„(rK,) ■ ■ ■ V„(rK„), G.4.14)
which follows from the Aleksandrov-Fenchel inequality, and the
inequality G.4.12) together yield a generalization of G.4.11), namely
L " L ^"(^'* ■ ■ ■ 'X„)dxi ■ ■ ■ dx„
^-^ v„(a:i)(''+"/'' ■. ■ v„(/:„)(''+i'/« G.4.15)
n + 1 k"„

7.4 Affinely associated bodies 419
for Ki,. . . , K„ eSo. The equality sign holds if and only \i Ki, . . . , K„
are homothetic centred ellipsoids.
Using the (n - l)-dimensional case of G.4.15) in G.4.8), Petty
obtained G.4.9) again,
Centroid bodies are also treated in Milman & Pajor A989) and
Lutwak A990b).
A further type of associated convex body, to be considered now, may
seem artificial at first sight, but is quite useful in establishing relations
between different associated bodies and in proving affine inequalities.
One says that the convex body K e X" has curvature function
f(K, ■): 5""' ^ R if its surface area measure 5„_,(/C, •) has f(K, ■) as
a density with respect to spherical Lebesgue measure or, equivalently, if
V{L, K,...,K) = - L.,/t(L, u)f{K, u) da(u)
ft •'•^
for all L € JC". If K is of class C+, then K has the curvature function
f = s„-\, the reciprocal Gauss curvature, viewed as a function of the
unit normal vector. By (^" we denote the set of all convex bodies in DC^
that have a positive continuous curvature function.
First we remark that one can define the affine surface area of
^(K) := l„J(K, «)"/<"+" da{u). G.4.16)
This notion comes originally from affine differential geometry (see
Blaschke 1923). Q is invariant under volume-preserving affine
transformations. The (extended) affine isoperimetric inequality says that
i2(A:)"+' ^ n''+'it^V„(A:)"-' G.4.17)
for K eQ^", with equality precisely if K is an ellipsoid. Under stronger
differentiability assumptions, this is a classical result, due to Blaschke
(for n = 2,3 see Blaschke 1916a, 1923; it was extended to higher
dimensions by Santal6 A949) and Deicke A953)), which he obtained by
means of Steiner symmetrization. The extension to '7^", including the
equality condition, is due to Petty A985).
For the definition of the envisaged associated body, the curvature
image, let /C € Zo be given. For ze'xniK, let
K' := {K - z)*.
As noted by Santalo A949), there is a unique point s e'lnXK such that
V„{K') ^ V„(K') for all z e int K.
This point is called the Santald point of K. The minimum property of s
implies that

420 Selected applications
J „.,/i(A: - s, m)-("+i> Mda(M) = o. G,4.18)
Since p(/C% m) =/i(/C - s, «)"' for u e S"~\ equation G.4,18) states
that K' has centroid o. Vice versa, if K* has its centroid at o, then
K = K** has its Santal6 point at o.
Equation G,4,18) says that the indefinite a-integral of
h(K - s, •)"<"+'' satisfies the sufficient condition of Minkowski's
existence theorem (Theorem 7,1.2), Hence, there exists a convex body
CK e X") with curvature function
f(CK, ■) = h(K - s, •)"*"*'^- G,4,19)
CK is unique up to a translation, and to make it unique we may choose
the translate that has its centroid at the origin. Clearly CK €<^". The
body C/C is called the curvature image of K. (Here we follow Lutwak
A990b) rather than [53], where the name 'curvature image' is reserved
for the body A/C, which in Lutwak A991) was called the 'polar
curvature image*. It is defined by
fiAK, ■) = -^^p(K, •)"^^ G.4,20)
where K e S" is a star body with centroid at the origin. For
comparison, we note that
CK = [V„(K')/k„Y/^"-^^\K' G.4,21)
for K e Zo, up to a translation,) The curvature image satisfies
CaK = aCK for a 6 SL(n)
(see Lutwak 1990b, G,12)),
Now
V„(K') = ^l p(K',urdo(u)
ft •'•^
= -^/,„.,/(CA:, «)"/("+'> da(«)
= j^[^^_MK-s,u)f(CK,u)da(u)
nV„{K') = Q(CK) G,4,22)
V„(K') = V(K, CK, ..., CK). G,4.23)
and hence
eind also

Notes for Section 7.4 421
Applying G,4,23), Minkowski's inequality F.2,2), the extended affine
isoperimetric inequality G,4,17) and G.4.22), we arrive at the inequality
V„(K)V„(K^) ^ Ki, G,4,24)
where equality holds if and only if K is an ellipsoid. The left-hand side
of G.4.24) is often called the volume product of the convex body K.
Inequality G,4,24) is known as the Blaschke-Santald inequality. The
approach sketched above is that of Santalo A949), completed by Petty's
A985) treatment of the equality condition for the extended affine
isoperimetric inequality.
For centrally symmetric convex bodies (where the Santalo point
coincides with the centre), simpler proofs of the Blaschke-Santal6
inequality, using Steiner symmetrization and the Brunn-Minkowski
theorem, have been found by Saint Raymond A981) and Meyer & Pajor
A990). The latter authors have obtained an improved version of G.4,24)
for general convex bodies (see Note 12 below).
There are many relations between the different associated bodies and
the affine inequalities mentioned in this section. For a discussion and for
generalizations we refer to Lutwak A985a, 1986a, 1990b) and Lutwak's
survey [53] and the literature quoted there.
Notes for Section 7.4
Volumes and quermassintegrals of projections. Extending definition G.4.1),
one may define a convex body TliK, for K e DC" and / = 1, , . ., n - 1, by
h{U,K, u) = Wn-.-iiK") = ^ j^Jiu, V )\AS,{K, v)
for u € 5"'', where w„-,^i(K") denotes the {n - i - l)th quermassintegral
of K" = K\Hu_o. Thus, the zonoid n,Jf comprises all the information on the
(n - i - l)th quermassintegrals of the orthogonal projections of K onto
hyperplanes. If conclusions are to be drawn from partial information about
volumes or other quermassintegrals of projections, this is equivalent to
obtaining information of K from partial knowledge of TliK. Investigations of
this kind, in several different directions, are found in Betke & McMullen
A983), Bourgain & Lindenstrauss A988a, b), Campi A986, 1988), Chakerian
A967b), Firey A965a, 1970d), Goodey A986), Goodey & Groemer A990),
Groemer A991a), Schneider A970c, 1977c) and Schneider & Weil A970);
see also the surveys on zonoids by Schneider & Weil [44], §10, and Goodey
& Weil [47], Section 5.
Mixed projection bodies. The definition of the projection body can be
extended as follows. For convex bodies Kj,.. ., K„-j € JC", there is a
convex body n(Ku ■ ■ ■, K„-i) such that
h(n(Ku ..., K„-,), u) = oiKt, . .. , K:_,) for u e 5"-',
where K" — Ki\Hu.o and v denotes the (n - l)-dimensional mixed volume in
Hu.o. n(Ku ■ ■ , K„-\) is called the mixed projection body of /C,,...,

422 Selected applications
K„-i. It was introduced and thoroughly studied by Lutwak A985b), The
mixed brightness a(Ku ■ ■ ■, K„-i;u) = viKf, . . ., K'^_i) also appears in an
investigation of Chakerian A967b). Lutwak A985b, 1986b) obtained a
number of inequalities for the volumes of polars of mixed projection bodies.
Further, Lutwak A989+a) proved inequalities for mixed projection bodies
that are analogous to the Brunn-Minkowski, Minkowski and Aleksandrov-
Fenchel inequalities, Quermassintegrals of (specialized) mixed projection
bodies are treated by Lutwak A990a),
3. The volume of the projection body and related inequalities. Petty's
conjectured inequality
v„(nK)v„(Ky-" g K"„.,,^-"
can be put into a more symmetric form. Lutwak A990c) showed that this
inequality would follow from (and would imply) the following conjecture: If
K, LeX^,then
\{u, v)\dudu ^ ^^^^^ [V„(K)V„(L)Y''-'Vn^
with equality if and only if K, L are homothetic to centred polar ellipsoids.
Here du, do denote the area elements on bdL, bd K whose outer unit
normal vectors are u, v and it is assumed that K, L are of class C\.
Lutwak proved the following formal analogue of the above inequality. If
K, Le X!^, then
)h6K)bi
ll\{., y)\dxdy m ^^ ^"^^;;-^,^„ [V„(K)V„(L)Y''^
l)/n
with equality if and only if K, L are dilates of centred polar ellipsoids. Here
dx, dy denote volume elements. Furthermore, Lutwak proved that
jJ (x, y)'dxdym " ^1^ [K„(J^)K„(L)r-^'/", G.4.25)
with equality if and only \i K, L are dilates of centred polar ellipsoids.
Let /C e ^0 t>e centrally symmetric with respect to o. Inequality G,4,25)
with L = K* gives
/X(^. y) '<^<iy S , ^ " 4/n [Vn{K)V„{K*)r^^^/'', G.4.26)
(n + z) K„
which was also proved by Ball A988). Ball conjectured that
" " S f f {x,y)^dxdy G.4.27)
for centrally symmetric K, with equality only for centred ellipsoids. He
proved this for K with the property that an affine image of K is symmetric
with respect to the coordinate hyperplanes. In view of G.4,26), inequality
G.4.27) would be stronger than the Blaschke-Santal6 inequality (for
symmetric bodies),
4. Volume estimates from projections and sections. Shephard A964b) has posed
the following question: If A^, L e XS (n ^ 3) are centrally symmetric and if
V„-,(K\H„fi) > V„-,(L\H,,o) for all u e 5"-', G.4.28)
does it follow that
V„(K) > V„(LI G.4.29)
(Without the assumption of central symmetry, a non-spherical body of
constant brightness and a suitable ball provide an immediate
counterexample.) Petty A967) and Schneider A967c) independently showed that the

Notes for Section 7.4 423
answer is negative, but that G.4.28) implies G.4.29) if /C is a zonoid and L
is an arbitrary (not necessarily symmetric) convex body. Schneider moreover
proved: If L is centrally symmetric, but not a zonoid, and is of class CJ*',
then there is a centrally symmetric convex body K such that G.4.28) holds,
but V„(K)<V„(L).
The analogous question for sections is much harder. It was posed by
Busemann & Petty A956): If K,LeX5(n^3) are centrally symmetric
with respect to the origin and if
V„-,(K n «„.rt) > V„.,(L n «„.o) for aUue 5"-', G.4.30)
does it follow that
V„(K) > V„(LO G.4.31)
The Busemann intersection inequality G.4.7), together with its equality
condition, implies an affirmative answer in the special case where L is an
ellipsoid. Busemann A960) gave a counterexample for the case where
convexity is relaxed to starshapedness with respect to the centre. For
three-dimensional coaxial bodies of revolution, affirmative answers were
given by Hadwiger A968c) and Giertz A969), both of whom admitted also
certain non-convex sets.
In analogy to the case of projections described above, Lutwak A988b)
proved that G.4.30) implies G.4.31) if L is an intersection body. He also
showed that if K is sufficiently smooth and not an intersection body, then
there exists a centred star body L such that G.4.30) holds, but inequality
G.4.31) is reversed. The analogy is remarkable; in particular, the role that
mixed volumes play in the treatment of projections is now played by
Lutwak's dual mixed volumes.
M. Meyer A990-I-) gave an affirmative answer for the case where L is a
crosspolytope, and asked whether the result holds true if L is the polar of a
zonoid.
A counterexample to the Busemann-Petty problem was first given by
Larman & Rogers A975). In their example, which works for dimensions
« = 12 only, the body /C is a ball, while the body L is not found explicitly;
its existence is established by means of a probabilistic argument. It came,
therefore, as a considerable surprise when Ball A988) observed that, for
n s 10, a suitable ball (for K) and a centred unit cube provide a
counterexample. This result, however, depended on the non-trivial fact, established by
Ball A986), that every hyperplane section of the unit cube in E" has
(n - l)-dimensional volume at most \/2. Replacing the cube by a suitable
cylinder, Giannopoulos A990) found counterexamples for n^7. Further
contributions and partial results are due to Tanno A987), Grinberg & Rivin
A990) and Bourgain A991). The original question remains open for
Returning to projections, we mention the following interesting theorem
due to Ball A991): For every convex body K e Xq, there exists an affine
image aK of volume 1 satisfying
V„.,(aK\H,,„) a 1 for all u e 5"-'.
The example of the cube shows that this result is best possible.
Ball A990-t-b) also provided a strong negative answer to Shephard's
question discussed above. He showed that G.4.28) does imply
\V^V„(K) a V„(L),
but that, up to the constant factor, no better estimate is possible for all n.
More precisely, there is a constant ^ > 0 such that for each n eN there is a

424 Selected applications
symmetric body K e Xq of volume 1 satisfying
V„.,(K\Hu.o)^S\^ for all « 6 5"-'.
Since for n-balls B'„'* of volume 1, V'„_,(B<,'*|Wuo) tends to a constant for
n —» 00, comparison with suitable balls verifies the claim.
For concentric sections, the situation seems to be different. It has been
asked whether there exists a constant c, not depending on n, such that
G.4.30) for centrally symmetric convex bodies K,L implies cV„(K)^
V„(L). The relations of this question to other problems on convex bodies
are discussed in Section 5 of Milman & Pajor A989).
5. Projection and intersection bodies. For K € Xoo(n^3), Martini A989)
observed that IK CTIK. Here equality holds if and only if /C is a centred
ellipsoid.
6. Monotoneity for centroid bodies. Two of the positive results mentioned in
the preceding notes can be formulated as follows. Let K, L € Xq.
(a) If UK criL and L is a projection body, then V„(K)^ V'„(L), with
equality only if K and L are translates.
(b) If IK C IL and K is an intersection body, then V„(K) S V„(L), with
equality only if K = L.
Lutwak A990b), Theorem (9.3), showed that there is a third result of a
similar type, this time for centroid bodies.
(c) If TKCTL and L is a polar projection body, then V„(K)^ V'„(L),
with equality only if K = L.
7. Extensions of the affine surface area. As mentioned, the notion of affine
surface area from affine differential geometry has been extended to the class
7" of convex bodies with positive continuous curvature functions. Further
extensions to arbitrary convex bodies were studied by LeichtweiB A986,
1988, 1989), Lutwak A991) and Schutt A990+). In particular, Lutwak's
theory includes an extension of the affine isoperimetric inequality, with the
same equality conditions, to arbitrary convex bodies. Schutt A991), Schutt
& Werner A990, 1990+) and Meyer & Reisner A991) investigated convex
floating bodies, which are closely related to an intuitive interpretation of
affine surface area.
Lutwak A987b) defined the mixed affine surface area of Jf i,.. .,
K„ € >7" by
Q(Ku ...,K„) := j^„.,[f(Ku u) ■ ■ ■ f(K„, «)]'•'(-') da(u)
and obtained a number of inequalities, for example,
Q(Ku ..., K„r^' S n'-^'KiViKu ..., K„r-\
with equality if and only if the Jf, are homothetic ellipsoids.
8. Petty's affine projection inequality. If the left-hand inequality of conjecture
G.4.2) were true, then this inequality together with the affine isoperimetric
inequality G.4.17) would yield
V„(nK)^-~^^Q(Kr-^
for K e'T^". This inequality is known to be true; it was proved by Petty
A967). Equality holds if and only if K is an ellipsoid.
9. The Petty projection inequality. It should be observed that the Petty
projection inequality (the right-hand inequality of G.4.5)) provides an
improvement of the classical isoperimetric inequality. Let K e X.^. Using the
Cauchy surface area formula (see E.3.27))
S{K) = -!- l„_ynMK\H,,o)da(u),
K"n-1

Notes for Section 7.4 425
the Holder (or Jensen) inequality and G.4.4), one obtains the inequality
By G.4.5), the right-hand side is § 1, so that the isoperimetric inequality
results.
Applications of the Petty projection inequality to slightly more remote
topics appear in Kiener A986) and Schneider A987a).
10. Geominimal surface area. Petty A972, 1974) introduced a new concept of
affine surface area, the geominimal surface area, which serves, as he
explained, as a connecting link between Minkowski geometry, relative
differential geometry and affine differential geometry. For KeX", the
geominimal surface area of K is defined by
G{K) := in{{nV(K, ...,K, Q*)\Q e JC^ Po(Q) = o, V„{Q) = k„},
where po denotes the centroid (Petty's definition is slightly different, but
equivalent). The geominimal surface area is invariant under
volume-preserving affine transformations; it is homogeneous of degree n - 1, monotone
and continuous, and G'"""'' is concave under Minkowski addition. Besides
many other results. Petty A974) proved the inequalities
G(Kr S n''K„V„(K)''-K
with equality if and only if K is an ellipsoid (see Petty A985) for the
equality case), and
Q{K)"*' g nK„G(K)"
for K € (J", where equality holds if and only if K is the curvature image of
some convex body. Extensions of these results are proved by Lutwak
A991+b). Variants of the geominimal surface area are obtained if the
infimum in its definition extends only over bodies Q restricted to an affine
invariant subclass of JCq such as zonoids, polar zonoids or ellipsoids.
11. A reverse isoperimetric inequality. The case of ellipsoids mentioned in the
preceding note amounts to considering, as a concept of affine surface area,
the infimum of the areas of aK, taken over all volume-preserving affine
transformations a. Petty A961b) found the necessary and sufficient
conditions for K € JCq to have minimal surface area among its affine transforms
of fixed volume. The analogous question for Minkowski surface area was
treated by Clack A990).
With this notion of affine surface area, a reverse isoperimetric inequality
is meaningful. The following theorem was proved by Ball A991a).
Theorem. If K € Xq and if T denotes a regular n-simplex, there exists an
affine image aK of K satisfying
V„(aK)=V„(T) and S(aK)^S(T).
If K is centrally symmetric and if Q denotes the n-cube, there exists an
affine image aK of K satisfying
V„(aK) = V„(Q) and S{aK) S S(Q).
12. The Blaschke-Santald inequality. Inequality G.4.24) was first proved, for
n = 2 and 3, by Blaschke A917); see also Blaschke A923). His approach,
for n = 3, can be sketched as follows. One starts with a convex body K of
class C+. For its boundary surface bd K one defines, in equi-affine
differential geometry, a certain Riemannian metric (a suitable multiple of the
Euclidean second fundamental form) that is invariant under unimodular
affine transformations. If A denotes the corresponding Laplace-Beltrami

426 Selected applications
operator, then Y := ^AX, where X is the position vector of bd /C, defines
the affine normal vector Y. It traces out a surface, which Blaschke calls the
curvature image (Kriimmungsbild) of bdK. Suppose (this is_an additional
assumption on K) that it is the boundary of a convex body K, and let K*
be the polar body of K. Then
3V(K, K, K) = 3Vi(K*) Q,
where Q is the affine surface area of K. Blaschke uses his affine isoperi-
metric inequality and Minkowski's inequality to deduce that
V'3(^)V'3(^*) ^ 4.
Under his assumptions, equality holds only if /C* is an ellipsoid.
In Euclidean terms, one computes that
h(K, ■)=f(K, •)-'/\
hence
PiK* ■) = f(K, ■y'\
Furthermore, K* has its centroid at the origin. It follows that the curvature
image as defined by G.4.19) leads from K to K (andjience is inverse to
Blaschke's curvature image), thus K is_a translate of CK. Comparison with
G.4.20) shows that Jf is a dilate of AK*. Blaschke himself pointed out that
Minkowski's existence theorem can be used to construct K, starting from a
star body K* with centroid at the origin.
Santal6 A949), in lus proof of the n-dimensional case of G.4.24), used
the curvature image CK of a given convex body K. The other ingredients of
his proof are essentially the same as in Blaschke's proof, but extended to
higher dimensions. Santal6 had to apply the classical affine isoperimetric
inequality, which requires differentiability properties, to a convex body CK
obtained as a solution of Minkowski's existence theorem. He did not
mention the necessary assumptions and regularity results for the Minkowski
problem that are necessary to justify such a procedure. The difficulties, in
particular in connection with the characterization of the equality case, were
overcome by Petty A985).
A different proof for the two-dimensional case of the Blaschke-Santal6
inequality was found by Heil A967b).
Somewhat surprisingly, rather simple and direct proofs of the Blaschke-
Santal6 inequality (including identification of the equality case) have been
found that use classical tools of convexity, namely Steiner symmetrization
and the Brunn-Minkowski theorem. For centrally symmetric bodies, such
proofs were given by Saint Raymond A981) and Meyer & Pajor A989, 1990).
For arbitrary convex bodies, Meyer & Pajor A989) obtained the following
result, which is more general than the Blaschke-Santal6 inequality and from
which they deduced that equality in the latter holds only for ellipsoids.
Theorem. Let K e JCq and let W be a hyperplane such that H n \nt K j^ 0.
Then there exists a point z € H Hint K such that
'^"<^)'^"<^^) = ^I^'
where A e @, 1) is defined by V„(K r\H*) = XV„(K)-
Applications of the Blaschke-Santal6 inequality to other problems from
the geometry of convex bodies can be found in Lutwak A983, 1989+c),
Schneider A969a, 1982b, 1987a) and Wieacker A986).

Notes for Section 7.4 427
13. Lower hounds for the volume product. It has repeatedly been conjectured
that
(n + 1)"+'
\ > g V„(K)V„(K*) G.4.32)
for K € XS with centroid at the origin, with equality precisely for simplices,
and that
4"
n!
<
V„(K)V„(K*) G.4.33)
for convex bodies K with centre o. Equality holds in G.4.33) for affine
transforms of cubes and crosspolytopes, but for n^4 not only for these.
The latter fact was first pointed out by Saint Raymond A981) and shows
that the treatment of Guggenheimer A973) is invalid. Conjectures G.4.32)
and G.4.33) are still open.
For n = 2, inequalities G.4.32) and G.4.33) were proved by Mahler
A939b). The equality case, however, was settled by him only under the
assumption that /C is a polygon.
For the special case where Jf is a zonoid, inequality G.4.33) was proved
by Reisner A985) (see also Saint Raymond 1986), and in Reisner A986) he
showed that equality holds only for parallelepipeds. A shorter proof was
later given by Gordon, Meyer & Reisner A988). The case n =2 also settles
the equality case in Mahler's inequality G.4.33) for n = 2.
Another case where G.4.33) has been proved is when K is the unit ball
of a normed space with a 1-unconditional basis. Geometrically, this means
that some affine transform of K is symmetric with respect to each of the
coordinate hyperplanes. For such K, inequality G.4.33) was proved by Saint
Raymond A981). The equality case was characterized independently by M.
Meyer A986) and Reisner A987). Equality is attained precisely by bodies
K of the following type: K can be obtained from n independent segments
with centres at the origin by successively taking either the Minkowski sum or
the convex hull of the union.
Non-sharp lower bounds for V„(K)V„(K*) in the case of centrally
symmetric convex bodies were obtained by Mahler A939a) and other
authors. The strongest result of this kind is due to Bourgain & Milman
A985a, 1987, announced in 1985b), who proved that there exists a universal
constant c > 0 such that
C S K-„^V„(K)V„(K*).
Different proofs were given by Pisier A989).

Appendix
Spherical harmonics
In several places in the geometry of convex bodies, spherical harmonics
prove quite useful. In the present book, they were applied in Sections
3.3, 3.5, 4.3, 6.6 and 7.2 and some of the notes describe results that
were obtained by means of this method. In this appendix, we briefly
collect, without proofs, the basic facts about spherical harmonics, as
they are needed in their applications to convexity. For introductory
material, including proofs, we refer to Miiller A966), Seeley A966) and
the literature quoted there; for the connections with representations of
the rotation group we refer also to Coifman & Weiss A968).
A spherical harmonic of degree m on S"~^ is, by definition, the
restriction to S"~^ of a homogeneous polynomial / (with respect to
Cartesian coordinates) of degree m (or identically zero) on E" that
satisfies A/ = 0; here A denotes the Laplace operator. The set S'" ol
spherical harmonics of degree m is a vector subspace of C{S"~^) of
dimension
.• cm x,/ N Bm + n - 2)r{n + m - 2) ^ ^ ,,
dim^- = Nin, m) = r(m-M)r(n - 1) ' ^^'^^
As a consequence of their definition, the spherical harmonics are
eigenfunctions of the spherical Laplace operator As; each Y^ e S„
satisfies
^sYm = -m(m + n- 2)Y„. (A.2)
By 1,2E""') we denote the Hilbert space of square-integrable real
functions on 5""' (with the usual identification of functions coinciding
almost everywhere) with scalar product (•, •) defined by
(/. g) ■■= L/i?da;

Spherical harmonics 429
here a denotes spherical Lebesgue measure on 5""'. The induced
L2-norm is denoted by || • Ha- Spherical harmonics of different degrees
are orthogonal, that is,
(/. g) = 0 ioxf^^\ geSi, k^j.
The system of spherical harmonics is complete, thus
(/, K) = 0 for y € S'" and all m e Nq
for some f e 1,2E""^) implies / = 0 (almost everywhere). Every
continuous real function on S"~^ can be approximated uniformly by finite
linear combinations of spherical harmonics. Consequently, if
I y d<p = 0 for y € S"' and all m e Nq
for some finite signed Borel measure <p on 5"~\ then qp = 0.
In each space S'" we choose an orthonormal basis (K^i, . . . ,
^mN{n,m))- Then {Y„j\m eNo, j = \, . . . , N(n, m)} is a complete ortho-
normal system in 1,2E""^). For / e 1,2E""^) we write
N{n.m)
^J-= 2 (/, Y„;)Y„f, (A.3)
/=1
then n„f does not depend on the choice of the orthonormal basis, and
the map n^„: L2{S"~^)-*S'" is the orthogonal projection. The Fourier
series of / € 1,2E""^) is given by
cc
m = 0
it converges to / in the L2-norm. The Parseval relation says that
ilk^lli = ll/lli (A.4)
m=0
or, more generally, that
<» N{n,m)
1 1 (f, Y„,)(g, y„,) = (/. g) (A.5)
m = 0 ;=1
for/, g€ £.2E"-').
We remark that the projections ttq and n^i have a simple meaning.
Since cV" contains only constant functions,
is the mean value of /. Let (c,, . . . , c„) be an orthonormal basis of E"
and put
•^;(") ■= . I <«;. ") for " € 5"-\ y = 1 , n.
Using A.7.5), we get

430 Appendix
(Si, Sj) = — L-/^'-' ")<^/' ")da(M) = {ei, c,).
Since dimcV^ = n, we see that (Sy, . . . , S„) is an orthonormal basis of
a' and hence that
(n^fKo)
;=i ;=i ■'•'f V *:„ V *:„
*/•= —J5„-,/(")"'^<^(")■
/■=l
with
Thus
In particular, for the support function Hk of a convex body K e X"
we have
TiohK = \b(K), (A.6)
n,hK= {s(K), ■), (A.7)
in view of A-7.2) and A.7.3).
If we require spherical harmonics in the form Y(u) = f{{e, u)) with
fixed e € 5""\ we are led to the Gegenbauer polynomials. Assuming
that
V := —-— > 0,
2
the Gegenbauer polynomial C^ of degree m and order v can be defined
by means of the generating function
A- /x + x2)-= 2CW)x'".
m=0
Explicitly,
C(/) = fl^d - '')"''"'^(^)'"(i - t'r^^-'^, (A.8)
where
o"^ = C;(l)l- ^1 , ^ - ',,, (A.9)

Spherical hannonics 431
The function C^((c, • )), where e e 5""' is fixed, is a spherical
harmonic of degree m on 5""'.
The addition theorem says that
/=! W„C„A)
for u,veS"~^ and m e No- As a consequence, one obtains that the
maximum norm |(y„|| of y„ € cy" can be estimated from
N{n, m)y^,
2 Y„i{u)Y„j(u) = ^^;;;^;c;«»,t;)) (A.ll)
y ^
I * mil —
@„
The following result is particularly useful.
\YJ2- (A.12)
Funk-Hecke theorem. Let / € Liii-l, 1]) 6c a function satisfying
/j/(Ol(i-'¥"-'>/'d/<«>.
IfY^eS"' is a spherical harmonic of degree m, then
/,„./««. v))Y„{v)da{v) = A„[/]y„(«) (A.13)
for u e S"~^, where
K[f] = co„-,[aM)]-' /_/(/)C;(/)(l - /2)('-3)/2d,. (A.14)
We conclude with a remark on the development of sufffciently
smooth support functions into series of spherical hannonics.
Let / be a real function of class C" on S"~^ and let
=0 N{n,m)
/- 2 2 OmiYmn (A. 15)
m=0 ;=1
with a„y = (/, Y„j), be its Fourier series with respect to the orthonormal
system {Y„j). It follows from Theorems 4 and 5 of Seeley A966) that
<» N{n,m)
D'f(x/\x\) =22 fl„/D»y„,(x/|x|), (A.16)
m=0 /=!
where D* = 3^"^/(dx i)"^ . . . (ax„)*", \a\ = a^ + ... + a„, and each of
these series is uniformly convergent on 5""'.
Let g be a function of class C^ on 5""'; let G be its positively
homogeneous extension. By ri(g, x) we denote the smallest eigenvalue
of the second differential of G at x € S"~', restricted to the orthogonal
complement of x; and we define
r,(g) := min{;;(g, x)|x € 5"-'}.
By Theorems 1.5.10 and 1.7.1, g is the restriction of a support function
if and only if r](g) ^ 0.

432 Appendix
Now let / be the support function of a convex body K of class C".
Then ;;(/) > 0 by Corollary 2.5.2. Consider the partial sum
k N{n,m)
fk ■= 2j Zj '^miYmj
of (A. 15). Using (A. 16) and uniform convergence, we may show that
r)(fk) -* r](f) for /:-»», hence r](fk) > 0 for almost all k. It follows that
the partial sum f^ is the restriction of a support function for all
sufficiently large k.
Appendix Note
Applications to convex bodies. Spherical harmonics in space and Fourier series
in the plane have been applied to a great variety of geometric problems on
convex bodies. The first applications of this kind were Hurwitz's A901)
well-known proof of the isoperimetric inequality for closed (not necessarily
convex) curves in the plane, the thorough investigation of Hurwitz A902) in the
plane and in three-space, and Minkowski's A904) characterization of convex
bodies of constant width. In the many different later contributions, two main
principles of application are dominant. This is either the use of the Parseval
relation to obtain quadratic inequalities, or the connection of spherical
harmonics with irreducible representations of the rotation group. In particular, this often
permits us to solve certain rotation invariant linear functional equations on the
sphere by finding the spherical harmonics that satisfy the equation, and a
number of geometric questions can be reduced to such equations.
Many authors have made geometric applications of Fourier series and
spherical harmonics in the domain of convexity. Among them are Aleksandrov
A937b), Anikonov & Stepanov A981), Berg A969b), Blaschke A914), [5], §23,
A916b, 1923), §90, A936), Bol A939), Bourgain A987, 1991), Bourgain &
Lindenstrauss A988a, b, 1989), Bourgain, Lindenstrauss & Milman A989),
Campi A981, 1984, 1986, 1988), Chemoff A969). Cie^lak & G6^dz A987),
Dinghas A939, 1940a), Falconer A983), Fillmore A969), Fisher A987), Focke
A969a), Fuglede A986b, 1989), Fujiwara A915, 1919), Fujiwara & Kakeya
A917), Geppert A937), Gericke A940a, b, 1941), Goodey & Groemer A990),
Gortler A937a, b, 1938), Groemer A990a, 1991a), Hayashi A924, 1926a, b,c),
Inzinger A949), Kameneckii A947), Knothe A957b), Kubota A918, 1920, 1922,
1925a, b), Matsumura A930), Meissner A909, 1918), Minoda A941), Nakajima
A920), Oishi A920), Petty A961a), ReJetnjak A968a), Sas A939), Schneider
A967c, 1970a, d, 1971a, b,c,d, 1974a, 1975c, 1977c, 1984, 1989c), Shephard
A968b), Su A927a,b). For details, see Groemer [49].

REFERENCES
A. Books on convexity
[1] Aleksandrov, A. D. 1955, Die innere Geomelrie der konvexen Flachen. Akademie-
Verlag, Berlin (Russian original: 1948).
[2] Aleksandrov, A. D. 1958, Konvexe Polyeder. Akademie-Verlag, Berlin, (Russian
original: 1950).
[3] Bair, J. & Fourneau, R. 1980. Etude g^omitrique des espaces vectorieh, II. Polyidres
et polytopes convexes. Lecture Notes in Math. 802, Springer, Berlin.
[4] Benson, R. V. 1966. Euclidean Geometry and Convexity. McGraw-Hill. New York.
[5] Blaschke. W. 1956, Kreis and Kugel, 2nd edn. W. de Gruyter, Berlin Ast edn:
1916).
[6] Boltjanski, V. G. & Soltan, P. S. 1978, Combinatorial Geometry of Several Classes
of Convex Sets (in Russian). "Stiinca', Kishinev.
[7] Bonnesen, T. 1929, Les problemes des isopirimitres et des isipiphanes. Gauthiers-
Villars, Paris.
[8] Bonnesen, T. & Fenchel, W. 1934, Theorie der konvexen Kdrper. Springer, Berlin.
Reprint: Chelsea Publ. Co., New York, 1948. English translation: BCS
Associates, Moscow, Idaho, 1987.
[9] Bourbaki, N. 1953, Espaces vectoriels topologiques, Chap. I, II. Hermann, Paris.
[10] Bourbaki, N. 1955, Espaces vectoriels topologiques, Chap. Ill, IV, V. Hermann,
Paris.
[11] Br0ndsted, A. 1983, An Introduction to Convex Polytopes. Springer, New York.
[12] Busemann, H. 1958, Convex Surfaces. Interscience, New York.
[13] Eggleston, H. G. 1958, Convexity. Cambridge Univ. Press.
[14] Fenchel, W. 1951, Convex Cones, Sets and Functions. Mimeographed lecture notes,
Princeton Univ.
[15] GrOnbaum, B. 1967, Convex Polytopes. Interscience, London.
[16] Guggenheimer, H. W. 1977, Applicable Geometry. Global and Local Convexity. R.
E. Krieger, Huntington, NY.
[17] Hadwiger, H. 1955, Altes und Neues iiber konvexe Korper. Birkhauser, Basel.
[18] Hadwiger, H. 1957, Vorlesungen iiber Inhalt, Oberflache und Isoperimetrie. Springer,
Berlin.
[19] Hadwiger, H., Debrunnner, H. E. & Klee V. 1964, Combinatorial Geometry in the
Plane. Holt, Rinehart & Winston, New York.
[20] Jaglom, I. M. & Boltjanski W. G. 1956, Konvexe Figuren. Deutsch. Verl. d. Wiss.,
Berlin (Russian original: 1951).

434 References
|21] Kelly P. J. & Weiss, M. L. 1979, Geometry and Convexity. John Wiley, New York.
B2] Lay, S. R. 1982, Convex Sets and Their Applications. John Wiley, New York.
B3] LeichtweiU, K. 1980, Konvexe Mengen. Springer, Berlin; VEB Deutsch. Verl. d.
Wiss., Berlin.
[24] Lyusternlk. L. A. 1963, Convex Figures and Polyhedra. Dover, New York (Russian
original: 1956).
[25] Marti, J. T. 1977, Konvexe Analysis. BirkhSuser Verlag, Basel.
[26] McMullen, P. & Shephard G. C. 1971, Convex Polytopes andjhe Upper Bound
Conjecture. Cambridge Univ. Press.
[27] Pogorelov, A. V. 1973, Extrinsic Geometry of Convex Surfaces. Translations of
Mathematical Monographs 35, Amer. Math. Soc, Providence, RI (Russian
original: 1969).
B8] Roberts, A. W. & Varberg, D. E. 1973, Convex Functions. Academic Press, New
York.
B9] Rockafellar, R. T. 1970, Convex Analysis. Princeton Univ. Press, Princeton, NJ.
(.•!0] Valentine, F. A. 1964, Convex Sets. McGraw-Hill, New York.
B. Some survey articles
C1] Chakerian, G. D. & Groemer, H. 1983, Convex bodies of con.stant width. In:
Convexity and its Applications (eds P. M. Gruber & J. M. Wills), Birkhauser,
Basel, pp. 49-96.
C2] Danzer, L., Griinbaum, B. & Klee, V. 1963, Helly's theorem and its relatives. In:
Convexity, Proc. Symposia Pure Math., vol. VII. Amer. Math. Soc, Providence,
RI,pp. 101-80.
[33] Gruber, P. M. 1983, Approximation of convex bodies. In: Convexity and its
Applications (eds P. M. Gruber & J. M. Wills), Birkhauser, Basel, pp. 131-62.
[34] Gruber, P. M. 1984, Aspects of convexity and its applications. Expo. Math. 2,
47-83.
[35] Gruber, P. & Hobinger, J. 1976, Kennzeichnungen von Ellipsoiden mit Anwen-
dungen. In: Jahrbuch Oberblicke Mathematik, Bibliographisches Institut,
Mannheim, pp. 9-29.
[36] Gruber, P. & Schneider, R. 1979, Problems in geometric convexity. In:
Contributions to Geometry, Proc. Geometry Symp., Siegen 1978 (eds J. Tolke & J. M.
Wills), Birkhauser, Basel, pp. 255-78.
[37] Griinbaum, B. 1963, Measures of symmetry for convex sets. In: Conve.tity, Proc.
Symposia Pure Math., vol. VII, Amer. Math. Soc., Providence, RI, pp. 233-70.
C8] Klee, V. 1971, What is a convex set? Amer. Math. Monthy 78. 616-31.
C9] McMullen, P. 1979, Transforms, diagrams, and representations. In: Contributions to
Geometry, Proc. Geometry Symp.. Siegen 1978 (eds J. Tolke & J. M. Wills),
Birkhauser, Basel, pp. 92-130.
D0] McMullen, P. & Schneider, R. 1983, Valuations on convex bodies. In: Convexity
and its Applications (eds P. M. Gruber & J. M. Wills). Birkhauser, Basel, pp.
170-247.
D1] Osserman. R. 1978. The isoperimetric inequality. Bull. Amer. Math. Soc. 84,
1182-238.
D2] Petty. C. M. 1983. Ellipsoids. In: Convexity and its Applications (eds P. M. Gruber
& J. M. Wills), Birkhauser, Basel, pp. 264-76.
D3] Schneider. R. 1979, Boundary structure and curvature of convex bodies. In:
Contributions to Geometry. Proc. Geometry Symp.. Siegen 1978 (eds J. Tolke & J.
M. Wills), Birkhauser, Basel, pp. 13-59.
[44] Schneider, R. & Weil, W. 1983. Zonoids and related topics. In: Convexity and its
Applications (eds P. M. Gruber & J. M. Wills), Birkhauser, Ba.sel. pp. 296-317.

References 435
|45) Zamfirescu, T. 1991, Baire categories in convexity. Atti Sem. Fis. Univ. Moderw 39,
139-64.
The following articles in the Handbook of Convex Geometry (eds P. M. Gruber & J. M.
Wills), North-Holland, Amsterdam (to appear 1993). are particularly relevant to the topic
of this book.
[46] Florian, A., Extremum problems for convex discs and polyhedra.
[47] Goodey, P. & Weil, W., Zonoids and generalizations.
[48] Groemer, H., Stability of geometric inequalities.
D9] Groemer, H., Fourier series and spherical harmonics.
E0] Gruber, P. M., The space of convex bodies.
E1] Gruber, P. M., Baire categories in convexity.
[52] LeichtweiU. K., Convexity and differential geometry.
[53] Lutwak, E., Selected affine isoperimetric inequalities.
[54] Sangwine-Yager, J. R., Mixed volumes.
E5) Schneider, R., Convex surfaces, curvature and surface area measures.
E6) Schneider, R. & Wieacker, J. A., Integral geometry.
C. Articles and monographs
A date followed by a plus sign, e.g. '1990+', indicates that the article to which it refers
appeared in preprint form in that year.
Ahrens, I. 1970. Uber unendliche Reihen von Punktmengen bei Minkowskischer Addition
und Subtraktion. Dis.sertation. Freie Universitat Berlin.
1972, Uber Folgen und Reihen von Punktmengen. Arch. Math. 13, 92-103.
Aleksandrov, A. D., 1933, A theorem on convex polyhedra (in Russian). Trudy Fiz.-Mat.
Inst. Im. Steklova 4, 87.
1937a, Zur Theorie der gemischten Volumina von konvexen Korpern, I. Verallge-
meinerung einiger Begriffe der Theorie der konvexen Korper (in Russian). Mat.
Shornik N.S. 2, 947-72.
1937b, Zur Theorie der gemischten Volumina von konvexen Korpern, II. Neue
Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen (in
Russian). Mat. Shornik N.S. 2, 1205-38.
1937c, Uber die Frage nach der Existenz eines konvexen Korpers, bei dem die Summe
der Hauptkriimmungsradien eine gegebene positive Funktion ist, welche den Bedin-
gungen der Geschlo.ssenheit geniigt. C.R. (Doklady) Acad. Sci. URSS 14, 59-60.
1938a, Zur Theorie der gemischten Volumina von konvexen Korpern. III. Die Er-
weiterung zweier Lehrsatze Minkowskis iiber die konvexen Polyeder auf beliebige
konvexe FlSchen (in Russian). Mat. Shornik N.S. 3, 27-46.
1938b, Zur Theorie der gemischten Volumina von konvexen Kiirpern, IV. Die
gemischten Diskriminanten und die gemischten Volumina (in Ru.ssian). Mat. Shornik N.S. 3,
227-51.
19.^9a, Anwendung des Satze.s uber die Invarianz des Gebiete.s auf Existenzbeweise (in
Russian). Izv. Akad. Nauk SSSR 3, 243-.'i6.
1939b, Uber die Oberflachenfunktion eines konvexen Kiirpers (in Russian). Mat.
Shornik N.S. 6, 167-74.
1939c, Almost everywhere existence of the .second differential of a convex function and
some properties of convex surfaces connected with it (in Russian). Uchenye Zapiiki
Leningrad. Gos. Univ., Math. Ser. 6, 3-35.
1942a, Existence and uniqueness of a convex surface with a given integral curvature.
C.R. (Doklady) Acad. Sci. URSS 35, 131-4.
1942b, Smoothness of the convex surface of bounded Gaussian curvature. C. R.
(Doklady) Acad. Sci. URSS 36, 195-9.

436 References
1956/8, Uniqueness theorems for surfaces in the large, I, II, III, IV, V (in Russian).
Vestnik Leningrad. Univ. 11 A956), No. 19, 5-17; 12 A957), No. 7, 15-44; 13 A958),
No. 7, 14-26; 13 A958), No. 13, 27-34; 13 A958), No. 19, 5-8. English translation:
Amer. Math. Soc. Transl., Ser. 2, 21 A962), 341-416.
1961, A congruence condition for closed convex surfaces (in Russian). Vestnik
Leningrad. Univ. 16, 5-7.
1967, On mean values of support functions (in Russian) Dokl. Akad. Nauk SSSR 172,
755-8. English translation: Soviet Math. Dokl. 8, 149-53.
Alexander, R. 1988, Zonoid theory and Hilbert's fourth problem. Geom. Dedicata 28,
199-211.
Alfsen, E. M. 1971, Compact Convex Set.<: and Boundary Integrals. Springer, Berlin.
Allendoerfer, C. B. 1948, Steiner's formula on a general 5"*'. Bull. Amer. Math. Soc. 54,
128-35.
Ambartzumian, R. V. 1987, Combinatorial integral geometry, metrics, and zonoids. Acta
Appl. Math. 9, 3-27.
Anderson, R. D. & Klee, V. L. 1952, Convex functions and upper .semi-continuous
collections. Duke Math. J. 19, 349-57.
Anikonov, Yu. E. & Stepanov, V. N. 1981, Uniqueness and .stability of the solution of a
problem of geometry in the large (in Russian). Mat. Sbornik 116 A981), 539-46.
Arnold, R. 1989, Zur L^-Bestapproximation eines konvexen Korpers durch einen beweg-
ten konvexen KOrper. Monatsh. Math. 108, 277-93.
Arrow, K. J. & Hahn, F. H. 1971, General Competitive Analysis. Holden-Day, San
Francisco.
Artstein, Z. 1974, On the calculus of closed set-valued functions, Indiana Univ. Math. J.
24,433-41.
1980, Discrete and continuous bang-bang and facial spaces or: Look for the extreme
points. SIAM Review 22, 172-85.
1987-t- Convergence of sums of random sets. Preprint.
Artstein, Z. & Vitale, R. A. 1975, A strong law of large numbers for random compact
sets. Ann. Prohah. 3, 879-82.
Ash, R. B. 1972, Measure, Integration, and Functional Analysis. Academic Press, New
York.
Asplund, E. 1963, A A:-extreme point is the limit of A:-exposed points. Israel J. Math. 1,
161-2.
1973, Differentiability of the metric projection in finite dimensional Euclidean space.
Proc. Amer. Math. Soc. 38, 218-19,
Assouad, P. 1980, Characterisations de sous-espaces norm^s de L' de dimension finie.
Siminaire d'analyse fonctionnelle (Ecole Polytechnique Palaiseau), 1979-80, expose
No. 19.
Baddeley, A. J. 1980, Absolute curvatures in integral geometry. Math. Proc. Camb. Phil.
Soc. 88. 45-58.
Baebler, F. 1957, Zum isoperimetrischen Problem. Arch. Math. 8, 52-65.
Bair, J. 1975, Une mise au point sur la decomposition des convexes. Bull. Soc. Roy. Sci.
Liige 44, 698-705.
1976a, Extension du th^orfime de Straszewicz. Bull. Soc. Roy. Sci. Liige 45, 166-8.
1976b, Une etude des sommands d'un polyfidre convexe. Bull. Soc. Roy. Sci. Liige 45,
307-11.
1977a, Description des sommands de quelques ensembles convexes. Rend. 1st. Mat.
Univ. Trieste 9, 83-92.
1977b, Decompositions d'un convexe a I'aide de certains de ses dilates. Bull. Soc. Roy.
Sci. Liige 46, 167-71.
1979, Sur la structure extremale de la somme de deux convexes. Canad. Math. Bull. 22,
1-7.
Bair, J., Fourneau, R. & Jongmans, F. 1977, Vers la domestication de Textremisme. Bull.

References 437
Soc. Roy. Sci. Liige 46, 126-32.
Bair, J. & Jongmans, F. 1977, Sur les graves questions qui naissent quand la
decomposition des ensembles atteint un stade avanc^. Bull. Soc. Roy. Sci. Liege 46, 12-26.
Ball. K. 1986, Cube slicing in R". Proc. Amer. Math. Soc. 97, 465-73.
1988, Some remarks on the geometry of convex sets. In: Geometric Aspects of
Functional Analysis (eds J. Lindenstrauss & V. D. Milman), Lecture Notes in Math.
1317, Springer, Heidelberg, pp. 224-31.
1991a, Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. 44,
351-9.
1991b, Shadows of convex bodies. Trans. Amer. Math. Soc. 327, 891-901.
Banchoff, T. F. 1967, Critical points and curvature for embedded polyhedra. J. Differential
Geom. 1, 245-56.
1970, Critical points and curvature for embedded polyhedral surfaces. Amer. Math.
Monthly 77, 475-85.
Bandle, C. 1980, Isoperimetric Inequalities and Applications. Pitman, Boston.
Bandt, Ch. 1986, On the metric structure of hyperspaces with Hausdorff metric. Math.
Nachr. 129, 175-83.
1988, Estimates for the Hausdorff dimension of hyperspaces. In: General Topology and
its Relations to Modern Analysis and Algebra VI, Proc. 6th Prague Topological Symp.
1986, Heldermann, Berlin, pp. 33-42.
Bandt, Ch. & Baraki, G. 1986, Metrically invariant mea.sures on locally homogeneous
spaces and hyperspaces. Pacific J. Math, 121, 13-28.
Bangert, V. 1977, Konvexitat in riemannschen Mannigfaltigkeiten. Dissertation, Univ.
Dortmund.
1979, Analytische Eigenschaften konvexer Funktionen auf Riemannschen
Mannigfaltigkeiten. J. reine angew. Math. 307/308, 309-24.
Bantegnie. R. 1975, Convexity des hyperespaces. Arch. Math. 26, 414-20.
Bdrdny, I. 1989, Intrinsic volumes and /-vectors of random polytopes. Math. Ann. 2SS,
671-99.
BSriiny, I. & Larman D. G. 1988, Convex bodies, economic cap coverings, random
polytopes. Mathematika 35, 274-91.
B^r^ny, I. & Zamfirescu, T. 1990, Diameters in typical convex bodies. Canad. J. Math.
42, 50-61.
Barthel, W., 1959a, Zum Busemannschen und Brunn-Minkowskischen Satz. Math. Z. 70
A958/59), 407-29.
1959b, Zur isodiametrischen und isoperimetrischen Ungleichung In der Relativgeome-
trie. Comment. Math. Helvet. 33, 241-57.
Barthel, W & Bettlnger, W. 1961, Die isoperimetrische Ungleichung fiir die innere
Minkowskische Relatlvoberflache. Math. Ann. 142 A960/61), 322-7.
1963, Bemerkungen zum isoperimetrischen Problem. Arch. Math. 14, 424-9.
Barthel, W & Franz, G. 1961, Eine Verallgemelnerung des Busemannschen Satzes vom
Brunn-Minkowskischen Typ. Math. Ann. 144, 183-98.
Batson, R. G. 1988, Necessary and sufficient conditions for boundedness of extreme points
of unbounded convex sets. J. Math. Anal. Appl. 130, 365-74.
Bauer, H. 1964, Konvexitat in topologischen VektorrSumen. Vorlesungsausarbeltung,
Hamburg.
1974, Wahrscheinlichkeitstheorie und Grundziige der Mafitheorie. 2nd edn, W. de
Gruyter, Berlin.
Baum, D. 1973a, Zur analytischen Darstellung des Randes konvexer Korper durch
Stiitzhyperboloide, 1. Halbachsenfunktion und Scheltelsatze. Manuscripta Math. 9,
113-41.
1973b, Zur analytischen Darstellung des Randes konvexer Korper durch
Stiitzhyperboloide, II. Die Randdarstellung durch die Halbachsenfunktion. Manuscripta Math. 9,
307-22.

438 References
1987, Bemerkungen zur Halbachsenfunktion. Analysis 7, 221-35.
1989, Randdarstellungen konvexer KOrper. Analysis 9, 185-94.
Beer, G. A. 1975, Starshaped sets and the Hausdorff metric. Pacific J. Math. 61, 21-7.
1989, Support and distance functional for convex sets. Numer. Fund. Anal. Optim. 10,
15-36.
Bensimon, D. 1987, Sur I'espace topologique des points extr^maux d'un convexe compact.
C.R. Acad. Sci. Paris, iVr. / Math. 304, 391-2.
Benson, D. C. 1970, Sharpened forms of the plane isoperimetric inequality. Amer. Math.
Monthly 77. 29-34.
Berg, Ch. 1969a. Shephard's approximation theorem for convex bodies and the Milman
theorem. Math. Scand. 25. 19-24.
1969b, Corps convexes et potentiels spheriques. Danske Vid. Selskah. Mat.-fys. Medd.
37, 6.
1971. Abstract Steiner points for convex bodies. J. London Math. Soc. B) 4. 176-80.
Bernstein, D. N. 1975, The number of roots of a system of equations. Functional Anal.
Appl. 9, 183-5.
Berwald, L. & Varga. O. 1937. Iiitegralgeometrie 24. Uber die Schiebungen im Raum.
Math. Z. 42. 710-36.
Besicovitch, A. S. 1963a. On singular points of convex surfaces. In: Convexity, Proc.
Symp. Pure Math., vol. 7. Amer. Math. Soc. Providence, RI, pp. 21-3.
1963b, On the set of directions of linear segments on a convex surface. In: Convexity,
Proc. Symp. Pure Math., vol. 7. Amer. Math. Soc. Providence, RI, pp. 24-5.
Betke. U. 1992, Mixed volumes of polytopes. Arch. Math. 58. 388-91.
Betke. U. & Goodey, P. R. 1984, Continuous translation invariant valuations on convex
bodies. Abh. Math. Sem. Univ. Hamburg 54, 95-105.
Betke. U. & Gritzmann. P. 1986. An application of valuation theory to two problems In
discrete geometry. Discrete Math. 58. 81-5.
Betke. U. & Kneser. M. 1985. Zerlegungen und Bewertungen von Gitterpolytopen. J.
reine angew. Math. 358, 202-8.
Betke. U. & McMullen, P. 1983. Estimating the sizes of convex bodies from projections.
J. London Math. Soc. 27, 525-38.
Betke. U. & Weil. W. 1991, Isoperimetric inequalities for the mixed area of plane convex
sets. Arch. Math. 57. 501-7.
Bjorck. G. 1958, The set of extreme points of a compact convex set. Ark. Mat. 3, 463-8.
Blackwell. D. 1951. The range of certain vector Integrals. Proc. Amer. Math. Soc. 2,
.390-5.
Blaschke, W. 1914, Bewelse zu Satzen von Brunn und Minkowski iiber die Minimalelgen-
schaft des Kreiser. Jahresber. Deutsche Math.-Ver. 23. 210-34. Ces. Werke A985b),
pp. 106-30.
1915. Konvexe Berelche gegebener konstanter Breite und kleinsten Inhalts. Math. Ann.
76. 504-13. Ces. Werke A985b). pp. 157-66.
1916a. Uber afflne Geometric I: Isoperimetrische Elgenscliaften von Ellipse und
Ellipsoid. Ber. Verh. Sachs. Akad. Leipzig, Math.-Phys. Kl. 68. 217-39. Ges. Werke
A985b). pp. 181-203.
1916b, Eine kennzelchnende Eigenschaft des Ellip.soids und eine Funktionalgleichung
auf der Kugel. Ber. Verh. Sachs. Akad. Leipzig, Math.-Phys. Kl. 68, 129-36. Ges.
Werke A985b), pp. 173-80.
1917, Affine Geometrie IX: Verschiedeiie Bemerkungen und Aufgaben. Ber. Verh.
.Sachs. Akad. Leipzig, Math.-Phys. Kl. 69. 412-20. Ges. Werke A985b), pp. 259-67.
1923, Vorlesungen iiber Differentialgeometrie, II. Affine Differentialgeometrie. Springer,
Berlin.
1924, Vorlesungen Uber Differentialgeometrie, I. Springer, Berlin.
1933, Uber die Schwerpunkte von Eibereichen. Math. Z. 36. 166. Ges. Werke A985b),
p. 350.

References 439
1936, Integralgeometrie 10: Eine isoperimetrische Eigenschaft der Kugel. Bull. Math.
Soc. Roum. Sci. 37, 3-7. Ges. Werke A985a), pp. 265-9.
1937a, Vorlesungen iiher Integralgeometrie. 3rd edn, VEB Deutsch. Verl. d. Wiss.,
Berlin, 1955 Ast edn: Part I, 1935; Part II, 1937).
1937b, Integralgeometrie 21, Liber Schiebungen. Math. Z. 42, 399-410. Ges. Werke
A985a), pp. 337-48.
1985a, Gesammelte Werke (eds W. Burau et al.), vol. 2, Kinematik and
Integralgeometrie. Thale.s, Essen.
1985b, Gesammelte Werke (eds W. Burau et al.). vol. 3, Konvexgeometrie. Thales,
Essen.
Blaschke, W. & Reidemeister, K. 1922, Uber die Entwicklung der Affingeometrie.
Jahresher. Deutsche Math.-Ver. 31. 63-82.
Blind, R. 1977, Eine Charakterisierung der SphSre im £'. Manuscripta Math. 21, 243-53.
Blumenthal. L. M. 1953, Theory and Applications of DLuance Geometry. Clarendon Press,
Oxford.
BoCek, L. 1983, Eine Verscharfung der Poincare-Ungleichung. Casopis pi.<:t. mat. 108,
78-81.
BOhm, J. & Hertel, E. 1980, Polyedergeometrie in n-dimensionalen R&umen konstanter
Kriimmung. VEB Deutsch. Verl. d. Wiss., Berlin.
Bokowski, J. 1973, Eine verschSrfte Ungleichung zwischen Volumen, OberflSche und
Inkugelradius im R". Elem. Math. 28, 43-4.
Bokowski, J., Hadwiger, H. & Wills, J. M. 1976, Eine Erweiterung der Croftonschen
Formeln fiir konvexe Kiirper. Mathematika 23, 212-19.
Bokowski, J. & Heil, E. 1986, Integral representations of quermassintegrals and Bon-
nesen-style Inequalities. Arch. Math. 47, 79-89.
Bokowski, J. & Mani-Levitska, P. 1984, Zur Approximation konvexer Korper im E".
Ann. Discrete Math. 20, 319-20.
1987, Approximation of convex bodies by polytopes with uniformly bounded valencies.
Monatsh. Math. 104, 261-4.
Bol, G. 1939, Zur Theorie der konvexen Korper. Jahresher. Deutsche Math.-Ver. 49,
113-23.
1940, Ein isoperimetrische Problem. Nieuw Arch. Wiskunde B) 20 A94), 171-5.
1941, Lsoperimetrische Ungleichungen fur Bereiche auf Flachen. Jahresher. Deutsche
Math.-Ver. 51, 219-57.
1942, Zur Theorie der Eikorper. Jahresher. Deutsche Math.-Ver. 52, 250-66.
1943a, Einfache Isoperimetrie-Beweise fiir Kreis und Kugel. Ahh. Math. Sem. Univ.
Hamhurg 15, 27-36.
1943b, Beweis einer Vermutung von H. Minkowski. Ahh. Math. Sem. Univ. Hamhurg
15, 37-56.
Bol, G. & Knothe, H. 1949, Uber konvexe Kiirper mit Ecken und Kanten. Arch. Math. 1
A948/1949), 427-31.
Bolker, E. D. 1969, A class of convex bodies. Trans. Amer. Math. Soc. 145, 323-46.
1971, The zonoid problem (Research Problem). Amer. Math. Monthly 78, 529-31.
Bonnesen, T. 1921, Uber eine Verscharfung der isoperimetrischen Ungleichhcit des
Kreises in der Ebene und auf der KugeloberflSche nebst einer Anwendung auf eine
Minkowskische Ungleichheit fiir konvexe Korper. Math. Ann. 84, 216-27.
1924, Uber das isoperimetrische Defizit ebener Figuren. Math. Ann. 91, 252-68.
1932, Beweis einer Minkowskischen Ungleichung. Mat. Tid^skr. B, 25-7.
Bonnice, W. & Klee, V. L. 1963, The generation of convex hulls. Math. Ann. 152, 1-29.
BOrOczky, K., B&T&ny, I., Makai, E. jr. & Pach, J. 1986, Maximal volume enclosed by
plates and proof of the chessboard conjecture. Discrete Math. 60, 101-20.
Borwein, J. M. & O'Brien, R. C. 1978, Cancellation characterizes convexity. Nanta Math.
n, 100-2.
Bose, R. C. & Roy, S. N. 1935a, Some properties of the convex oval with reference to its

440 References
perimeter centroid. Bull. Calcutta Math. Soc. 27, 79-86.
1935b, A note on the area centroid of a closed convex oval. Bull. Calcutta Math. Soc.
27, 111-18.
1935c, On the four centroids of a closed surface. Bull. Calcutta Math. Soc. 27, 119-46.
Botts, T. 1942, Convex sets. Amer. Math. Monthly 49, 527-35.
Bourgain, J. 1987, Remarques sur les zonoides (Projection bodies, etc.). In: Simin.
d'analysefonctionnelle, Paris 1985/1986/1987, Puhl. Math. Univ. Paris VII 28, 171-86.
1991 On the Busemann-Petty problem for perturbations of the ball. Geom. Functional
Anal. I, 1-13.
Bourgain, J. & Lindenstrauss, J. 1988a, Nouveaux r^sultats sur les zonoides et les corps de
projection. CR. Acad. Sci. Paris 306, 377-80.
1988b, Projection bodie.s. In; Geometric Aspects of Functional Analysis (eds J. Linden-
.strauss & V. D. Milman), Lecture Notes in Math. 1317, Springer, Berlin, pp. 250-70.
1988c, Distribution of points on spheres and approximation by zonotopes. Israel J.
Math. 64. 25-31.
1989 Almost euclidean sections in spaces with a symmetric basis. In: Geometric Aspects
of Functional Analysis (eds J. Lindenstrauss & V. D. Milman), Lecture Notes in Math.
1376, Springer, Berlin, pp. 278-88.
Bourgain, J., Lindenstrauss, J. & Milman, V. D. 1986, Sur I'approximation de zonoides
par des zonotopes. CR. Acad. Sci. Paris 303, 987-8.
1988, Minkowski sums and symmetrization.s. In: Geometric Aspects of Functional
Analysis (eds. J. Lindenstrauss & V. D. Milman), Lecture Notes in /Hath. 1317,
Springer, Berlin, pp. 44-66.
1989, Approximation of zonoids by zonotopes. Acta Math. 162, 73-141.
Bourgain, J. & Milman, V. 1985a, On Mahler's conjecture on the volume of a convex
symmetric body and its polar. Preprint IHES/M/85/17.
1985b, Sections euclidiennes et volume des corps sym^triques convexes dans R". CR.
Acad. Sci. Paris 300, 435-8.
1987, New volume ratio properties for convex symmetric bodies in R". Invent. Math. 88,
319-40.
Brascamp, H. J. & Lieb, E. H. 1976, On extensions of the Brunn-Minkowski and
Pr^kopa-Leindler theorems, including inequalities for log concave functions, and with
an application to the diffusion equation. J. Functional Anal. 22, 366-89.
Brehm, U. & Kiihnel, W. 1982, Smooth approximation of polyhedral surfaces regarding
curvatures. Geom. Dedicata 12, 435-61.
Br0ndsted, A. 1966, Convex sets and Chebyshev sets, II. Math. Scand. 18, 5-15.
1977, The inner aperture of a convex set. Pacific. J. Math. 72, 335-40.
1986, Continuous barycenter functions on convex polytopes. Expo. Math. 4, 179-87.
Bronshtein, E. M. 1976, «-entropy of convex sets and functions (in Russian). Sibirskii Mat.
Zh. 17, 508-14. English translation: Siberian Math. J. 17, 393-8.
1978a. e-entropy of the affine-equivalent convex bodies and Minkowski compacta (in
Russian). Optimizatsiya 22, 5-11.
1978b, Extremal convex functions (in Russian). Sihir.^kii Mat. Zh. 19, 10-18. English
translation: Siberian Math J. 19, 6-12.
1978c, Extremal convex functions and sets (in Russian). Optimizatcija 22, 12-23.
1979, Extremal H-convex bodies (in Russian). Sibirskii Mat. Zh. 20 A979), 412-15.
English translation: Siberian Math. J. 20, 295-7.
1981, On extreme boundaries of finite-dimensional convex compacts (in Russian).
Optimizatsiya 26 D3), 119-28.
Brooks, J. N. & Strantzen, J. B. 1989, Blaschke's rolling theorem in R". Memoirs Amer.
Math. Soc. 80, 101 pp.
Brown, A. L. 1980, Chebyshev sets and facial systems of convex sets in finite-dimensional
spaces. Proc. London Math. Soc. 41, 297-339.
Brunn, H. 1887, Ober Ovale und EiflSchen. Dissertation, Miinchen.

References 441
1889, Ober Curven ohne Wendepunkte. Habilitationsschrift, Miinchen.
1894, Referat Uber eine Arbeit: Exacte Grundlagen fiir eine Theorie der Ovale. S.-B.
Bayer. Akad. Wiss., 93-111.
Budach, L. 1989, Lipschitz-Killing curvatures of angular partially ordered sets. Adv.
Math. 78, 140-67.
Buldygin, V. V. & Kharazishvili, A. B. 1985, The Brunn-Minkowski Inequality and its
Applications (in Russian), "Naukova Dumka', Kiev.
Bunt, L. N. H. 1934, Bijdrage tot de theorie der convexe puntverzamelingen. Thesis,
Univ. Groningen, Amsterdam.
Burago, Ju. D. & Zalgaller, V. A. 1978, Sufficient criteria of convexity. J. Soviet Math.
10, 395-435.
1988, Geometric Inequalities. Springer, Berlin (Russian original: 1980).
Burckhardt, J. J. 1940, Uber konvexe Kiirper mit Mittelpunkt. Vierteljahresschr. Naturf.
Ges. Zurich 85, 149-54.
Burton, G. R. 1976, On the sum of a zonotope and an ellipsoid. Comment. Math. Helvet.
51,369-87.
1979a, On the nearest-point map of a convex body. J. London Math. Soc. 19, 147-52.
1979b, The measure of the ^-skeleton of a convex body. Mathematika 26 A979),
290-301.
1980a, Skeleta and sections of convex bodies. Mathematika 27, 97-103.
1980b, Subspaces which touch a Borel subset of a convex surface. J. London Math. Soc.
21, 167-70.
Burton, G. R. & Larman D. G. 1981, An inequality for skeleta of convex bodies. Arch.
Math. 36, 378-84.
Busemann, H. 1947, The Lsoperimetrlc problem in the Minkowski plane. Amer. J. Math.
69, 863-71.
1949a, A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat. Acad.
Sci. USA. 35,27-31.
1949b, The isoperimetric problem for Minkowski area. Amer. J. Math. 71, 743-62.
1950, The foundations of Minkowskian geometry. Comment. Math. Helvet. 24, 156-86.
1953, Volume in terms of concurrent cross-sections. Pacific J. Math. 3, 1-12.
1959, Minkow.ski's and related problems for convex surfaces with boundaries. Michigan
Math. J. 6, 259-66.
1960, Volumes and areas of cross-sections. Amer. Math. Monthly 67, 248-50.
Correction: ibid., p. 671.
Busemann, H. & Feller, W. 1935a, Bemerkungen zur Differentialgeometrie der konvexen
HSchen, I. Kiirzeste Linien auf differenzierbaren HSchen. Mat. Tidsskr. B, 25-36.
1935b, Bemerkungen zur Differentialgeometrie der konvexen Flachen, II. Ober die
Kriimmungsindikatrizen. Mat. Tidsskr. B, 87-115.
1936a, KrOmmungseigen.schaften konvexer HSchen. Acta Math. 66, 1-47.
1936b, Bemerkungen zur Differentialgeometrie der konvexen HSchen, II. Uber die
Gaussche KrOmmung. Mat. Tidsskr. B, 41-70.
Busemann, H. & Petty, 1956, Problems on convex bodies. Math. Scand. 4, 88-94.
Campi, S. 1981, On the reconstruction of a function on a sphere by its integrals over great
circles. Boll. Un. Mat. Ital., Ser. V, 18, 195-215.
1984, On the reconstruction of a star-shaped body from its 'half-volumes'. J. Austral.
Math. Soc, Ser. A, 37, 243-57.
1986, Reconstructing a convex surface from certain measurements of its projections.
Boll. Un. Mat. Ital. B F) 5, 945-59.
1988, Recovering a centred convex body from the areas of its shadows: A stability
estimate. Ann. Mat. Pura Appl. 151, 289-302.
Cas.sels, J. W. S. 1975, Measures of the non-convexity of sets and the Shapley-Folkman-
-Starr theorem. Math. Proc. Camb. Phil. Soc. 78, 433-6.
Chakerian, G. D. 1966, Sets of constant width. Pacific J. Math. 19, 13-21.

442 References
1967a, Inequalities for the difference body of a convex body. Proc. Amer. Math. Soc.
18, 879-84.
1967b, Sets of constant relative width and constant relative brightness. Trans. Amer.
Math. Soc. 129, 26-37.
1971, Higher dimensional analogue.s of an isoperimetric inequality of Benson. Math.
Nachr. 48. 33-41.
1972, The mean volume of boxes and cylinders circumscribed about a convex body.
IsraelJ. Math. 12,249-56.
1973, Isoperimetric inequalities for the mean width of a convex body. Geom. Dedicata
1, 356-62.
Chakerian, G. D. & Sangwine-Yager, J. R. 1979, A generalization of Minkowski's
inequality for plane convex sets. Geom. Dedicata 8, 437-44.
Cheeger, J.. Miiller, W. & Schrader, R. 1984, On the curvature of piecewise flat spaces.
Commun. Math. I'hys. 92, 405-454.
1986, Kinematic and tube formulas for piecewise linear spaces. Indiana Univ. Math. J.
35, 737-54.
Cheng, S.-Y. & Yau, S.-T. 1976, On the regularity of the solution of the n-dimensional
Minkowski problem. Commun. Pure Appl. Math. 29, 495-516.
Chern, S. S. 1959, Integral formulas for hypersurfaces in euclidean space and their
applications to uniqueness theorems. J. Math. Mech. 8, 947-55.
Chern, S. S. & Lashof, R. K. 1958, On the total curvature of immersed manifolds. II.
Michigan Math. 7. 5, 5-12.
Chernoff, P. R. 1969, An area-width inequality for convex curves. Amer. Math. Monthly
76. 34-5.
Choquet, G. 1969a, Lectures on Analysis, vol. II. W. A. Benjamin, Reading MA.
1969b, Lectures on Analysis, vol. Ill, W. A. Benjamin, Reading, MA.
1969c, Mesures coniques, affines et cylindriques. In: Proc. Symposia Math., INDAM,
Rome 1968, vol. II, Academic Press, London, pp. 145-82.
Choquet, G., Corson, H. & Klee, V. 1966, Exposed points of convex sets. Pacific J. Math.
17, 33-43.
Christoffel, E. B. 1865, Uber die Bestimmung der Gestalt einer krummen Oberflache
durch lokale Messungen auf derselben. J. reine angew. Math. 64, 193-209. Ges. Math.
Abh., vol. I, pp. 162-77. Teubner, Leipzig, 1910.
CieSlak, W. & G6idt, S. 1987, On curves which bound special convex sets. Serdica 13,
281-6.
Clack, R. 1990, Minkowski surface area under affine transformations. Mathematika 37,
232-8.
Coifman, R. R. & Weiss, G. 1968, Representations of compact groups and spherical
harmonics. L'Einseignement math. 14, 121-73.
Collier, J. B. 1975, On the set of extreme points of a convex body. Proc. Amer. Math.
Soc. 47 A975), 184-6.
1976, On the facial structure of a convex body. Proc. Amer. Math. Soc. 61, 367-70.
Corson, H. H. 1965, A compact convex set in £' whose exposed points are of the first
category. Proc. Amer. Math. Soc. 16, 1015-21.
Coxeter, H. S. M. 1962, The classification of zonohedra by means of projective diagrams.
J. Math, pures appl. 41, 137-56.
1963, Regular Polytopes, 2nd edn. Macmillan, New York.
Cre.ssie, N. 1978, A strong limit theorem for random sets. Suppl. Adv. Appl. Prob. 10,
36-46.
Curtis, D. W., Quinn, J. & Schori, R. M. 1977, The hyperspace of compact convex
subsets of a polyhedral 2-cell. Houston J. Math. 3, 7-15.
Czipszer, J. 1962, Uber die Parallelbereiche nach innen von Eibereichen. Publ. Math. Inst.
Hiingar. Acad. Sci. 7A, 197-202.
Dalla, L. 1985, The n-dimensional Hausdorff measure of the n-skeleton of a convex

References 443
w-compact set (body). Math. Nachr. 123, 131-5.
1986, Increasing paths on the one-skeleton of a convex compact set in a normed space.
Pacific J. Math. 124, 289-94.
1987, On the measure of the one-skeleton of the sum of convex compact sets. J. Austral.
Math. Soc. Ser. A 42, 385-9.
Dalla, L. & Larman, D. G. 1980, Convex bodies with almost all A:-dimensional sections
polytopes. Math. Proc. Camh. Phil. Soc. 88, 395-401.
Das Gupta, S. 1980. Brunn-Minkowski inequality and its after-math. J. Multivariate Anal.
10, 296-318.
Davis, C. 1954, Theory of positive linear dependence. Amer. J. Math. 76, 733-46.
Davy, P. J. 1976, Projected thick sections through multidimensional particle aggregates. J.
Appl. Proh. 13, 714-22. Correction: J. Appl. Proh. IS, 456.
1978, Stereology - a statistical viewpoint. The.sis, Austral. Nat. Univ., Canberra.
1980a. The estimation of centroids in stereology. Mikro.<:kopie (Wien) 37 (Suppl.),
94-100.
1980b, The stereology of location. J. Appl. Proh. 17, 860-4.
I98I, Interspersion of phases in a material. J. Microscopy 121, 3-12.
Debrunner, H. 1955, Zu einem maUgeometrischen Satz iiber Korper konstanter Breite.
Math. Nachr. 13, 165-7.
Debs, G. 1978, Applications affines ouvertes et convexes compacts stables. Bull. Sc. math.
B) 102,401-14.
Deicke, A. 1953, Liber die Finsler-Raume mit A, = 0. Arch. Math. 4, 45-5L
Delgado, J. A. 1979, Blaschke's theorem for convex hypersurfaces. J. Differential Geom.
14, 489-96.
de Rham, G. 1957, Sur quelques courbes d^finies par des equations fonctionnelles. Rend.
Sem. Mat. Univ. Politecn. Torino 16 A956/57), 101-13.
Dierolf, P. 1970, Uber den Auswahlsatz von Blaschke und ein Problem von Valentine.
Manuscripta Math. 3, 289-304.
Dinghas, A. 1939, Zur Theorie der konvexen Korper im n-dimensionalen Raum. Ahh.
Preufi. Akad. Wiss., Phys.-Math. Kl. 4, 30 pp.
1940a, Geometrische Anwendungen der Kugelfunktionen. Nachr. Ges. Wiss. Gott.,
Math.-Phys. W., 1, 213-35.
1940b, Verallgemeinerung eine Blaschkeschen Satzes iiber konvexe Korper konstanter
Breite. Rev. Math. Union Interhalkan. 3, 17-20.
1940c, VerschSrfung der isoperimetrischen Ungleichung fiir konvexe Korper mit Ecken.
Math. Z. 47, 669-75.
1942, Isoperimetrische Ungleichung fiir konvexe Bereiche mit Ecken. Math. Z. 48,
428-40.
1943a, Ober die lineare isoperimetrische Ungleichung fiir konvexe Polygone und Kurven
mit Ecken. Monatsh. Math. Phys. 51. 35-45.
1943b, VerschSrfung der Minkowskischen Ungleichungen fiir konvexe Kiirper. Monatsh.
Math. Phys. 51, 46-56.
1944, Uber einen geometrischen Satz von Wulff fiir die Gleichgewichtsform von
Kristallen. Zeitschr. Kristallographie 105, 304-14.
1948, Bemerkung zu einer Verscharfung der isoperimetrischen Ungleichung durch H.
Hadwiger. Math. Nachr. 1. 284-6.
1949a, Zur Theorie der konvexen Rotationskiirper im n-dimensionalen Raum. Math.
Nachr. 2, 124-40.
1949b, Neuer Beweis einer verscharften Minkowskischen Ungleichung fiir konvexe
Korper. Math. Z. 51, 306-16.
1949c, Neuer Beweis einer isoperimetrischen Ungleichung von Bol. Math. Z. 51,
469-73.
I949d, Uber eine neue isoperimetrische Ungleichung fUr konvexe Polyeder. Math. Ann.
120. 533-8.

444 References
1956, Zum Minkowskischen Integralbegriff abgeschlossener Mengen. Math. Z. 66,
173-88.
1957a, Uber das Verhalten der Entfemung zweier Punktmengen bei gleichzeitiger
Symmetrisierung derselben. Arch. Math. 8, 46-57.
1957b, Uber eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lus-
ternikschem Typus. Math. Z. 68, 111-25.
I96I, Minkowskische Summen und Integrate, superaddiiive Mengenfunktionale,
isoperimetrische Ungleichungen. Gauthier-Villars, Paris.
Diskant, V. I. 1964, Stability in Liebmann's theorem (in Russian). Dokl. Akad. Nauk
SSSR 158, 1257-59. English translation: Soviet Math. 5, 1387-90.
1965, Theorems of stability for surfaces close to the sphere (in Russian). Sihirskii Mat.
Zh. 6. 1254-66.
1968, Stability of a sphere in the class of convex surfaces of bounded specific curvature
(in Russian). Sihirskii Mat. Zh. 9, 816-24. English translation: Siberian Math. J. 9,
610-15.
1971a, Bounds for convex surfaces with bounded curvature functions (in Russian).
Sihirskii Mai. Zh. 12. 109-25. English translation: Siberian Math. J. 12, 78-89.
I97Ib, Convex surfaces with bounded mean curvature (in Russian). Sihirskii Mat. Zh.
12, 659-63. English translation: Siberian Math. J. 12, 469-72.
1972, Bounds for the discrepancy between convex bodies in terms of the isoperimetric
difference (in Russian). Sihirskii Mat. Zh. 13, 767-72. English translation: Siberian
Math. J. 13.529-32A973).
1973a, Stability of the solution of the Minkowski equation (in Russian). Sihirskii Mat.
Zh. 14, 669-73. English translation: Siberian Math. J. 14, 466-69.
1973b. Strengthening of an isoperimetric inequality (in Russian). Sihirskii Mat. Zh. 14,
873-77. English translation: Siberian Math. J. 14, 608-11.
1973c, A generalization of Bonnesen's inequalities (in Russian). Dokl. Akad. Nauk
SSSR 213, 519-21. English translation: Soviet Math. Dokl. 14, 1728-31.
1975, The stability of the solutions of the generalized Minkowski equations for a ball (in
Russian). Ukrain. Geom. Sh. 18, 53-9.
1976, Stability of a convex body under a change in the (n - 2)nd curvature function (in
Russian). Ukrain. Geom. Sh. 19, 22-33.
1979. On the question of the order of the stability function in Minkowski's problem (in
Russian). Ukrain. Geom. Sh. 22, 45-7.
1982, Stability of the solution of the Minkowski equation for the surface area of convex
bodies (in Russian). Ukrain. Geom. Sh. 25,43-51.
1984, A counterexample to an assertion of Bonnesen and Fenchel (in Russian). Ukrain.
Geom. Sh. 27,31-3.
1985. Stability in Aleksandrov's problem for a convex body, one of whose projections is
a ball (in Russian). Ukrain. Geom. Sh. 28, 50-62, English translation: J. Sov. Math.
48 A990), 41-9.
Dor, L. E. 1976, Potentials and isometric embeddings in L,. Israel. J. Math. 24, 260-8.
DreJevid, M. 1970, A certain generalization of Blaschke's theorem to the class of
m-convex sets (in Serbo-Croatian). Mat. Vesnik 7, 223-6.
Dubins. L. E., 1962, On extreme points of convex sets. J. Math. Anal. Appl. 5, 237-44.
Dubois, C. & Jongmans, F. 1982, Invitation 4 la gestion des stocks de cdnes. Bull. Soc.
Roy. Sci. Liige 51, 107-20.
Dudley, R. 1974, Metric entropy of some classes of sets with differentiable boundaries. J.
Approximation Th. 10, 227-36. Corrigendum: ihid. 26, 192-3.
1977, On second derivatives of convex functions. Math. Scand. 41, 159-74.
Duporcq, E. 1896, Sur les centres de gravity des courbes parallfeles. Bull. Soc. Math.
France 24, 192-4.
1897, Sur les centres de gravity des surfaces parallfeles a une surface ferm^e. C.R. Acad.
Sci. Paris 124, 492-3.

References 445
Dziechciiiska-Halamoda, Z. & Szwiec, W. 1985, On critical sets of convex polyhedra.
Arch. Math. 44,461-6.
Eckhoff, J. 1979, Radon's theorem revisited. In: Contributions to Geometry, Proc.
Geometry Symp. Siegen 1978 (eds J. Tiilke & J. M. Wills), Birkhauser, Basel, pp.
164-85.
1980, Die Euler-Charakteristik von Vereinigungen konvexer Mengen im /?''. Abh. Math.
Sem. Univ. Hamburg SO, 133-44.
Edelstein, M. 1965, An example concerning exposed points. Canad. Math. Bull. 8, 323-7.
Edelstein, M. & Fesmire, S. 1975, On the extremal structure and closure of sums of
convex sets. Bull. Soc. Roy. Sci. Liege 44, 590-9.
Efimow, N. W. 1957, FlUchenverbiegung im Grofien. Akademie-Verlag, Berlin.
Eggleston, H. G., Griinbaum, B. & Klee, V. 1964, Some semicontinuity theorems for
convex polytopes and cell complexes. Comment. Math. Helvet. 39, 165-88.
Eifler, L. Q. 1977, Semi-continuity of the face-function for a convex set. Comment. Math.
Helvet. 52, 325-8.
Ekeland, I. & Temam, R. 1976, Convex Analysis and Variational Problems. North-
Holland, Amsterdam.
Ewald, G. 1964, Cber die Schattengrenzen konvexer Korper. Abh. Math. Sem. Univ.
Hamburg 27, 167-70.
1965, Von Klassen konvexer Korper erzeugte HilbertrSume. Math. Ann. 162, 140-6.
1967, On Busemann's theorem of the Brunn-Minkowski type. In; Proc. Colloquium
Convexity, Copenhagen 1965, K0benhavns Univ. Mat. Inst., pp. 69-71.
1988, On the equality case in Aleksandrov-Fenchel's inequality for convex bodies.
Geom. Dedicata 28, 213-20.
Ewald, G., Larman, D. G. & Rogers, C. A. 1970, The direction.s of the line segments and
of the r-dimensional balls on the boundary of a convex body in Euclidean space.
Mathematika 17, 1-20.
Ewald, G. & Shephard, G. C. 1966, Normed vector spaces consisting of classes of convex
sets. Math. Z. 91, I-19.
Falconer, K. J. 1983, Applications of a result on spherical integration to the theory of
convex sets. Amer. Math. Monthly 90, 690-3.
Faro Rivas, R. 1986, Aproximacion de cuerpos convexos simetricos. Thesis, Univ. de
Extremadura, 1986, 147 pp.
FSry, I. I96I, Functional related to mixed volumes. Illinois J. Math. 5, 425-30.
1983, Expectation of the isoperimetric deficit: R^nyi's integral. Z. Wahrscheinlichkeitsth.
verw. Geb. 63, 289-95.
1986, The isoperimetric deficit as the integral of a positive function. Bull. Sci. Math. B)
110, 315-34. (This work is erroneous.)
F5ry, I. & Makai, E. jr. 1982, Isoperimetry in variable metric. Studia Sci. Math. Hungar.
17, 143-58.
Favard, J. 1929, Problfemes d'extremums relatifs aux courbes convexes, I. Ann. Sci. Ec.
Norm. Sup. 46, 345-69.
1930, Sur les inegalitds de Minkowski, Mat. Tidsskr. B, 33-40.
1933a, Sur les corps convexes. J. Math, pures appl. (9) 12, 219-82.
1933b, Sur la determination des surfaces convexes. Bull. Acad. Roy. Belgique (CI. Sci.)
19, 65-75.
1938, Sur la determination des surfaces convexes. CR. Acad. Sci. Paris 208, 871-3.
Federer, H. 1959, Curvature measures. Trans. Amer. Math. Soc. 93, 418-91.
1969, Geometric Measure Theory. Springer, Berlin.
Fedotov, V. P. 1978a, On the notions of faces of a convex compactum (in Russian).
Ukrain. Geom. Sb. 21, 131-41.
1978b, On the sum of pth surface functions (in Russian). Ukrain. Geom. Sb. 21,
125-31.
1979a, Isolated faces of a convex compactum (in Russian). Mat. Zametki 25, 139-47.

446 References
English translation: Math. Notes 25, 73-7.
1979b, A counterexample to a hypothesis of Firey (in Russian). Mat. Zametki 26,
269-75. English translation: Math. Notes 26, A980), 625-9.
1982, Polar representation of a convex compactum (in Russian). Ukrain. Geom. Sb. 25,
137-8.
Fejes T6th, L. 1948, The isepiphan problem for n-hedra. Amer. J. Math. 70, 174-80.
1950, Elementarer Beweis enier isoperimetrischen Ungleichung. Acta Math. Acad. Sci.
Hangar. 1, 273-5.
Fenchel, W. 1936a, Indgalitds quadratiques entre les volumes mixtes des corps convexes.
C.R. Acad. Sci. Paris 203, 647-50.
1936b, Gdndralisation du thdorfime de Brunn et Minkowski concemant les corps
convexes. C.R. Acad. Sci. Paris 203, 764-6.
1938, Uber die neuere Entwicklung der Brunn-Minkowskischen Theorie der konvexen
Korper. In: Proc. 9th Congr. Math. Scand., Helsingfors 1938, pp. 249-72.
Fenchel, W. & Jessen, B. 1938, Mengenfunktionen und konvexe Kiirper. Danske Vid.
Selskab. Mat.-fys. Medd. 16, 3.
Ferrers, N. M. 1861, Note on a geometrical theorem of Mr. Steiner. Quarterly J. Math. 4,
92-4.
Fet, A, I. 1963, Stability theorems for convex, almost spherical surfaces (in Russian).
Dokl. Akad Nauk SSSR 153, 537-9. English translation: Soviet Math. 4, 1723-5.
Figiel, T., Lindenstrauss, J. & Milman, V. 1977. The dimension of almost spherical
sections of convex bodies. Acta Math. 129, 53-94.
Filliman. P. 1988, Extremum problems for zonotopes. Geom. Dedicata 27, 251-62.
Fillmore, J. R. 1969. Symmetries of surfaces of constant width. J. Differential Geom. 3,
103-10.
Firey, W. J. 1961a, Polar means of convex bodies and a dual to the Brunn-Minkowski
theorem. Canad. J. Math. 13, 444-53.
1961b. Mean cross-section measures of harmonic means of convex bodies. Pacific J.
Math. 11. 1263-6.
1962, p-means of convex bodies. Math. Scand. 10. 17-24.
1963, The mixed area of a convex body and its polar reciprocal. Israel J. Math. 1,
201-2.
1964, A generalization of an inequality of H. Groemer. Monatsh. Math. 68, 393-5.
1965a, The brightness of convex bodies. Technical Report no. 19, Oregon State Univ.
1965b, Lower bound for volumes of convex bodies. Arch. Math. 16, 69-74.
1966, Lower bounds for qucrmassintegrals of a convex body. Portugaliae Math. 25,
141-6.
1967a, Blaschke sums of convex bodies and mixed bodies. In: Proc. Colloquium
Convexity, Copenhagen 1965, Kpbenhavns Univ. Mat. Inst. 1967, pp. 94-101.
1967b, The determination of convex bodies from their mean radius of curvature
functions. Mathematika 14, I-13.
1967c, Generalized convex bodies of revolution. Canad. J. Math. 14. 972-96.
1968, Christoffel's problem for general convex bodies. Mathematika 15, 7-21.
1969, An upper bound for volumes of convex bodies. J. Austral. Math. Soc. 9, 503-10.
1970a, Local behaviour of area functions of convex bodies. Pacific J. Math. 35, 345-57.
1970b, The determination of convex bodies by elementary symmetric functions of
principal radii of curvature. Mimeographed manuscript.
1970c, Intermediate Christoffel-Minkowski problems for figures of revolution. Israel J.
Math. 8, .•!84-90.
I970d, Convex bodies of constant outer p-measure. Mathematika 17, 21-7.
1972, An integral-geometric meaning for lower order area functions of convex bodies.
Mathematika 19, 205-12.
1973, Support flats to convex bodies. Geom. Dedicata 2, 225-48.
1974a, Kinematic measures for sets of support figures. Mathematika 21, 270-81.

References 447
1974b, Approximating convex bodies by algebraic ones. Arch. Math. 25, 424-5.
1974c, Extension of some integral formulas of Chem. Geom. Dedicata 3, 325-33.
I974d, Shapes of worn stones. Mathematika 21, 1-11.
1975, Some open questions on convex surfaces. In: Proc. Int. Congr. Math., Vancouver
1974 (ed. R. D. James), Canad. Mathematical Congress, pp. 479-84.
1976, A functional characterization of certain mixed volumes. Israel J. Math. 24,
274-81.
1977, Addendum to R. E. Miles' paper on the fundamental formula of Blaschke in
integral geometry. Austral. J. Statist. 19, 155-6.
1979, Inner contact measures. Mathematika 26, 106-12.
1981, Subsequent work on Christoffel's problem about determining a surface from local
measurements. In: E. B. Christoffel, Aachen/Monschau 1979 (eds P. Butzer & F.
Fehdr), BirkhSuser, Basel, pp. 721-3.
Firey, W. J. & GrUnbaum, B. 1964, Addition and decomposition of convex polytopes.
hraelj. Math. 2,91-100.
Firey, W. J. & Schneider, R. 1979, The size of skeletons of convex bodies. Geom.
Dedicata 8,99-103.
Fisher. J. C. 1987, Curves of constant width from a linear viewpoint. Math. Mag. 60,
131-40.
Flaherty, F. J. 1973, Curvature measures for piecewise linear manifolds. Bull. Amer.
Math. Soc. 79, 100-2.
Flanders, H. 1966, The Steiner point of a closed hypersurface. Mathematika 13, 181-8.
1968, A proof of Minkowski's inequality for convex curves. Amer. Math. Monthly 75,
581-93.
Flatto, L. & Newman, D. J. 1977, Random coverings. Acta Math. 138, 241-64.
Fiorian, A. 1986, Approximation of convex discs by polygons. Discrete Comput. Geom. 1
A986), 241-63.
1989, On a metric for the class of compact convex sets. Geom. Dedicata 30, 69-80.
Focke, J. 1969a, Symmetrische n-Orbiformen kleinsten Inhahs. Acta Math. Acad. Sci.
Hung. 20. 39-68.
1969b, Die beste Ausbohrung eines regularen n-Ecks. Z. Angew. Math. Mech. 49,
235-48.
Focke, J. & Gensel, B. I97I, n-Orbiformen maximaler Randknickung. Beitr. Anal. 2,
7-16.
Fourneau, R. 1979, Espaces mdtriques constituds de classes de polytopes convexes lids aux
problemes de ddcomposition. Geom. Dedicata 8, 463-76.
1985, Choquet simplices in finite dimensions. In: Discrete Geometry and Convexity (eds
J. E. Goodman, E. Lutwak, J. Malkevitch & R. Pollack), Ann. New York Acad. Sci.
440. 147-62.
Fu, J. H. G. 1989. Curvature measures and generalized Morse theory. J. Differential
Geom. 30, 619-42.
1990, Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39, 1115-54.
1990+a, Curvature of singular spaces via the normal cycle. Preprint.
1990+b, Curvature measures of subanalytic sets. Preprint.
Fuglede, B. 1986a, Continuous selection in a convexity theorem of Minkowski. Expo.
Math. 4, 163-78.
1986b, Stability in the isoperimetric problem. Bull. London Math. Soc. 18, 599-605.
1989, Stability in the isoperimetric problem for convex or nearly spherical domains in
R". Trans. Amer. Math. Soc. 314, 619-38.
1991, Bonnesen's inequality for the isoperimetric deficiency of closed curves in the
plane. Geom. Dedicata 38, 283-300.
Fujiwara, M., I9I5, Uber die einem Vielecke eingeschriebenen und umdrehbaren kon-
vexen geschlossenen Kurven. Sci. Rep. Tdhoku Univ. 4, 43-55.
I9I6, Cber die Anzahl der Kantenlinien einer geschlossenen konvexen Hache. Tdhoku

448 References
Math. J. 10, 164-6.
I9I9, Uber die innen-umdrehbare Kurve eines Vielecks. Sci Rep. Tdhoku Univ. 8,
221-46.
Fujiwara, M. & Kakeya, S. I9I7, On some problems of maxima and minima for the curve
of constant breadth and the in-revolvable curve of the equilateral triangle. Tdhoku
Math. J. 11,92-110.
Gage, M. E. 1990, Positive centers and Bonnesen's inequality. Proc. Amer. Math. Soc.
110, I04I-8.
Gale, D. 1954, Irreducible convex sets. In: Proc. Int. Congr. Math., Amsterdam 1954, vol.
II, Noordhoff, Groningen; North-Holland, Amsterdam, pp. 217-18.
Gale, D. & Klee, V. 1959, Continuous convex sets. Math. Scand. 7, 379-91.
1975, Unique reducibility of subsets of commutative topological groups and semigroups.
Math. Scand. 36, 174-98.
Gallivan, S. 1979, On the number of disjoint increasing paths in the one-skeleton of a
convex body leading to a given exposed face. Israel J. Math. 32, 282-8.
Gallivan, S. & Gardner, R. J. I98I, A counterexample to a 'simplex algorithm' for convex
bodies. Geom. Dedicata 11, 475-88.
Gallivan, S. & Larman, D. G. 1981, Further results on increasing paths in the
one-skeleton of a convex body. Geom. Dedicata 11, 19-29.
Geivaerts, M. 1972, Enkele eigenschappen van de relatie "Homothetisch aanpasselijk' in de
ruimte der konvexe lichamen. Mededelingen Kon. Ac. voor Wetenschappen, Letteren
en Schone kunsten van Belgie 34, 6, 19 pp.
1974a, Classes of homothetically adaptable convex bodies. Acad. Roy. Belgique, Bull.
Classe Sci. E) 60, 409-29.
1974b, Vector spaces consisting of classes of convex bodies. In: Journ^e de Convexity,
Univ. de Lifege, Inst, de Math.
1976, Vector spaces consisting of classes of convex bodies. Geom. Dedicata S, 175-87.
Geppert, H. 1937, Uber den Brunn-Minkowskischen Satz. Math. Z. 42, 238-54.
Gericke, H. 1937, Uber eine Ungleichung fiir gemischte Volumina. Integralgeometrie 23.
Deutsche Math. 2, 61-7.
1940a, Uber stiitzbare Flachen und ihre Entwicklung nach Kugelfunktionen. Math. Z.
46.55-61.
1940b, Stiitzbare Bereiche in komplexer Fourierdarstellung. Deutsche Math. S, 279-99.
I94I, Zur Relativgeometrie ebener Kurven. Math. Z. 47, 215-28.
Giannopoulos, A. A. 1990, A note on a problem of H. Busemann and C. M. Petty
concerning sections of symmetric convex bodies. Mathematika 37, 239-42.
Giertz, M. 1969, A note on a problem of Busemann, Math. Scand. 25, 145-8.
Gind, E. & Hahn, M. G. 1985a, The L^vy-HinCin representation for random compact
convex subsets which are infinitely divisible under Minkow.ski addition. Z. Wahr-
scheinlichkeitsth. verw. Geb. 70, 271-87.
1985b, M-infinitely divisible random compact convex sets. In: Probability in Banach
Spaces V, Proc. 5th Int. Conf., Medford MA 1984, Lecture Notes in Math. 1153,
Springer, pp. 226-48.
1985c, Characterization and domains of attraction of p-stable random compact sets.
Ann. Probab. 13, 447-68.
Gind, E., Hahn, M. G. & Zinn, J. 1983, Limit theorems for random sets: An application
of probability in Banach space results. In: Probability in Banach Spaces IV, Lecture
Notes in Math. 990, Springer, pp. 112-35.
Gluck, H. 1972, The generalized Minkowski problem in differential geometry in the large.
Ann. Math. 96, 245-76.
1975, Manifolds with preassigned curvature - a survey. Bull. Amer. Math. Soc. 81,
313-29.
Godbersen, C. 1938, Der Satz vom Vektorbereich in RSumen beliebiger Dimensionen.
Dissertation, Giittingen.

References 449
Godet-Thobie, Ch. & The Lai, Ph. 1970, Sur le plongement de I'ensemble des convexes.
fermds, bornds d'un espace vectoriel topologique localement convexe. C.R. Acad. Sci.
Paris, S&. A 271, 84-7.
Goikhman, D. M. 1974, The differentiability of volume in Blaschke lattices (in Russian).
Sihirskii Mat. Zh. IS, 1406-8. English translation: Siberian Math. J. IS, 997-9.
Goldberg, M. 1948, Circular-arc rotors in regular polygons. Amer. Math. Monthly SS,
393-402.
1957, Trammel rotors in regular polygons. Amer. Math. Monthly 64, 71-8.
1960, Rotors in polygons and polyhedra. Math. Computation 14, 229-39.
1966, Rotors of variable regular polygons. Elem. Math. 21, 25-7.
Goodey, P. R. 1977, Centrally symmetric convex sets and mixed volumes. Mathematika
24, 193-8.
1981, Limits of intermediate surface area measures of convex bodie.s. Proc. London
Math. Soc. CL3, 151-68.
1982, Connectivity and freely rolling convex bodies. Mathematika 29, 249-59.
1984a, Centrally symmetric convex bodies. Mathematika 31, 305-22.
1984b, Degrees of symmetry of convex bodies. Mitt. Math. Ges. Giefien 166, 39-48.
1986, Instability of projection bodies. Geom. Dedicata 20, 295-305.
Goodey, P. R. & Groemer, H. 1990, Stability re.sults for first order projection bodies.
Proc. Amer. Math. Soc. 109, 1103-14.
Goodey, P. R. & Howard, R. 1990, Proce.sses of flats induced by higher dimensional
processes. Adv. Math. 80, 92-109.
Goodey, P. R. & Schneider, R. 1980, On the intermediate area functions of convex
bodies. Math. Z. 173, 185-94.
Goodey, P. R. & Weil, W. 1984, Distributions and valuations. Proc. London Math. Soc.
CL9,504-16.
1987, Translative integral formulae for convex bodies. Aequationes Math. 34, 64-77.
1990+, Centrally symmetric convex bodies and the spherical Radon transform. Preprint.
1991, Centrally symmetric convex bodies and Radon transforms on higher order
Grassmannians. Mathematika 38, 117-33.
I99I + , The determination of convex bodies from the mean of random sections.
Preprint.
Goodman, A. W. 1985, Convex curves of bounded type. Int. J. Math. Math. Sci. 8,
625-33.
Gordon, Y. 1985, Some inequalities for Gaussian processes and applications. Israel J.
Math. SO, 265-89.
Gordon, Y., Meyer, M. & Reisner, S. 1988, Zonoids with minimal volume product - a
new proof. Proc. Amer. Math. Soc. 104, 273-6.
Giirtler, H. 1937a, Zur Addition beweglicher ebener Eibereiche. Math. Z. 42, 313-21.
1937b, Erzeugung stUtzbarer Bereiche I. Deutsche Math. 2, 454-66.
1938, Erzeugung stiitzbarer Bereiche II. Deutsche Math. 3, 189-200.
Gray, A. 1990. Tubes. Addison-Wesley, Redwood City, CA.
Gray, A. & Vanhecke, L. I98I, The volume of tubes in a Riemannian manifold. Rend.
Sem. Mat. Univ. PoUtecn. Torino C) 39, 1-50.
Green, J. W. 1950, Sets .subtending a constant angle on a circle. Duke Math. J. 17, 263-7.
Grinberg, E. L. 1986, Isoperimetric inequalities for A:-dimensional sections of convex
bodies. Preprint IHES/M/89/28.
I99I, Isoperimetric inequalities and identities for A:-dimensional cross-sections of convex
bodies. Math. Ann. 291, 75-86.
Grinberg, E. L. & Rivin, I. 1990, Infinitesimal aspects of the Busemann-Petty problem.
Bull. London Math. Soc. 22, 478-84.
Gritzmann, P. 1988, A characterization of all loglincar inequalities for three quermas.s-
integrals of convex bodies. Proc. Amer. Math. Soc. 104, 563-70.
Gritzmann, P. & Sturmfels, B. 1990, Minkowski addition of polytopes: Computational

450 References
complexity and applications to Griibner bases. Report No. 212, Inst. Math. Univ.
Augsburg.
Gritzmann. P., Wills, J. M. & Wrase, D. 1987. A new isoperimetric inequality. J. reine
angew. Math. 379, 22-30.
Groemer, H. 1965, Eine neue Ungleichung fiir konvexe Korper. Math. Z. 86, 361-4.
1972, Eulersche Charakteristik, Projektionen und QuermaBintegrale. Math. Ann. 198,
23-56.
1973, Uber einige Invarianzeigenschaften der Eulerschen Charakteristik. Comment.
Math. Helvet. 48, 87-99.
1974, On the Euler characteristic in spaces with a separability property. Math. Ann. 211,
315-21.
1975, The Euler characteristic and related functional on convex surfaces. Geom.
Dedicata 4, 91-lOA.
1977a, Minkowski addition and mixed volumes. Geom. Dedicata 6, 141-63.
1977b, On translative integral geometry. Arch. .Math. 29, 324-30.
1978. On the extension of additive functionaU on classe.s of convex sets. Pacific J. Math.
75. 397-410.
1980a, The average measure of the intersection of two sets. Z. Wahrscheinlichkeitsth.
venv. Geb. 54, 15-20.
1980b, The average di.stance between two convex sets. J. Appl. Proh. 17, 415-22.
1987a, Stability theorems for projections of convex sets. Israel. J. Math. 60, 177-90.
1987b, On rings of sets associated with characteristic functions. Arch. Math. 49, 91-3.
1988a. On the Brunn-Minkowski theorem. Geom. Dedicata 27, 357-71.
1988b. Stability theorems for convex domains of constant width. Canad. Math. Bull. 31,
328-37.
1990a. Stability properties of geometric inequalities. Amer. Math. Monthly 97, 382-94.
1990b. On an inequality of Minkowski for mixed volumes. Geom. Dedicata 33, 117-22.
1991a. Stability properties of Cauchy's surface area formula. Monatsh. Math. 112,
43-60.
1991b, Stability theorems for projections and central symmetrization. Arch. Math. 51,
394-9.
Groemer, H. & Schneider, R. 1991, Stability estimates for some geometric inequalities.
Bull. London Math. Soc. 23, 67-74.
Gromov. M. 1988, Convex sets and Kahler manifolds. Preprint IHES/M/88./I0.
Gromov, M. & Milman, V. D. 1987, Generalization of the spherical isoperimetric
inequality to uniformly convex Banach spaces. Compositio Math. 62, 263-82.
Gruber, P. M. 1969, Zur Charakterisierung konvexer KOrper. Uber einen Satz von Rogers
und Shephard I. Math. Ann. 181, 189-200.
1970a, Zur Charakterisierung konvexer Korper. Uber einen Satz von Rogers und
Shephard II. Math. Ann. 184, 79-105.
1970b, Uber die Durchschnitte von translationsgleichen Polyedern. Monatsh. Math. 74,
223-38.
I97I, Uber eine Kennzeichnung der Simplices des R". Arch. Math. 22. 94-102.
1977, Die meisten konvexen Korper sind glatt, aber nicht zu glatt. Math. Ann. 229,
259-66.
1978, Isometries of the space of convex bodies of £■*. Mathematika 25, 270-8.
1980a, The space of compact subsets of E''. Geom. Dedicata 9, 87-90.
1980b, Isometrien des Konvexringes. Colloq. Math. 43, 99-109.
1982, Isometries of the space of convex bodies contained in a Euclidean ball. Israel J.
Math. 42, 277-83.
1983, In most cases approximation is irregular. Rend. Sem. Mat. Univ. Politecn. Torino
41, 19-33.
1984, Aspects of convexity and its applications. Expo. Math. 2, 47-83.

References 451
1985, Results of Baire category type in convexity. In: Discrete Geometry and Convexity
(eds J. E. Goodman, E. Lutwak, J. Malkevitch & R. Pollack), Ann. New York Acad.
Sci. 440, 163-9.
1988, Volume approximation of convex bodies by inscribed polytopes. Math. Ann. 281,
229-45.
1989, Dimension and structure of typical compact sets, continua and curves. Monatsh.
Math. 108, 149-64.
1990+, Asymptotic best and step by step approximation of convex bodies. Preprint.
I99Ia, A typical convex surface contains no closed geodesic. J. reine angew. Math. 416,
195-205.
1991b, Volume approximation of convex bodies by circumscribed polytopes. In: Applied
Geometry and Discrete Mathematics (The Victor Klee Festschrift, eds P. Gritzmann &
B. Sturmfels), DIMACS Series, vol. 4, Amer. Math. Soc, Providence, RI, pp.
309-17.
Gruber, P. M. & Kenderov, P. 1982, Approximation of convex bodies by polytopes.
Rend. Circ. Mat. Palermo 31, 195-225.
Gruber, P. M. & Lcttl, G. 1979, Isometries of the space of compact subsets of E''. Studia
Sci. Math. Hung. 14, 169-81.
1980, Isometries of the space of convex bodies in Euclidean space. Bull. London Math.
Soc. 12, 455-62.
Gruber, P. M. & Sorger, H. 1989, Shadow boundaries of typical convex bodies. Measure
properties. Mathematika 36, 142-52.
Gruber, P. M. & Tichy, R. 1982, Isometries of spaces of compact or compact convex
subsets of metric manifold.s. Monatsh. Math. 93, 116-26.
Grza^lewicz, R. 1984, Extreme convex sets in R^. Arch. Math. 43, 377-80.
Guggenheimer, H. 1969, Nearly spherical surfaces. Aequationes Math. 3, 186-93.
1973, Polar reciprocal convex bodies. Israel J. Math. 14, 309-16. Correction: ibid. 29
A978), 312 (Both works are erroneous.)
Guggenheimer, H. & Lutwak. E. 1976, A characterization of the n-dimensional parallel-
otope. Amer. Math. Monthly 83, 475-8.
Hadamard, J. 1897, Sur certaines propriet^s des trajectoires en dynamique. J. Math. Pures
Appl. 3,331-87.
Hadwiger, H. 1941, Gegenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie.
Vierteljahresschr. Naturf. Ges. Ziirich 86, 152-6.
1944, Eine elementare Ableitung der isoperimetrischen Ungleichung fiir Polygone.
Comment. Math. Helvet. 16 A943/44), 305-9.
1945, Die erweiterten Steinerschen Formeln fiir ebene und spharische Bereiche.
Comment. Math. Helvet. 18, 59-72.
1946a, Ober die erweiterten Steinerschen Formeln fiir Parallelflachen. Rev. Hisp.-Amer.
DN, 1-4.
1946b, Uber das Volumen der Parallelmengen. Mitt. Naturf. Ges. Bern. 3, I2I-5.
1946c, Eine Erweiterung des Steiner-Minkowskischen Satzes fiir Polyeder. Experientia
2.2, 1-2.
I946d, Inhaltsungleichungen fur innere und SuQere Parallelmengen. Experientia, 2, 12,
1-2.
1947. Uber eine symbolisch-topologische Formel. Elem. Math. 2, 35-41 (Portugue.se
translation: Gazeta Mat. 35, 6-9).
1948, Kurzer Beweis der isoperimetrischen Ungleichung fur konvexe Bereiche. Elem.
Math. 3, 111-12.
1949a, Ein Auswahlsatz fiir abgeschlossene Punktmengen. Portugaliae Math. 8, 13-15.
1949b. Uber konvexe Korper mit Flachstellen. Math. Z. 52, 212-16.
1949c, Kurze Herleitung einer verscharften isoperimetrischen Ungleichung fur konvexe
KOrper. Revue Fac. Sci. Univ. Istanbul, S6r. A 14, 1-6.

452 References
1950a, Einige Anwendungen eines Funktionalsatzes fiir konvexe Korper in der rSum-
lichen Integralgeometrie. Monatsh. Math. 54, 345-53.
1950b, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die
Theoreme von Erhard Schmidt. Math. Z. S3, 210-18.
1951, Beweis eines Funktionalsatzes fiir konvexe KOrper. Ahh. Math. Sem. Univ.
Hamburg 17, 69-76.
1952, Mittelpunktspolyeder und translative Zerlegungsgleichheit. Math. Nachr. 8, 53-8.
1955a, Eulers Charakteristik und kombinatorische Geometrie. J. reine angew. Math.
194, 101-10.
1955b- Konkave Eikbrperfunktionale. Monatsh. Math. 59, 230-7.
1956, IntegralsStze im Konvexring. Abh. Math. Sem. Univ. Hamburg 20, 136-54.
1959, Normale KOrper im euklidischen Raum und ihre topologischen und metrischen
Eigenschaften. Math. Z. 71, 124-40.
1960, Zur Eulerschen Charakteristik euklidischer Polyeder. Monatsh. Math. 64, 49-60.
1968a, Eine Schnittrekursion fiir die Eulersche Charakteristik euklidischer Polyeder mit
Anwendungen innerhalb der kombinatorischen Geometrie. Elem. Math. 13, 121-32.
1968b, Halbeikorper und Isoperimetrie. Arch. Math. 19, 659-63.
1968c, Radialpotenzintegrale zentralsymmetrischer Rotationskiirper und Ungleichheits-
aussagen Busemannscher Art. Math. Scand. 23, 193-200.
1969a, Zur axiomatischen Charakterisierung des Steinerpunktes konvexer Kiirper. Israel
J. Math. 7, 168-76. (This work is erroneous.)
1969b, Notiz zur Eulerschen Charakteristik offener und abgeschlossener Polyeder. Studia
Sci. Math. Hungar. 4, 385-7.
1969c, Eckenkriimmung beliebiger kompakter euklidischer Polyeder und Charakteristik
von Euler-Poincard. L'Enseignement Math. IS, 147-51.
1971, Zur axiomatischen Charakterisierung des Steinerpunktes konvexer Korper; Berich-
tigung und Nachtrag. Israel J. Math. 9, 466-72.
1972, Polytopes and translative equidecomposability. Amer. Math. Monthly 79, 275-6.
1973, Erweiterter Polyedersatz und Euler-Shephardsche Additionstheoreme. Abh.
Math. Sem. Univ. Hamburg 39, 120-9.
1974, Begriindung der Eulerschen Charakteristik innerhalb der ebenen Elementar-
geometrie. L' Enseignement Math. 20, 33-43.
1975a, Eine Erweiterung der kinematischen Hauptformel der Integralgeometrie. Abh.
Math. Sem. Univ. Hamburg 44, 84-90.
1975b, Eikiirperrichtungsfunktionale und kinematische Integralformeln. Studienvor-
le.sung (Manuscript), Univ. Bern.
1975c, Das Wills'sche Funktional. Monatsh. Math. 79, 213-21.
1979, Gitterpunktaozahl im Simplex und Wills'sche Vermutung. Math. Ann. 239,
271-88.
Hadwiger, H. & Mani, P. 1972, On the Euler characteristic of spherical polyhedra and the
Euler relation. Mathematika 19, 139-43.
1974, On polyhedra with extremal Euler characteristic. J. Combinat. Theory, Ser. A 17,
345-9.
Hadwiger, H. & Meier, Ch. 1973, Studien zur vektoriellen Integralgeometrie. Math.
Nachr. 56, 261-8.
Hadwiger, H. & Schneider, R. I97I, Vektorielle Integralgeometrie. Elem. Math. 26,
49-57.
Halmos, P. R. 1948, The range of a vector measure. Bull. Amer. Math. Soc. 54, 416-21.
Hammer, P. C. 1951, Convex bodies associated with a convex body. Proc. Amer. Math.
Soc. 2, 781-93.
1963, Approximation of convex surfaces by algebraic surfaces. Mathematika 10, 64-71.
Haralick, R. M., Sternberg, S. R. & Zhuang, X. 1987, Image analysis using mathematical
morphology. IEEE Trans. Pattern Anal. Machine Intell. 9, 532-50.
Hardy, G. H., Littlewood, J. E. & P61ya, G. 1934, Inequalities. Cambridge Univ. Press.

References 453
Hartman. P. & Wintner, A. 1953, On pieces of convex surfaces. Amer. J. Math. 75,
477-87.
Hausdorff, F. 1914, Grundziige der Mengenlehre. Verlag von Veil, Leipzig.
1927, Mengenlehre. W. de Gruyter, Berlin.
Hayashi, T. 1924, On Steiner's curvature centroid. Sci. Rep. Tohoku Univ. 13, 109-32.
1926a, Some geometrical applications of the Fourier series. Rend. Circ. Mat. Palermo
50, 96-102.
1926b, On in-revolvable and circum-revolvable curves of a regular polygon, I. Sci. Rep.
Tdhoku Univ. 15, 261-3.
1926c, On in-revolvable and circum-revolvable curves of a regular polygon, II. Sci. Rep.
Tohoku Univ. 15, 499-502.
Heil, E. 1967a, Ober die AuswahlsStze von Blaschke und ArzelS-Ascoli. Math.-Phys.
Semesterber. 14, 169-75.
1967b, Abschatzungen fiir einige Affininvarianten konvexer Kurven. Monatsh. Math. 71,
405-23.
1976, Ungleichungen fiir die QuermaBintegrale polarer Kiirper. Manuscripta Math. 19,
143-9.
1987, Extensions of an inequality of Bonnesen to D-dimensional space and curvature
conditions for convex bodies. Aequationes Math. 34, 35-60.
Heine, R. 1937, Der Wertevorrat der gemischten Inhalte von zwei, drei und vier ebenen
Eibereichen. Math. Ann. 115, 115-29.
Herglotz, G. 1943, Uber die Steinersche Formel fiir Parallelflachen. Abh. Math. Sem.
Univ. Hamburg 15, 165-77.
Hilbert, D. 1910, Minkowskis Theorie von Volumen und Oberflache. Nachr. Ges. Wiss.
Gottingen, 388-406. Also in: Grundziige einer allgemeinen Theorie der Hnearen
Integralgleichungen. Teubner, Leipzig u. Berlin, 1912, Kap. 19.
Hildenbrand, W. 1974, Core and Equilibria of a Large Economy. Princeton Univ. Press.
1981, Short-run production functions based on microdata. Econometrica 49, 1095-125.
Hildenbrand, W. & Neyman, A. 1982. Integrals of production sets with restricted
substitution. J. Math. Econom. 9, 71-82.
Hille, E. & Phillips, R. S. 1957, Functional Analysis and Semi-groups. AMS Colloquium
Publications vol. 31, Amer. Math. Soc, Providence, RI.
Hiriart-Urruty, J.-B. 1986, A new set-valued second order derivative for convex functions.
In: Mathematics for Optimization (ed. J.-B. Hiriart-Urruty), Mathematical Studies
Series 129, North-Holland, pp. 157-82.
Hiriart-Urruty, J.-B. & Seeger, A. 1989, The second order subdifferential and the Dupin
indicatrices of a nondifferentiable convex function. Proc. London Math. Soc, III.
Ser., 58, 351-65.
Hirose, T. 1965, On the convergence theorem for star-shaped sets in E". Proc. Japan.
Acad. 41,209-11.
Holmes, R. B. 1975, Geometric Functional Analyus and its Applications. Springer, New
York.
Holmes, R. D. & Thompson, A. C. 1979, A/-dimensional area and content in Minkowski
spaces. Pacific J. Math. 85, 77-110.
Hormander, L. 1955, Sur la fonction d'appui des ensembles convexes dans un espace
localement convexe. Arkiv Mat. 3, 181-6.
Horn, B. K. P. 1986, Robot Vision. MIT Press, Cambridge, MA; McGraw-Hill, New
York.
Howe, R. 1982, Most convex functions are smooth. J. Math. Econom. 9, 37-9.
Huck, H., Roitzsch, R., Simon, U., Vortisch, W., Walden, R., Wegner, B. & Wendland,
W. 1973, Beweismethoden der Differentialgeometrie im Grofien. Lecture Notes in
Math. 335, Springer, Berlin.
Hurwitz, A. I90I, Sur le problfeme des isop^rimfetres. C.R. Acad. Sci. Paris 132, 401-3.
Math. Werke, vol. I, pp. 490-1, BirkhSuser, Basel, 1932.

454 References
1902, Sur quelques applications g^om^triques des series de Fourier. Ann. Ecole Norm.
C) 13, 357-408. Math. Werke. vol. I, pp. 509-54. Birkhauser, Basel, 1932.
Husain, T. & Tweddle, I. 1970, On the extreme points of the sum of two compact convex
sets. Math. Ann. 188. 113-22.
Inzinger, R. 1949, Stiitzbare Bereiche. trigonometrische Polynome und Defizite hOherer
Ordnung. Monatsh. Math. S3, 302-23.
Ivanov, B. A. 1973. Uber geradlinige Abschnitte auf der Berandung eines konvexen
Kiirpers (in Russian). Ukrain. Geom. Shomik. 13, 69-71.
1976, Exceptional directions for a convex body (in Russian). Mat. Zametki 20, 365-71.
English translation: Math. Notes 20, 763-6.
Jacobs, K. 1971, Extremalpunkte konvexer Mengen. In: Selecta Mathematica III, Springer,
Berlin, pp. 90-118.
Jessen, B. 1929, Om konvekse Kurvers Krumning. Mat. Tidsskr. B, 50-62.
Johnson, K. & Thompson, A. C. 1987, On the isoperimetric mapping in Minkowski
spaces. In: Intuitive Geometry, Si6fok 1985, Colloquia Math. Soc. Jdnos Bolyai 48,
North-Holland, Amsterdam, pp. 273-87.
Jongmans, F. 1968, Thtorime de Krein-Milman et programtnation mathematique. Bull.
Soc. Roy. Sci. Liige 37, 261-70.
1973, Sur les complications d'une loi de simplification dans les espaces vectoriels. Bull.
Soc. Roy. Sci. Liige 42, 529-34.
1976, Reflections sur Part de sauver la face. Bull. Soc Roy. Sci. Liege 45, 294-306.
1979, De Part d'gtre a bonne distance des ensembles dont la decomposition atteint un
stade avanc6. Bull. Soc. Roy. Sci. Liege 48, 237-61.
1981a, Contribution aux fondements de la calvitie mathematique. Bull. Soc. Roy. Sci.
Liige SO, 8-15.
1981b, Etude des cdnes associ^s & un ensemble. In: Seminaires dirigis par F. Jongmans
et J. Varlet. Univ. Lifige 1980-1, 220 pp.
Jourlin. M. & Laget, B. 1987, Measure of asymmetry of plane convex bodies. Acta
Stereologica 6, 115-20.
1988. Convexity and symmetry. Part 1. In: Image Analysis and Mathematical
Morphology, vol. 2 (ed. J. Serra), Academic Press, London, pp. 343-57.
Kahn. J. & Saks, M. 1984, Balancing poset extensions. Order 1, 113-26.
Kallay, M. 1974. Reconstruction of a plane convex body from the curvature of its
boundary. Israel J. Math. 17. 149-61.
1975, The extreme bodies in the set of plane convex bodies with a given width function.
Israel J. Math. 22, 203-7.
1982, Indecomposable polytopes. Israel. J. Math. 41, 235-43.
1984, Decomposability of polytopes is a projective invariant. In: Convexity and Graph
Theory, Jerusalem 1981, North-Holland Math. Stud. 87, North-Holland, Amsterdam,
pp. 191-6.
Kalman, J. A. 1961, Continuity and convexity of projections and barycentric coordinates in
convex polyhedra. Pacific J. Math. 11, 1017-22.
Kameneckii, I. M. 1947, Die Losung eines von Ljusternik gestellten geometrischen
Problems (in Russian). Uspechi mat. nauk 2, 199-202.
Karcher, H. 1968, Umkreise und Inkreise konvexer Kurven in der sphiirischen und der
hyperbolischen Geometrie. Math. Ann. 177, 122-32.
Katsurada, Y. 1962, Generalized Minkowski formulas for closed hypersurfaces in Rieman-
nian space. Ann. Mat. pura appl., IV. Ser., 57, 283-93.
Kellerer, A. M. 1983. On the number of clumps resulting from the overlap of randomly
placed figures in a plane. J- Appl. Prob. 20, 126-35.
1985, Counting figures In planar random configurations. J. Appl. Prob. 22, 68-81.
Kellerer, H. G. 1984, Minkowski functional of Poisson processes. Z. Wahrscheinlich-
keitsth. verw. Geb. 67, 63-84.
Khovanskli, A. G. 1984, Analogues of Aleksandrov-Fenchel inequalities for hyperbolic

References 455
forms (in Russian). Dokl. Akad. Nauk SSSR 276, 1332-4. English translation: Soviet
Math. Dokl. 29.710-13.
Kiener, K. 1986, Extremalitat von Ellipsoiden und die Faltungsungleichung von Sobolev.
Arch. Math. 46, 162-8.
Kineses, J. 1987, The classification of 3- and 4-Helly dimensional convex bodies. Geom.
Dedicata 22, 283-301.
Kiselman, C. O. 1986, How .smooth is the shadow of a smooth convex body? J. London
Math. Soc. 33, I0I-9; Serdica 12, 189-95.
1987. Smoothness of vector sums of plane convex sets. Math. Scand. 60, 239-52.
Klee, V. 1953, The critical set of a convex body. Amer. J. Math. 75, 178-88.
1955, A note on extreme points. Amer. Math. Monthly 62. 30-2.
1957a, Research problem No. 5. Bull. Amer. Math. Soc. 63. 419.
1957b, Extremal structure of convex sets. Arch. Math. 8, 234-40.
1958, Extremal structure of convex sets II. Math. Z. 69, 90-104.
1959a, Some characterizations of convex polyhedra. Acta Math. 102, 79-107.
1959b, Some new results on smoothness and rotundity in normed linear spaces. Math.
Ann. 139, 51-63.
1963a, On a theorem of Dubins. J. Math. Anal. Appl. 7, 425-7.
1963b, The Euler characteristic in combinatorial geometry. Amer. Math. Monthly 70,
119-27.
1968, Maximal separation theorems for convex sets. Trans. Amer. Math. Soc. 134.
133-47.
1969a, Can the boundary of a d-dimensional convex body contain segments in all
directions? Amer. Math. Monthly 76, 408-10.
1969b. Separation and support properties of convex sets - a survey. In: Control Theory
and the Calculus of Variations (ed. A. Balakrishnan), Academic Press, New York, pp.
23.')-304.
Klee, V. L. & Martin K. 1971, Semi-continuity of the face-function of a convex set.
Comment. Math. Helvet. 46, 1-12.
Klein, E. & Thompson. A. C. 1984, Theory of Correspondences. Wiley, New York.
Klima, V. & Netuka, I. I98I, Smoothness of a typical convex function. Czech. Math. J.
31.569-72.
Klotzler, R. 1975, Beweis einer Vermutung iiber n-Orbiformen kleinsten Inhalts. Z.
Angew. Math. Mech. 55, 557-70.
Kneser. H. 1970, Die Stutzfunktion eines Durchschnitts konvexer Korper. Arch. Math. 21.
221-4.
Kneser, H. & Siiss, W. 1932, Die Volumina in linearen Scharen konvexer Korper. Mat.
Tidsskr. B, 19-25.
Knothe, H. 1949, Uber eine Vermutung H. Minkowskis. Math. Nachr. 2, 380-5.
1957a, Contributions to the theory of convex bodies. Michigan Math. J. 4, 39-52.
1957b, Inversion of two theorems of Archimedes. Michigan Math. J. 4, 53-6.
Kohlmann, P. 1988, KriimmungsmaBe, Minkowskigleichungen und Charakterisierungen
metrischer BSlle in Raumformen. Dissertation. Univ. Dortmund.
Koutroufiotis, D. 1971, Ovaloids which are almost spheres. Commun. Pure Appl. Math.
24, 289-300.
1972, On Blasche's rolling theorems. Arch. Math. 23, 655-60.
Krein, M. & Milman, D. 1940. On extreme points of regularly convex sets. Studia Math.
9, 133-8.
Kruskal, J. B. 1969, Two convex counterexamples: A discontinuous envelope function and
a nondifferentiable nearest-point mapping. Proc. Amer. Math. Soc. 23, 697-703.
Kubota, T. 1918, Uber die Schwerpunkte der geschlossenen konvexen Kurven und
Flachen. Tohoku Math. J. 14, 20-7.
1920, Einige Probleme iiber konvex-geschlossene Kurven und FlSchen. Tohoku Math. J.
17, 351-62.

456 References
1922, Beweise einiger Satze Uber Eiflachen. Tohoku Math. J. 21, 261-4.
1925a, Uber die konvex-geschlossenen Mannigfaltigkeiten im n-dimensionalen Raume.
Set. Rep. Tdhoku Univ. 14, 85-99.
1925b, Uber die Eibereiche im n-dimensionalen Raume. Sci. Rep. Tdhoku Univ. 14,
399-402.
Kuiper, N. H. I97I, Morse relations for curvature and tightness. In: Proc. Liverpool
Singularities Symp. (ed. C. T, C. Wall), Lecture Notes in Math. 209, Springer, Berlin,
1971, pp. 77-89.
Kuratowski, K. 1968, Topology, vol. II. Academic Press, New York (French original:
1961).
Kutateladze, S. S. 1973, Die Blaschkesche Struktur in der Programmierung isoperime-
trischer Probleme (in Russian). Mat. Zametki 14, 745-54. English translation: Math.
Notes 14, 985-9.
1976, Symmetry measures (in Russian). Mat. Zametki 19, 615-22. English translation:
Math. Notes 19, 372-5.
Kutateladze, S. S. & Rubinov, A. M. 1969, Problems of isoperimetric type in a space of
convex bodies (in Russian). Optimalnoe Planirovanie 14, 61-79.
Lafontaine, J. 1987, Mesures de courbure des vari^t^s lisses et des polyedres (S^minaire
Bourbaki, vol. 1985/86) Asterisque 145-6, 241-56.
Lagarias, J. C. 1982. The van der Waerden conjecture: Two Soviet solutions. Notices
Amer. Math. Soc. 29, 130-3.
Laget, B. 1987, Sur I'ensemble critique d'un ensemble convexe. Arch. Math. 49, 333-4.
Larman, D. G. 1971, On a conjecture of Klee and Martin for convex bodies. Proc.
London Math. Soc. 23, 668-82. Corrigendum: ibid. 36 A978), 86,
1974, On the inner aperture and intersections of convex sets. Pacific J. Math. 55,
219-32.
1977, On the one-skeleton of a compact convex set in a Banach space. Proc. London
Math. Soc. CK4, 117-44.
1980, The d -2 skeletons of polytopal approximations to a convex body in E''.
Mathematika 27, 122-33,
Larman, D. G. & Mani, P. 1970, Gleichungen und Ungleichungen fiir die Geriiste von
konvexen Polytopen und Zellenkomplexen. Comment. Math. Helvet. 45, 199-218.
Larman, D. G. & Rogers, C. A. 1970, Paths in the one-skeleton of a convex body.
Mathematika 17, 293-314.
1971, Increasing paths on the one-skeleton of a convex body and the directions of line
segments on the boundary of a convex body. Proc. London Math. Soc. 23, 683-98.
1973, The finite dimensional skeletons of a compact convex set. Bull. London Math.
Soc. 5, 145-53.
1975, The existence of a centrally symmetric convex body with central sections that are
unexpectedly small. Mathematika 22, 164-75.
Lebesgue, H. 1914, Sur le problfime des isop^rimfitres et sur les domaines de largeur
constante. Bull. Soc. Math. France C.R., 72-6.
LeichtweiB, K. 1965, Uber eine analytische Darstellung des Randes konvexer Korper.
Arch. Math. 16, 300-19.
1981, Zum Beweis eines Eindeutigkeitssatzes von A. D. Aleksandrov. In: E. B.
Christoffel, Aachen/Monschau 1979 (Eds P. Butzer & F. Feh^r), BirkhSuser, Basel,
pp. 636-52.
1986, Zur Affinoberflache konvexer Korper. Manuscripta Math. 56, 429-64.
1988, Uber einige Eigenschaften der Affinoberflache beliebiger konvexer Korper.
Results Math. 13, 255-82.
1989, Bemerkungen zur Definition einer erweiterten Affinoberflache von E. Lutwak.
Manuscripta Math. 65, 181-97.
Leindler, L. 1972, On a certain converse of Holder's inequality II. Acta. Sci. Math. Szeged
33. 217-23.

References 457
Lenz, H. 1970, Mengenalgebra und Eulersche Charakteristik. Abh. Math. Sent. Univ.
Hamburg 34, 135-47.
Letac, G. 1983, Mesures sur le cercle et convexes du plan. Ann. Sci. Univ. Clermont-
Ferrand II, Prob. Appl. 1, 35-65.
Lettl, G. 1980, Isometrien des Raumes der konvexen Teilmengen der SphSre. Arch. Math.
35, 471-5.
Levy, P. 1937, Thiorie de ('addition des variables aliatoires. Gauthier-Villars, Paris.
Lewis, J. E. 1975, A Banach space whose elements are classes of sets of constant width.
Canad. Math. Bull. 18, 679-89.
Lewy, H. 1938, On differential geometry in the large, I (Minkowski's problem). Trans.
Amer. Math. Soc. 43, 258-70.
Liapounoff, A. A. 1940, On completely additive vector functions (in Russian). Izv. Akad.
Nauk SSSR 4, 465-78.
Lindeliif, L. 1869, Propri^t^s generates des polyfidres qui, sous une 6tendue superficielle
donn^e, renferment le plus grand volume. Bull. Acad. Sci. St. Pitersbourg 14,
257-69. Extract: Math. Ann. 2 A870), 150-9.
Lindenstrauss, J. 1966, A short proof of Liapounoffs convexity theorem. J. Math. Mech.
IS, 971-2.
Lindquist, N. F. 1975a, Approximation of convex bodies by sums of line segments.
Portugaliae Math. 34, 233-40.
1975b, Support functions of central convex bodies. Portugaliae Math. 34, 241-52.
Linhart, J. 1986, Extremaleigenschaften der regulSren 3-Zonotope. Studia Sci. Math.
Hungar. 21, I8I-8.
1987, An upper bound for the intrinsic volumes of equilateral zonotopes. In: Intuitive
Geometry, Siofok 1985, Coll. Math. Soc. Jdnos Bolyai 48, North-Holland,
Amsterdam, pp. 339-45.
1989, Approximation of a ball by zonotopes using uniform distribution on the sphere.
Arch. Math. 53, 82-6.
Linke, Ju. E. 1980, Application of Steiner's point for investigation of a class of sublinear
operators (in Russian). Dokl. Akad. Nauk SSSR 254, 1069-79. English translation:
Soviet Math. Dokl. 22, 522-5.
Little, J. J. 1983, An iterative method for reconstructing convex polyhedra from extended
Gaussian images. In: Proc. AAAI-83, Washington, DC, pp. 247-50.
LjaSenko, N. N. 1979, Limit theorems for sums of independent compact random subsets of
Euclidean space (in Russian). Zap. Nauin. Sem. Leningrad. Otdel. Mat. Inst. Steklov
85, 113-28.
Lutwak, E. 1975a, Dual mixed volumes. Pacific J. Math. 58, 531-8.
1975b, Width-integrals of convex bodies. Proc. Amer. Math. Soc. 53, 435-9.
1975c, A general Bieberbach inequality. Proc. Camb. Phil. Soc. 78, 493-6.
I975d, Dual cross-sectional measures. Atti Accad. Naz. Lincei 5S, 1-5.
1976, On cross-sectional measures of polar reciprocal convex bodies. Geom. Dedicata 5,
79-80.
1977, Mixed width-integrals of convex bodies. Israel J. Math. 28, 249-53.
1979a, Mean dual and harmonic cross-sectional measures. Ann. Mat. Pura Appl. 119,
139-148.
1979b, On a complementary Minkowski inequality. J. Math. Anal. Appl. 72, 70-4,
1983, A width-diameter inequality for convex bodies. J. Math. Anal. Appl. 93, 290-5.
1984, A general isepiphanic inequality. Proc. Amer. Math. Soc. 90, 415-21.
1985a, On the Blaschke-Santal6 inequality. In: Discrete Geometry and Convexity (eds J.
E. Goodman, E. Lutwak, J. Malkevitch & R. Pollack), Ann. New York Acad. Sci.
440, pp. 106-12.
1985b, Mixed projection inequalities. Trans. Amer. Math. Soc. 287, 91-106.
1986a, On some affine isoperimetric inequalities. J. Differential Geom. 23, I-I3.
1986b, Volume of mixed bodies. Trans. Amer. Math. Soc. 294, 487-500.

458 References
1987a, Rotation means of projections. Israel J. Math. 58, I6I-9.
1987b, Mixed affine surface area. J. Math. Anal. Appl. 125, 351-60.
1988a. Inequalities for Hadwiger's harmonic quermassintegrals. Math. Ann. 280,
165-75.
1988b, Intersection bodies and dual mixed volumes. Adv. Math. 71, 232-61.
1989+a. Inequalities for mixed projection bodies. Trans. Amer. Math. Soc. (to appear).
1989+b, Larger objects with smaller projections. Preprint.
I989+C, A minimax inequality for inscribed cones. J. Math. Anal. Appl. (to appear).
1990a. On quermassintegrals of mixed projection bodies. Geom. Dedicata 33, 51-8.
1990b. Centroid bodies' and dual mixed volumes. Proc. London Math. Soc. C) 60,
365-91.
1990c, On a conjectured projection inequality of Petty. Contemp. Math. 113, 171-82.
1991, Extended affine surface area. Adv. Math. 85, 39-68.
1991+ a. The Brunn-Minkowski-Firey theory, I. J. Differential Geom. (to appear).
1991+b. The Brunn-Minkowski-Firey theory, II. Preprint.
Macbeath, A. M. 1951, A compactness theorem for affine equivalence-classes of convex
regions. Canad. J. Math. 3, 54-61.
Maehara, H. 1984, Convex bodies forming pairs of constant width. J. Geom. 22, 101-7.
Mahler, K. 1939a, Ein Ubertragungsprinzip fiir konvexe Kiirper. Casopis Pist. Mat. Fys.
68.93-102.
1939b, Ein Minimalproblem fiir konvexe Polygone. Mathematica (Zutphen) B7, 118-27.
Makai, E. 1959, Steiner type inequalities in plane geometry. Period. Polytech. Elec.
Engrg. 3, 345-355.
Makai, E. jr. 1974, Research Problem. Periodica Math. Hungar. 5, 353-4.
Mani, P. (Mani-Levitska, P.) 1971. On angle sums and Steiner points of polyhedra. Israel.
J. Math. 9, 380-8.
1988, A simple proof of the kinematic formula. Monatsh. Math. 105, 279-85.
Martini, H. 1987, Some results and problems around zonotopes. In: Intuitive Geometry,
Si6fok 1985 (eds K. BOroczky & G. Fejes Toth), Coll. Math. Soc. Jdnos Bolyai 48,
pp. 383-418.
1989, On inner quermasses of convex bodies. Arch. Math. 52, 402-6.
1990, A new view on some characterizations of simplices. Arch. Math. 55, 389-93.
1991, Convex polytopes whose projection bodies and difference sets are polars. Discrete
Comput. Geom. 6, 83-91.
Mase, S. 1979. Random compact convex sets which are infmitely divisible with respect to
Minkowski addition. Adv. Appl. Proh. 11, 834-50.
Matheron, G. 1974a, Un th^oreme d'unicite pour les hyperplans poissoniens. J. Appl.
Proh. 11, 184-9.
1974b, Hyperplans poissoniens et compacts de Steiner. Adv. Appl. Proh. 6, 563-79.
1975, Random Sets and Integral Geometry. Wiley, New York.
1978a, The infinitesimal erosions. In: Geometrical Prohability and Biological Structures:
Buffon's 200th Anniversary (eds R. E. Miles & J. Serra), Lecture Notes in Biomath.
23, Springer, Berlin, pp. 251-69.
1978b, La formule de Steiner pour les erosions. J. Appl. Proh. 15, 126-35.
Matsumura, S. 1930, EiflSchenpaare gleicher Breiten und gleicher UmfSnge. Japan. J.
Math. 7. 225-6.
1932, Uber Minkowskis gemischten HScheninhalt. Japan. J. Math. 9. 161-3-
Mazur, .S. 1933, Uber konvexe Mengen in linearen normierten RSumen. Studia Math. 4,
70-84.
McKinney, R. L. I%2, Positive bases for linear spaces. Trans. Amer. Math. Soc. 103,
131-48.
McMinn, T. J. I960, On the line segments of a convex surface in £,. Pacific J. Math. 10,
943-6.
McMullen, P. 1970, Polytopes with centrally symmetric faces. Israel J. Math. 8, 194-6.
I97I. On zonotopes. Trans. Amer. Math. Soc. 159, 91-109.

References 459
1973, Representations of polytopes and polyhedral sets. Geom. Dedicata 2, 83-99.
1974a, A dice probability problem. Mathematika 21, 193-8.
1974b, On the inner parallel body of a convex body. Israel J. Math. 19, 217-19.
1975a, Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc.
Camh. Phil. Soc. 78, 247-61.
1975b, Space tiling 2»notopes. Mathematika 22, 202-211.
1975c, Metrical and combinatorial properties of convex polytopes. In: Proc. Int. Congr.
Math.. Vancouver 1974 (ed. R. D. James), Canadian Mathematical Congress, pp.
491-5.
1976a, On support functions of compact convex sets. Elem. Math. 31, 117-19.
1976b, Polytopes with centrally symmetric facets. Israel J. Math. 23, 337-8.
1977, Valuations and Euler type relations on certain classes of convex polytopes. Proc.
London Math. Soc. 35. 113-35.
1979, Transforms, diagrams and repre.sentations. In: Contributions to Geometry, Proc.
Geometry Symp.. Siegen 1978 (eds J. Tolke & J. M. Wills), BirkhSuser, Basel, pp.
92-130.
1980, Continuous translation invariant valuations on the space of compact convex .sets.
Arch. Math. 34, 377-84.
1983a, Weakly continuous valuations on convex polytopes. Arch. Math. 41, 555-64.
1983b, Notes on Asplund's theorem (Private communication).
1984, The Hausdorff distance between compact convex sets. Mathematika 31, 76-82.
1987, Indecomposable convex polytopes. IsraelJ. Math. 58, 321-3.
1989, The polytope algebra. Adv. Math. 78, 76-130.
1990, Monotone translation invariant valuations on convex bodies. Arch. Math. 55.
595-8.
McMullen, P., Schneider, R. & Shephard, G. C. 1974, Monotypic polytopes and their
intersection properties. Geom. Dedicata 3,99-129.
Mecke, J., Schneider, R., Stoyan, D. & Weil, W. 1990, Slocha-itische Ceometrie. DMV
Seminar 1989, vol. 16, BirkhSuser. Basel.
Meier, Ch. 1977, MultilinearitSt bei Polyederaddition. Arch. Math. 29, 210-17.
Meissner, E. 1909, Uber die Anwendung von Fourier-Reihen auf einige Aufgaben der
Geometrie und Kinematik. Vierteljahres.schr. Naturf. Ges. Ziirich 54, 309-29.
1918, Uber die durch regulSre Polyeder nicht stiitzbaren Korper. Vierteljahresschr.
Naturf. Ges. Zurich 63, 544-51.
Melzak, Z. A. 1958, Limit sections and universal points of convex surfaces. Proc. Amer.
Math. Soc. 9, 729-34.
Menger, K. 1928, Untersuchungen iiber allgemeine Metrik. Math. Ann. 100, 75-163.
Meschkowski, H. & Ahrens, I. 1974, Theorie der Punktmengen. Bibliographisches Institut.
Mannheim.
Meyer, M. 1986. Une characterisation volumique de certains espaces normes de dimension
finie. IsraelJ. Math. 55, 317-26.
1990+ On a problem of Busemann and Petty. Preprint.
Meyer, M. & Pajor, A. 1989, On Santalo's inequality. In: Geometric Aspects of Functional
Analysis (eds J. Lindenstrauss & V. D. Milman), Lecture Notes in Math. 1376,
Springer, Berlin, pp. 261-3.
1990, On the Blaschke-Santal6 inequality. Arch. Math. 55, 82-93.
Meyer, M. & Reisner, S. I99I, A geometric property of the boundary of symmetric
convex bodies and convexity of flotation surfaces. Geom. Dedicata 37. 327-37.
Meyer, P. 1977, Eine Verallgemeinerung der Steinerschen Formeln. Abh. Braunschweig.
Wi.t.i. Ge.%. 28, 119-23.
Meyer, W. 1969, Minkowski addition of convex sets. Thesis, Univ. of Wisconsin.
1970a, Characterization of the Steiner point. Pacific J. Math. 35. 717-25.
1970b, Indecomposable polytopes. In: Combinatorial Structures and Their Applications,
Calgary 1969 (eds R. Guy et al.), Gordon & Breach, New York, pp. 271-2.
1972, Decomposing plane convex bodies. Arch. Math. 23, .'i34-6.

460 References
1974, Indecomposable polytopes. Trans. Amer. Math. Soc. 190, 77-86.
Michael, E. 1951, Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71, 152-82.
Miles, R. E. 1974, The fundamental formula of Blaschke in integral geometry and
geometric probability, and its iteration, for domains with fixed orientations. Austral.
J. Statist. 16, 111-18.
1976, Estimating aggregate and overall characteristics from thick sections by transmission
microscopy. J. Microscopy 107, 227-33.
Milka, A. D. 1973, Indecomposability of a convex surface (in Russian). Ukrain. Geom.
Sb. 13, 112-29. English translation: Selected Transl. in Math., Statist, and Proh. IS,
129-43.
Milman, D. 1948, Isometry and extremal points. Dokl. Akad. Nauk SSSR 59, I24I-4.
Milman, V. D. 1986, In^galitd de Brunn-Minkowski inverse et applications i la thtorie
locale des espaces norm^s. C.R. Acad. Set. Paris, Six. I, 302, 25-8.
Milman, V. D. & Pajor A. 1989, Isotropic position and inertia ellipsoids and zonoids of
the unit ball of a normed n-dimensional space. In: Geometric Aspects of Functional
Analysis (eds J. Lindenstrauss & V. D. Milman), Lecture Notes in Math. 1376,
Springer, Berlin, pp. 64-104.
Minkowski, H. 1897, Allgemeine LehrsStze iiber die konvexen Polyeder. Nachr. Ges.
Wiss. Gdttingen, 198-219. Gesammelte Ahhandlungen, vol. 11, Teubner, Leipzig,
1911, pp. 103-21.
I90Ia, Ober die Begriffe Lange, Oberflache und Volumen. Jahresber. Deutsche Math.-
Ver. 9, 115-21. Gesammelte Ahhandlungen, vol. II, Teubner, Leipzig, 1911, pp.
122-27.
I90Ib, Sur les surfaces convexes ferm^es. C.R. Acad. Sci. Paris 132, 21-24. Gesammelte
Ahhandlungen, vol. II, Teubner, Leipzig, I9II, pp. 128-30.
1903, Volumen und Oberflache. Math. Ann. 57, 447-95. Gesammelte Ahhandlungen,
vol. II, Teubner, Leipzig, I9I1, pp. 230-76.
1904, Uber die Kiirper konstanter Breite (in Russian). Mat Shornik 25, 505-8. German
translation: Gesammelte Ahhandlungen, vol. II, Teubner, Leipzig, 1911, pp. 277-9.
1910, Geometrie der Zahlen. Teubner, Leipzig.
1911, Theorie der konvexen Korper, insbesondere Begrundung ihres OberflSchenbe-
griff.s. Gesammelte Ahhandlungen, vol. II, Teubner, Leipzig, pp. 131-229.
Minoda, T. I94I, On certain ovals. Tdhoku Math. J. 48, 312-20.
Miranda, C. 1939, Su un problema di Minkowski. Rend. Sem. Mat. Roma 3, 96-108.
Mityagin, B. S. 1969, Two inequalities for volumes of convex bodies (in Russian). Mat.
Zametki 5, 99-106. English translation: Math. Notes 5, 61-5.
Molter, U. M. 1986, Tangential measure on the set of convex infinite cylinders. J. Appl.
Proh. 23, 961-72.
Moore, J. D. 1973, Almost spherical convex hypersurfaces. Trans. Amer. Math. Soc. 180,
347-58.
Motzkin, T. S. 1935, Sur quelques propri^t^s caract^ristiques des ensembles convexes. Atti
Real. Accad. Naz. Lincei, Rend. CI. Sci. Fis.. Mat., Natur., Seiie VI, 21, 562-7.
Miiller, C. 1966, Spherical Harmonics. Lecture Notes in Math. 17, Springer, Berlin.
Miiller, H. R. 1953, Ober Momente ersten und zweiten Grades in der Integralgeometrie.
Rend. Circ. Mat. Palermo, II. Ser., 2, 119-40.
Miirner, P. 1975, Translative Parkettierungspolyeder und Zerlegungsgleichheit. Elem.
Math. 30. 25-7.
1977, Translative Zerlegungsgleichheit von Polytopen. Arch. Math. 29, 218-24.
NidenTk, Z. 1965, Die VerschSrfung einer Ungleichung von Frobenius fiir den gemischten
Fliicheninhalt der konvexen ebenen Bereiche. Casopis PSst. Mat. 90, 220-5.
1967, Die Ungleichungen fiir die Oberflache, das Integral der mittleren Krummung und
die Breite der konvexen Kiirper. Casopis Pist. Mat. 92, 133-45.
1968, Erste Kriimmungsfunktion der RotationseiflSchen. Casopis Pist. Mat. 93, 127-33.
Nadler, S. B. 1978, Hyperspaces of Sets. Marcel Dekker, New York.

References 461
Nadler, S., Quinn, J. & Stavrakas, N. M. 1975, Hyperspaces of compact convex sets I.
Bull. Acad. Polon. Sci. 23, 555-9.
1977, Hyperspaces of compact convex sets II. Bull. Acad. Polon. Sci. 25, 381-5.
1979, Hyperspaces of compact convex sets. Pacific J. Math. 83, 441-62.
Nakajima, S. 1920, On some characteristic properties of curves and surfaces. Tdhoku
Math. J. 18, 272-87.
I92I, The circle and the straight line nearest to n given points, n given straight lines or
a given curve. Tdhoku Math. J. 19, 11-20.
Neveu, J. 1969, Mathematische Grundlagen der Warscheinlichkeitstheorie. Oldenbourg,
Miinchen.
Nikliborc, W. 1932, Uber die Lage des Schwerpunktes eines ebenen konvexen Bereiches
und die Extrema des logarithmischen Flachenpotentials eines konvexen Bereiches.
Math. Z. 36, I6I-5.
Nirenberg, L. 1953, The Weyl and Minkowski problems in differential geometry in the
large. Commun. Pure Appl. Math. 6, 337-94.
Oda, T. 1988, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of
Toric Varieties. Springer, Berlin.
Ohmann, D. 1952a, Eine Minkowskische Ungleichung fiir beliebige Mengen und ihre
Anwendung auf Extremalprobleme. Math. Z. 55, 299-307.
1952b, Ungleichungen zwischen den QuermaBintegralen beschrSnkter Punktmengen, I.
Math. Ann. 124, 265-76.
1953, Ein vollstandiges Ungleichungssystem fiir Minkowskische Summe und Differenz.
Comment. Math. Helvet. 27, I5I-6.
1954, Ungleichungen zwischen den QuermaBintegralen beschrankter Punktmengen, II.
Math. Ann. 127, 1-7.
1955a, Eine lineare VerschSrfung des Brunn-Minkowskischen Satzes fiir abgeschlossene
Mengen. Arch. Math. 6, 33-5.
1955b, Eine Verallgemeinerung der Steinerschen Formel. Math. Ann. 129, 209-12.
1956a, Ungleichungen fiir die Minkowskische Summe und Differenz konvexer Kiirper.
Comment. Math. Helvet. 30, 297-304.
1956b, Ungleichungen zwischen den QuermaBintegralen beschrSnkter Punktmengen, III.
Math. Ann. 130, 386-93.
1958, Ein allgemeines Extremalproblem fiir konvexe Kiirper. Monatsh. Math. 62,
97-107.
Ohshio, S. 1955, Volume, surface area and total mean curvature. Sci. Rep. Kanazawa
Univ. 4,21-8.
1958, Parallel series to a closed convex curve and surface and the differentiability of
their quantities. Sci. Rep. Kanazawa Univ. 6, 15-24.
1962, On an isoperimetric sequence. Sci. Rep. Kanazawa Univ. 8, 13-23.
Oishi, K. 1920, A note on the closed convex surface. Tdhoku Math. J. 18, 288-90.
Oliker, V. I. 1973, The uniqueness of the .solution in Christoffel and Minkowski problems
for open surfaces (in Russian). Mat. Zametki 13, 41-9. English translation: Math.
Notes 13, 26-30.
1979a, Eigenvalues of the Laplacian and uniqueness in the Minkowski problem. J.
Differential Geom. 14, 93-8.
1979b, On certain elliptic differential equations on a hypersphere and their geometric
applications. Indiana Univ. Math. J. 28, 35-51.
1982, On the linearized Monge-Ampfire equations related to the boundary value
Minkowski problem and its generalizations. In: Monge-Ampire Equations and
Related Topics, Florence 1980, 1st. Naz. Alta Mat. Francesco Severi, Roma, pp, 79-112.
1986, The problem of embedding S" into R"*' with prescribed Gauss curvature and its
solution by variational methods. Trans. Amer. Math. Soc. 295, 291-303.
Osserman, R. 1978, The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182-238.
1979, Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly 86, 1-29.

462 References
1987, A strong form of the isoperimetric inequality in R". Complex Variables 9, 241-9.
Overhagen, T. 1975, Zur Gitterpunktanzahl konvexer KOrper im 3-dimensionalen euklidi-
schen Raum. Math. Ann. 216, 217-24.
Panina, G. Y. 1988, Representation of an n-dimensional body in the form of a sum of
{n - I)-dimensional bodies (in Russian), hv. Akad. Nauk Armyan. SSR, Ser. Mat.
23, 385-95. English translation; Soviet J. Contemp. Math. Anal. 23, 91-103.
1989a, Convex bodies integral representations. In: Geobild '89 (eds A. Hiibler, W.
Nagel, B, D. Ripley & G. Werner), Akademie-Verlag, Berlin, pp. 201-4.
1989b, Integral representations of convex bodies (in Russian). Dokl. Akad. Nauk SSSR
307. English translation: Soviet Math. Dokl. 40 A990), II6-I8.
Panov, A. A. 1985a, On mixed discriminants connected with positive semidefmite
quadratic forms (in Russian). Doklady Akad. Nauk SSSR 282, 273-6. English
translation: Soviet Math. Dokl. 31, 465-7.
1985b, On some properties of mixed discriminants (in Russian). Mat. Sbornik 128,
291-305. English translation: Math. USSR (Sbornik) 56, 279-93.
Papaderou-Vogiatzaki, I. & Schneider, R. 1988, A collision probability problem. J. Appl.
Prob. 25, 617-23.
Papadopoulou, S. 1977, On the geometry of stable compact convex sets. Math. Ann. 229,
193-200.
1982. Stabile konvexe Mengen, Jahresber. Deutsche Math.-Ver. 84, 92-106.
Pays, R. & Yost. D. 1988, The two-ball property; Transitivity and examples. Mathematika
35, 190-7.
Perles, M, A. & Sallee, G, T. 1970, Cell complexes, valuation.s, and the Euler relation.
Canad. J. Math. 22, 235-41.
Petermann, E. 1967, Some relations between breadths and mixed volumes of convex
bodies in R". In: Proc. Colloq. Convexity. Copenhagen 1965, K0benhavns Univ. Mat.
Inst., pp. 229-33.
Petty, C. M. I96Ia, Centroid surfaces. Pacific J. Math. 11, 1535-47.
196lb. Surface area of a convex body under affine transformations. Proc. Amer. Math.
Soc. 12, 824-8.
1967, Projection bodies. In; Proc. Colloq. Convexity, Copenhagen 1965, Kpbenhavns
Univ. Mat, Inst., 234-41.
1972, Isoperimetric problems. In: Proc. Conf. Convexity Combinat. Geom., Univ.
Oklahoma 1971, pp. 26-41.
1974, Geominimal surface area. Geom. Dedicata 3, 77-97.
1985, Affine isoperimetric problems. In: Dbcrete Geometry and Convexity (eds J. E.
Goodman, E. Lutwak, J. Malkevitch & R. Pollack), Ann. New York Acad. Sci. 440,
113-27.
Phelp.s, R. R. 1966. Lectures on Choquet's Theorem. Van Nostrand, Princeton.
1980, Integral representatioas for elements of convex sets. Studies in Functional Analysis
(ed. R. G. Bartle), MAA Studies in Math. 21, Math. A.ssoc. Amer., Washington DC,
pp. 115-57.
Pisier, G. 1989, The Volume of Convex Bodies and Banach Space Geometry. Cambridge
Univ. Press.
Pogorelov, A. V. 1953, On the question of the existence of a convex surface with a given
sum of the principal radii of curvature (in Russian). Uspehi Mat. Nauk 8. 127-30.
1967, Nearly spherical surfaces. J. d'Analyse Math. 19, 313-21.
1975, The Minkowski Multidimensional Problem. Winston & Sons. Washington DC,
1978 (Russian original; 1975).
P61ya, G. & Szegi), G. 1951, Isoperimetric Inequalities in Mathematical Physics. Princeton
Univ. Press, Princeton, NJ.
Pontrjagin, L. S. 1957, Topologische Gruppen, vol 1. Teubner, Leipzig (English translation
of first Russian edn: 1946).
Positsel'skii, E. D. 1973, Characterization of Steiner points (in Russian). Mat. Zametki 14,

References 463
243-7. English translation: Maih. Notes 14, 698-700.
Pranger, W. 1973, Extreme points of convex sets. Math. Ann. 205, 299-302.
Pr^kopa, A. I97I, Logarithmic concave measures with applications to stochastic
programming. Acta. Sci. Math. Szeged 32, 301-16.
1973, On logarithmic concave measures and functions. Acta Sci. Math. Szeged 34,
335-43.
Przestawski, K. 1985, Linear and Lipschitz continuous selectors for the family of convex
sets in Euclidean vector spaces. Bull. Polish. Acad. Sci. Math. 33, 31-3.
1988, Linear algebra of convex sets and the Euler characteristic. Report, Institute of
Math., Phys. and Chem., Wyzsza Szkola Inzynierska, Zielona G6ra, 65 pp.
Rademacher, H. I92I, Zur Theorie der Minkowskischen StUtzebenenfunktion. Sit-
zungsber. Berliner Math. Ges. 20, 14-19.
Radon, J. 1936, AnnSherung konvexer KSrper durch analytisch begrenzte. Monatsh.
Math. Phys. 43, 340-4. Gesammelte Ahhandlungen, vol. I (eds P. M. Gruber et al.),
Birkhauser, Wien, 1987, pp. 370-4.
RSdstriim, H. 1952, An embedding theorem for spaces of convex sets. Proc. Amer. Math.
Soc. 3, 165-9.
Ratschek, H. & Schroder, G. 1977, Representation of semigroups as systems of compact
convex sets. Proc. Amer. Math. Soc. 65, 24-8,
Rauch, J. 1974, An inclusion theorem for ovaloids with comparable second fundamental
forms. J. Differential Geom. 9, 501-5.
Reay, J. R. 1965, Generalizations of a theorem of Carath^odory. Amer. Math. Soc.
Memoirs 54.
Reidemeister, K. I92I, Ober die singuliiren Randpunkte eines konvexen Korpers. Math.
Ann. 83, 116-18.
Reisner, S. 1985, Random polytopes and the volume-product of symmetric convex bodies.
Math. Scand. 57, 386-92.
1986, Zonoids with minimal volume product. Math. Zeitschr. 192, 339-46.
1987, Minimal volume-product in Banach spaces with a 1-unconditional basis. J. London
Math. Soc. 36, 126-36.
Reiter, H. B. & Stavrakas, N. M". 1977, On the compactness of the hyperspace of faces.
Pacific J. Math. 73, 193-6.
R^nyi, A. 1946, Integral formulae in the theory of convex curves. Acta Sci. Math. 11,
158-66.
ReJetnjak, Ju. G. 1968a, Some estimates for almost umbilical surfaces (in Russian).
Sihirskii Mat. Zh. 9, 903-17. English translation: Siberian Math. J. 9, 671-82,
1968b, Generalized derivatives and differentiability almost everywhere (in Russian).
Mat. Sbornik 75, 323-34. English translation: Math. USSR (Sbornik) 4, 293-302.
Reuleaux, F. 1875, Theoretbche Kinematik, vol. I. Vieweg, Braunschweig.
Ricker, W. 1981, A new class of convex bodies. In: Papers in Algebra, Analysis and
Statbtics, Hobart 1981, Contemp. Math. 9, Amer. Math. Soc, Providence, RI, pp.
333-40.
Rickert, N. W. 1967a, Measures whose range is a ball. Pacific J. Math. 23, 361-71.
1967b, The range of a measure. Bull. Amer. Math. Soc. 73, 560-3.
Rogers, C. A. 1970, Hausdorff Measures. Cambridge Univ. Press.
Rogers, C. A. & Shephard G. C. 1957, The difference body of a convex body. Arch.
Math. 8, 220-33.
1958a, Convex bodies associated with a given convex body. J. London Math. Soc. 33,
270-81.
1958b, Some extremal problems for convex bodies. Mathematika 5, 93-102.
Rota, G.-C. 1964, On the foundations of combinatorial theory, I. Theory of Mobius
functions, Z. Wahrscheinlichkeitsth. verw. Geb. 2, 340-68.
I97I, On the combinatorics of the Euler characteristic. In: Studies in Pure Math. (Papers
presented to Richard Rado), Academic Press, London, pp. 221-33.

464 References
Rother, W. & Ziihle, M. 1990, A short proof of a principal kinematic formula and
extensions. Trans. Amer. Math. Soc. 321, 547-58.
1992, Absolute curvature measures II. Geom. Dedicaia 41, 229-40.
Roy, A. K. 1972, Facial structure of the sum of two compact convex sets. Math. Ann. 197,
189-96.
Roy, N. M. 1987, Extreme points of convex sets in infinite dimensional spaces. Amer.
Math. Monthly 94, 409-22.
Roy, S. N. 1936, On the vector derivation of the invariants and centroid formulae for
convex surfaces. Bull. Calcutta Math. Soc. 28, 79-88.
Sacksteder, R. 1960, On hypersurfaces with no negative .sectional curvatures. Amer. J.
Math. 82, 609-30.
Saint Pierre, J. 1985, Point de Steiner et sections lipschitziennes. In: Simin. Anal.
Convexe, Univ. Sci. Tech. Languedoc 15, Exp. No. 7. 42 pp.
Saint Raymond, J. I98I, Sur le volume des corps convexes sym^triques. In: Seminaire
Initiation a I'Analyse, Univ. Pierre et Marie Curie, Paris 1980/81, vol. II, 25 pp.
1986, Nombre de sommets d'un polyfedre al^atoire. In: Siminaire Initiation d I'Analyse,
Univ. Pierre et Marie Curie, Paris 1985/86, vol. II, 13 pp.
Salinetti, G. & Wets, R. J.-B. 1979, On the convergence of sequences of convex sets in
finite dimensions. SIAM Rev. 21, 18-33.
Sallee, G. T. 1966, A valuation property of Steiner points. Mathematika 13, 76-82.
1968, Polytopes, valuations, and the Euler relation. Canad. J. Math. 20, 1412-24.
1971, A non-continuous 'Steiner point'. Israel J. Math. 10, 1-5.
1972, Minkowski decomposition of convex sets. Israel J. Math. 12, 266-76.
1974, On the indecomposability of the cone. J. London Math. Soc. 9, 363-7.
1982, Euler's theorem and where it led. In: Convexity and Related Combinatorial
Geometry (eds D. C. Kay & M. Breen), Marcel Dekker, New York, pp. 45-55.
1987, Pairs of sets of constant relative width. J. Geom. 29, I-11.
Sandgren, L. 1954, On convex cones. Math. Scand. 2, 19-28.
Sangwine-Yager, J. R. 1980, The mean value of the area of polygons circumscribed about
a plane convex body. Israel J. Math. 37, 351-63.
1983, The mean quermassintegral of simplices circumscribed about a convex body.
Geom. Dedicata 15, 47-57.
1988a, A Bonnesen-style inradius inequality in 3-space. Pacific J. Math. 134, 173-8.
1988b, Bonnesen-style inequalities for Minkowski relative geometry. Trans. Amer. Math.
Soc. 307, 373-82.
1989, The missing boundary of the Blaschke diagram. Amer. Math. Monthly 96, 233-7.
I99I, Stability for a cap-body inequality. Geom. Dedicata 38, 347-56.
Santal6, L. A. 1946, Sobre los cuerpos convexos de anchura constante en £„. Portugaliae
Math. 5, 195-201.
1949, Un invariante afin para los cuerpos convexos del espacio de n dimensiones.
Portugaliae Math. 8, 155-61.
1950, On parallel hypersurfaces in the elliptic and hyperbolic n-dimensional space. Proc.
Amer. Math. Soc. 1, 325-30.
1953, Introduction to Integral Geometry. Hermann & Cie., Paris.
I96I, Sobre los sistemas completes de desigualdades entre los elementos de una figura
convexa plana. Math. Notae 17, 1959/61, 82-104.
1976, Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.
Sas, E. 1939, Uber eine Extremaleigenschaft der Ellipsen. Compositio Math. 6, 309-29.
Sa^kin, Ju. A. 1973, Convex sets, extreme points, and simplexes (in Russian). Itogi Nauki
i Tekhniki (Mat. Analiz) 11, 5-50. English translation: J. Soviet Math. 4, 625-55.
Schaal, H. 1962, Priifung einer Kreisform mit Hohlwinkel und Taster. Elem. Math. 17,
33-8.
1963, Gleitkurven in regulSren Polygonen. Z. angew. Math. Mech. 43, 459-76.
Schmidt, K. D. 1986, Embedding theorems for classes of convex sets. Acta Appl. Math. 5,
209-37. Acknowledgement of priority: Acta Appl. Math. 11, 295.

References 465
Schmitt, K.-A. 1967, Hilbert spaces containing subspaces consisting of symmetry classes of
convex bodies. In: Proc. Colloq. Convexity, Copenhagen 1965. K0benhavn.s Univ.
Mat. Inst., pp. 278-80.
1968, Kennzeichnung des Steinerpunktes konvexer Kiirper. Math. Z. 105, 387-92. (This
work is erroneous.)
Schneider, R. 1966a, Ahnlichkeits- und TranslationssStze fiir EiflSchen. Arch. Math. 17,
267-73.
1966b, On A. D. Aleksandrov's inequalities for mixed discriminants. / Math. Mech. 15.
285-90. (This work is erroneous.)
1967a, Ober die Durchschnitte translationsgleicher konvexer Korper und eine Klasse
konvexer Polyeder. Abh. Math. Sem. Univ. Hamburg 30, 118-28.
1967b, Eine allgemeine Extremaleigenschaft der Kugel. Monatsh. Math. 71, 231-7.
1967c, Zu einem Problem von Shephard iiber die Projektionen konvexer Korper. Math.
Z. 101.71-82.
I967d, Zur affinen Differentialgeometrie im GroBen I. Math. Z. 101, 375-406.
1969a, Ober die FinslerrSume mit 5,,j, = 0. Arch. Math. 19, 656-8.
1969b, Characterization of certain polytopes by intersection properties of their
translates. Mathematika 16, 276-82.
1970a, Ober eine Integralgleichung in der Theorie der konvexen Korper. Math. Nachr.
44, 5.5-75.
1970b, Eine Verallgemeinerung des Differenzenkorpers. Monatsh. Math. 74, 258-72.
1970c, On the projections of a convex polytope. Pacific J. Math. 32, 799-803.
1970d, Functional equations connected with rotations and their geometric applications.
L'Enseignement Math. 16, 297-305.
1971a, On Steiner points of convex bodies. Israel J. Math. 9, 241-9.
I97Ib, Gleitkorper in konvexen Polytopen. J. reine angew. Math. 248, 193-220.
1971c, Zwei Extremalaufgaben fiir konvexe Bereiche. Acta Math. Hungar. 22, 379-83.
I97Id, The mean surface area of the boxe.s circumscribed about a convex body. Ann.
Polon. Math. 25, 325-8.
1972a, Krijmmungsschwerpunkte konvexer Kiirper, I. Abh. Math. Sem. Univ. Hamburg
37, 112-32.
1972b, Krijmmungsschwerpunkte konvexer Kiirper, II. Abh. Math. Sem. Univ.
Hamburg 31,204-17.
1973, Volumen und Schwerpunkt von Polyedern. Elem. Math. 28, 137-41.
1974a, Equivariant endomorphisms of the space of convex bodies. Trans. Amer. Math.
Soc. 194, 53-78.
1974b, Summanden konvexer Korper. Arch. Math. 25, 83-5.
1974c, BewegungsSquivariante, additive und stetige Transformationen konvexer
Bereiche. Arch. Math. 25, 303-12.
I974d, Additive Transformationen konvexer Korper. Geom. Dedicata 3, 221-8.
I974e, On asymmetry classes of convex bodies. Mathematika 21, 12-18.
1975a, Isometrien des Raumes der konvexen Kiirper. Colloquium Math. 33, 219-24.
1975b, Remark on a conjectured characterization of the sphere- Ann. Polon. Math. 31,
187-90.
1975c, Zonoids whose polars are zonoids. Proc. Amer. Math. Soc. SO, 365-8.
I975d, A measure of convexity for compact sets. Pacific J. Math. 58, 617-26.
1975e, Kinematische BeriihrmaBe fiir konvexe Kiirper. Abh. Math. Sem. Univ.
Hamburg 44, 12-23.
1975f, Kinematische BeriihrmaBe fiir konvexe KSrper und Integralrelationen fiir Ober-
flachenmaBe. Math. Ann. 218, 253-67.
1976, Bestimmung eines konvexen KSrpers durch gewisse BeriihrmaBe. Arch. Math. 27,
99-105.
1977a, Das Christoffel-Problem fUr Polytope. Geom. Dedicata 6, 81-5.
1977b, Eine kinematische Integralformel fiir konvexe KOrper. Arch. Math. 28, 217-20.
1977c, Rekonstruktion eines konvexen Kiirpers aus seinen Projektionen. Math. Nachr.

466 References
79, 325-9.
I977d, Kritische Punkte und Kriimmung flir die Mengen des Konvexringes. L'Enseigne-
ment Math. 23, 1-6.
1977e. Eine Charakterisierung der Kugel. Arch. Math. 29, 660-5.
1978a, Curvature measures of convex bodies. Ann. Mat. Pura Appl. 116, 101-34.
1978b, On the skeletons of convex bodies. Bull. London Math. Soc. 10, 84-5.
1978c, Ober Tangentialkorper der Kugel. Manuscripta Math. 23, 269-78.
1978d, Kinematic measures for sets of colliding convex bodies. Mathematika 25, I-I2.
1979a, Bestimmung konvexer Korper durch KriimmungsmaBe. Comment Math. Helvet.
54, 42-60.
1979b, On the curvatures of convex bodies. Math. Ann. 240, 177-81.
1980a. Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata 9,
111-27.
1980b, Curvature measures and integral geometry of convex bodies. Rend. Sem. Mat.
Univ. Politecn. Torino 38, 79-98,
I98Ia, Crofton's formula generalized to projected thick sections. Rend. Circ. Mat.
Palermo 30. 157-60.
1981b. A local formula of translative integral geometry. Arch. Math. 36, 149-56.
198Ic, Pairs of convex bodies with unique joining metric segments, Bull. Soc. Roy. Sci.
Liige 50, 5-7.
1982a, Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. verw.
Geb. 61.379-87,
1982b, Random polytopes generated by anisotropic hyperplanes. Bull. London Math.
Soc. 14, 549-53.
1983, Nonparametric convex hypersurfaces with a curvature restriction. Ann. Polon.
Math. 51. 57-61.
1984, Smooth approximation of convex bodies. Rend. Circ. Mat. Palermo, Ser. II 33,
436-40.
1985a, On the Aleksandrov-Fenchel inequality. In: Discrete Geometry and Convexity
(eds J, E. Goodman. E. Lutwak, J. Malkevitch & R. Pollack). Ann. New York Acad.
Sci. 440. 132-41.
1985b, Inequalities for random flats meeting a convex body. J. Appl. Proh. 22, 710-16.
1986a. Affine-invariant approximation by convex polytopes. Studia Sci. Math. Hungar.
21,401-8.
1986b, Curvature measures and integral geometry of convex bodies II. Rend. Sem. Mat.
Univ. Politecn. Torino 44, 263-75.
1987a, Geometric inequalities for Poisson processes of convex bodies and cylinders.
Results Math. 11, 165-85.
1987b, Polyhedral approximation of smooth convex bodies. J. Math. Anal. Appl. 128,
470-4,
1987c, Equidecomposable polyhedra. In: Intuitive Geometry, Si6fok 1985, Colloquia
Math. Soc. Jdnos Bolyai 48. North-Holland. Amsterdam, pp, 481-501,
1988a, On the Aleksandrov-Fenchel inequality involving zonoids. Geom. Dedicata 27,
113-26.
1988b. Closed convex hypersurfaces with curvature restrictions. Proc. Amer. Math. Soc.
103. 1201-4.
1988c, Random approximation of convex sets, J. Microscopy 151, 211-27.
1989a. On a morphological transformation for convex domains. J. Geom. 34. 172-80.
1989b, Gemischte Volumina in Kanalscharen. Geom. Dedicata 30, lli-iA.
1989c, Stability in the Aleksandrov-Fenchel-Jessen theorem. Mathematika 36, 50-9.
1990a. On the Aleksandrov-Fenchel inequality for convex bodies, I Results Math. 17,
287-95.
1990b, A stability estimate for the Aleksandrov-Fenchel inequality, with an application
to mean curvature. Manuscripta Math. 69, 291-300.

References 467
1990c, Curvature measures and integral geometry of convex bodies III. Rend. Sem. Mat.
Univ. Poliiecn. Torino 46, II1-23.
I99I + , Equality in the Aleksandrov-Fenchel inequality - present state and new results.
Preprint.
Schneider, R. & Weil, W. 1970, Ober die Bcstimmung eines konvexen Korpers durch die
Inhalte seiner Projektionen. Math. Z. 116. 338-48.
1986, Translative and kinematic integral formulae for curvature measures. Math. Nachr.
129, 67-80.
Schneider, R. & Wieacker, J. A. 1981, Approximation of convex bodies by polytope.s.
Bull. London Math. Soc. 13, 149-56.
1984, Random touching of convex bodies. In: Proc. Conf. Stochastic Geom., Geom.
Stati.u., Stereology, Oberwolfach 1983 (eds R. V. Ambartzumian & W. Weil),
Teubner. Leipzig, pp. 154-69.
Schiirger, K. 1983, Ergodic theorems for subadditive superstationary families of convex
compact random sets. Z. Wahrscheinlichkeitsth. venv. Geh. 62, 125-35.
Schiitt, C. 1990+, On the affine surface area. Preprint.
1991, The convex floating body and polyhedral approximation. Israel J. Math. 73,
65-77.
Schutt, C. & Werner, E. 1990. The convex floating body. Math. Scand. 66, 275-90.
1990+, The convex floating body of almost polygonal bodies. Preprint.
Schwarz. T. & Zamfirescu. T. 1987, Typical convex sets of convex sets. J. Austral. Math.
Soc (A) 43, 287-90.
Seeley, R. T. 1966, Spherical harmonics. Amer. Math. Monthly 73, 115-21.
Sen'kin, E. P. 1966a, Stability of the width of a general closed convex surface depending
on the change of the integral mean curvature (in Russian). Ukrain. Geom. Sh. 2,
88-9.
1966b, Stability of the width of a general closed convex surface depending on the change
of the integral mean curvature, II (in Russian), Ukrain. Geom. .Sb. 3, 93-4,
Serra, J, 1982, Image Analysis and Mathematical Morphology. Academic Press, London.
1988, (ed.) Image Analysis and Mathematical Morphology. Vol. 2: Theoretical Advances.
Academic Press, London.
Shahin, J. K. 1968, Some integral formulas for closed hypersurfaces in Euclidean space.
Proc. Amer. Math. Soc. 19, 609-13.
Shephard, G. C. 1960, Inequalities between mixed volumes of convex set.s. Mathematika 7,
125-38.
1963, Decomposable convex polyhedra. Mathematika 10, 89-95.
1964a, Approximation problems for convex polyhedra. Mathematika 11, 9-18.
1964b. Shadow systems of convex bodies. IsraelJ. Math. 2, 229-36.
1966a, The Steiner point of a convex polytope. Canad. J. Math. 18, 1294-300.
1966b, Reducible convex sets. Mathematika 13, 49-50.
1966c, A pre-Hilbert space consisting of classes of convex sets. Israel J. Math. 4. I-10.
1967, Polytopes with centrally symmetric faces. Canad. J. Math. 19, 1206-13.
1968a. The mean width of a convex polytope. J. London Math. Soc. 43. 207-9.
1968b, A uniqueness theorem for the Steiner point of a convex region, J. London Math.
Soc. 43. 439-44.
1968c, Euler-type relations for convex polytopes. Proc. London Math. Soc. C) 18,
597-606.
1971, Diagram.s for positive bases. J. London Math. Soc. B) 4, 165-75.
1974a. Combinatorial properties of associated zonotopes. Canad. J. Math. 26, 302-21.
1974b, Space-filling zonotopes. Mathematika 21. 261-9.
Shephard, G. C. & Webster, R. J. 1965, Metrics for sets of convex bodies. Mathematika
12, 73-88.
Silverman, R. 1973a, Decomposition of plane convex sets. Part I. Pacific J. Math. 47.
521-30.

468 References
1973b, Decomposition of plane convex sets. Part II: Sets associated with a width
function. Pacific J. Math. 48, 497-502. (This work is erroneous.)
Simon, U. 1967, Minkowskische Integralformeln und ihre Anwendungen in der Differen-
tialgeometrie im GroBen. Math. Ann. 173, 307-21.
Smilansky, Z. 1986, An indecomposable polytope all of whose facets are decomposable.
Mathematika 33, 192-6.
1987, Decomposability of polytopes and polyhedra. Geom. Dedicata 24, 29-49.
Sorokin, V. A. 1968, Classes of convex sets as generalized metric spaces (in Russian).
Mat. Zametki 4 A968), 45-52. English translation: Math. Notes 4, 517-21.
Spiegel, W. 1975, Ein Konvergenzsatz fiir eine gewisse Klasse kompakter Punktmengen. J.
reine angew. Math. 277, 218-20.
1976a, Zur Minkowski-Additivitat bestimmter Eikorperabbildungen. J. reine angew.
Math. 2«6/2«7, 164-8.
1976b, Ein Zerlegungssatz fUr spezielle Eikorperabbildungen in den euklidischen Raum.
J. reine angew. Math. 2*3/284, 282-6.
1978, Ein Beitrag ijber additive, translationsinvariante, stetige EikOrperfunktionale.
Geom. Dedicata 7, 9-19.
1982, Nonnegative, motion-invariant valuations of convex polytopes. In: Convexity and
Related Combinatorial Geometry (eds D. C. Kay & M. Breen), Marcel Dekker, New
York, pp. 67-72.
Stach6. L. L. 1979, On curvature measures. Acta. Sci. Math. 41, 191-207.
Stanley, R. P. 1981, Two combinatorial applications of the Aleksandrov-Fenchel
inequalities. J. Combinatorial Theory, Ser. A 31, 56-65.
1986, Two poset polytopes. Discrete Comput. Geom. 1, 9-23.
Starr, R. 1969, Quasi-equilibria in markets with nonconvex preferences. Econometrica 37,
25-38.
1981, Approximation of points of the convex hull of a sum of sets by points of the sum:
An elementary approach. J. Econom. Theory 25. 314-17.
Steenaerts, P. 1985, Mittlere SchattengrenzenlSnge konvexer Kiirper. Results Math. 8,
54-77.
Steiner, J. 1840a, Von dem Kriimmungsschwerpunkte ebener Curven. J. reine angew.
Math. 21, 33-63. Ges. Werke, vol. 2, Reimer, Berlin, 1882, pp. 99-159.
1840b, Ober parallele Flachen. Monatsber. preufi. Akad. Wbs., Berlin, II4-I8. Ges.
Werke, vol. 2, Reimer, Berlin, 1882, pp. I7I-6.
Stoka, M. I. 1968, Geomitrie integrale. Gauthier-Villars, Paris.
Stoker, J. J. 1936, Ober die Gestalt der positiv gekriimmten offenen Flachen. Compositio
Math. 3, 55-88.
1950, On the uniqueness theorems for the embedding of convex surfaces in
three-dimensional space. Commun. Pure Appl. Math. 3, 231-57.
Stoyan, D., Kendall, W. S. & Mecke, J. 1987, Stochastic Geometry and its Applications.
Akademie-Verlag, Berlin.
Straszewicz, S. 1935 Uber exponierte Punkte abgeschlossener Punktmengen. Fund. Math.
24, 139-43.
Streit, F. 1970, On multiple integral geometric integrals and their applications to
probability theory. Canad. J. Math. 22, 151-63.
1973, Mean-value formulae for a class of random sets. J. R. Statist. Soc. B 35, 437-44.
1975, Results on the intersection of randomly located sets. J. Appl. Prob. 12, 817-23.
Su, B. 1927a, On Steiner's curvature centroid. Japan. J. Math. 4, 195-201.
1927b, On Steiner's curvature centroid, II. Japan. J. Math. 4, 265-9.
Sung, C. H. & Tam, B. S. 1987, On the cone of a finite-dimensional compact convex set
at a point. Linear Algebra Appl. 90, 47-55.
Suss, W. 1929, Zur relativen Differentialgeometrie V: Ober Eihyperfiachen im /?"*'.
Tohoku Math. J. 30, 90-5.
193la, Bestimmung einer ge-schlossenen konvexen Flache durch die GauBsche Krtim-
mung. Sitzungsber. Preufi. Akad. Wiss. 686-95.

References 469
I93Ib, Die Isoperimetrie der mehrdimensionalen Kugel. Siizungsber. Preufi. Akad.
Wiss. 342-4.
1932a, Zusammensetzung von EikSrpern und homothetische Eifliichen. Tdhoku Math. J.
35,47-50.
1932b, Eindeutigkeitsstitze und ein Existenztheorem in der Theorie der Eifliichen im
GroBen. Tdhoku Math. J. 35, 290-3.
1933, Bestimmung einer geschlossenen konvexen HSche durch die Summe ihrer Haupt-
krilmmungsradien. Math. Ann. 108, 143-8.
Sz.-Nagy, B. v. (SzOkefalvi-Nagy, B.) 1940, Ober ein geometrisches Extremalproblem.
Acta. Set. Math. 9, 253-7.
1959, Ober Parallelmengen nichtkonvexer ebener Bereiche. Acta Set. Math. Szeged 20,
36-47.
Sz.-Nagy, G. v. 1949, Schwerpunkt von konvexen Kurven und von konvexen HSchen.
Portugaliae Math. 8, 17-22.
Tanno, S. 1976, C-approximation of continuous ovals of constant width. J. Math. Soc.
Japan 28, 384-95.
1987, Central sections of centrally symmetric convex bodies. Kodai Math. J. 10, 343-61.
Teissier, B. 1979, Du thtor^me de I'index de Hodge aux in^galit^s isop^rim^triques. C.R.
Acad. Sci. Paris 288, 287-9.
1981, Vari^t^s toriques et polytopes. In: Siminatre Bourbaki, vol. 1980/81, Lecture
Notes in Math. 901, Springer, Berlin, pp. 71-84.
1982, Bonnesen-type inequalities in algebraic geometry I: Introduction to the problem.
In: Seminar on Differential Geometry (ed. S. T. Yau), Ann. Math. Studies 102,
Princeton Univ. Press, pp. 85-105.
Tennison, R. L. 1967, An almost-measure of symmetry. Amer. Math. Monthly 74, 820-3.
Thomas, C. 1984, Extremum properties of the intersection densities of stationary Poisson
hyperplane processes. Math. Operationsforsch. Statist. (Ser. Statbt.) 15, 443-9.
Tolstonogov, A. A. 1977, Support functions of convex compacta (in Rus.sian). Mat.
Zametki 22, 203-13. English translation: Math. Notes 22, 604-9.
Treibergs, A. 1985, Existence and convexity for hyperspheres of prescribed mean
curvature. Ann. Scuola Norm. Sup. Pisa, CI. Sci.. Ser. IV, 12, 225-41.
1990, Bounds for hyperspheres of prescribed Gaussian curvature. J. Differential Geom.
31,913-26.
Tverberg, H. 1966, A generalization of Radon's theorem. J. London Math. Soc. 41,
123-8.
1981, A generalization of Radon's theorem, II. Bull. Austral. Math. Soc. 24, 321-5.
Uhrin, B. 1984, Sharpenings and extensions of Brunn-Minkowski-Lusternik inequality.
Technical Report No. 203, Stanford Univ., Dept. of Statistics, Stanford, CA.
1987, Extensions and sharpenings of Brunn-Minkowski and Bonnesen inequalities. In:
Intuitive Geometry, Si6fok 1985, Colloquia Math. Soc. Jdnos Bolyai 48, North-
Holland, Amsterdam, pp. 551-71.
Urbanski, R. 1976, A generalization of the Minkowski-Radstriim-Hiirmander theorem.
Bull. Acad. Polon. Sci., Sir. sci., math., astr., phys. 24, 709-15.
Valette, G. 1971, Couples de corps convexes dont la difference des largeurs est constante.
Bull. Soc. Math. Belg. 23, 493-9.
1974, Subadditive affine-invariant transformations of convex bodies. Geom. Dedicata 2,
461-5.
van Heijenoort, J. 1952, On locally convex manifolds. Commun. Pure Appl. Math. 5,
223-42.
Vidal Abascal, E. 1947, A generalization of Steiner's formulae. Bull. Amer. Math. Soc.
53, 841-4.
Vincensini, P. 1935, Sur les corps convexes admettant un domaine vectoriel donn6. C. K.
Acad. Sci. Paris 201, 761-3.
1936, Sur les domaines vectoriels des corps convexes. J. Math, pures appl. IS, 373-83.
1937a, Sur le prolongement des series lin^aires de corps convexes. Applications. Rend.

470 References
Circ. Mat. Palermo 60, 361-72.
1937b, Les domaines vectoriels et la theorie des corps convexes. L'Enseignemeni Math.,
69-80.
1937c, Sur une transformation des corps convexes et son application a la construction de
lensemble des corps convexes de I'espace a n dimensions c» partir de certains
sous-ensembles bases. Bull. Soc. Math. France 65. 175-89.
1938, Corps convexes. Series IMaires. Domaines vectoriels. Mem. des Sci. Math. 44,
Gauthier-Villars, Paris.
1956. Sur I'application d"une m^thode g^om^trique a I'etude de certains ensembles de
corps convexes. In: Colloque sur les questions de rialiti en ge'om^irie. Li^ge 1955.
Paris, pp. 77-94.
1957, Sur un invariant de partition de Pensemble des corps convexes de I'espace
euclidien k n dimensions. C. R. Acad. Sci. Paris 245, 132-3.
1965, Questions li^es k la notion de g^om^trie diff^rentielle globale. Rev. Roum. Math.
Pure.i Appl. 10. 877-93.
Vincensini, P. & Zamfirescu, T. 1967, Sur une fibration de I'espace des corps convexes. C.
R. Acad. Sci. Paris 264, 510-11.
Vitale, R. A. 1979, Approximation of convex set-valued functions. J. Approx. Theory 26,
301-16.
1984, On Gaussian random sets. In: Proc. Oherwolfach Conf. Stochastic Geom., Geom.
Statist.. Stereology (eds R. V. Ambartzumian & W. Weil). Teubner, Leipzig, pp.
222-4.
1985a, L^ metrics for compact, convex sets. J. Approx. Theory 45, 280-7.
1985b, The Steiner point in infinite dimensions. IsraelJ. Math. 52, 245-50.
1990, The Brunn-Minkowski inequality for random sets. J. Multivariate Anal. 33,
286-93.
1991, The translative expectation of a random set. J. Math. Anal. Appl. 160, 556-62.
Vlasov, L. P. 1973, Approximative properties of sets in normed linear spaces (in Russian).
i7.jp. Mat. Nauk 28, 3-66. English translation: Russ. Math. Surveys 28, 1-66.
Vodop'yanov, S. K. 1970. Estimates of the deviation from a sphere of quasi-umbilical
surfaces. Sibir.skii Mat. Zh. U, 971-87. English translation; Siberian Math. J. II,
724-3.').
Voiculescu. D. 1966, O ecuatie privind corpurile convexe si aplicatii la corpurile asociate
unui corp convex. Stud. Cere. Mat. 18, 741-.').
Volkov, Yu. A. 1963. Stability of the solution of Minkowski's problem (in Russian).
Vestnik Leningrad. Univ., Ser. Mat. Meh. Astronom. 18, 33-43.
Volkov, Yu., A. & Oliker, V. I. 1970, Uniqueness of the solution of Christoffel's problem
for nonclosed surfaces (in Russian). Mat. Zametki 8, 251-7. English translation:
Math. Notes. 8,611-14.
Volland, W. 1957, Ein Fortsetzungssatz fiir additive Eipolyederfunktionale im euklidischen
Raum. Arch. Math. 8, 144-9.
Waksman. Z. & Epelman, M. S. 1976, On point classification in convex sets. Math.
Stand. 38, 83-96.
Wallen, L. J. 1987, All the way with Wirtinger: A short proof of Bonne.sen's inequality.
Amer. Math. Monthly 94, 440-2.
Webster, R. J. 1965, The space of affine-equivalence classes of compact subsets of
Euclidean space. J. London Math. Soc. 40, 425-432.
Wegmann. R. 1980, Einige MaBzahlen fiir nichtkonvexe Mengen Arch. Math. 34, 69-74.
Wegner, B. 1977, Analytic approximation of continuous ovals of constant width. J. Math.
Soc. Japan 29, 537-40.
Weil, W. I97I, Uber die Projektionenkorper konvexer Polytope. Arch. Math. 11, 664-72.
1973, Ein Approximationssatz fUr konvexe KOrper. Manuscripta Math. 8, 335-62.
1974a, Uber den Vektorraum der Differenzen von Stiitzfunktionen konvexer KOrper.
Math. Nachr. 59, 353-69.
1974b, Decomposition of convex bodies. Mathematika 21, 19-25.

References 471
1975a, On mixed volumes of nonconvex sets. Proc. Amer. Math. Soc. 53, I9I-4.
1975b. Einsctiachtelung konvexer Korper. Arch. Math. 26, 666-9.
1976a, Kontinuierliche Linearkombination von Strecken. Math. Z. 148, 71-84.
1976b, Centrally symmetric convex bodies and distributions. Israel J. Math. 24. 352-67.
1977, Blaschkes Problem der lokalen Charakterisierung von Zonoiden. Arch. Math. 29,
655-9.
1979a, Centrally symmetric convex bodies and distributions, II. Israel J. Math. 32,
173-82.
1979b, Beriihrwahrscheinlichkeiten fiir konvexe Korper. Z. Wahrscheinlkhkeitsth. verw.
Geb. 48, 327-38.
1979c, Kinematic integral formulas for convex bodies. In: Contributions to Geometry,
Proc. Geometry Symp., Siegen 1978 (eds J. Tolke & J, M. Wills), BirkhSuser, Basel,
pp. 60-76.
1980a, On surface area measures of convex bodies. Geom. Dedicata 9, 299-306.
1980b, Eine Charakterisierung von Summanden konvexer Korper. Arch. Math. 34,
283-8.
1981a, Zufallige Beriihrung konvexer Korper durch 4-dimensionale Ebenen. Resultate
Math. 4, 84-101.
I98Ib, Das gemischte Volumen als Distribution. Manuscripta Math. 36, I-18.
1982a, Inner contact probabilities for convex bodies. Adv. Appl. Prob. 14, 582-99.
1982b, An application of the central limit theorem for Banach space-valued random
variables to the theory of random sets. Z. Wahrscheinlichkeitsth. verw. Geb. 60,
203-8.
1982c, Zonoide und verwandte Klassen konvexer KOrper. Monatsh. Math. 94, 73-84.
1983a, Stereology: A survey for geometers. In: Convexity and its Applications (eds P.
M. Gruber & J. M. Wills), BirkhSuser, Basel, pp. 360-412.
1983b, Stereological results for curvature measures. Bull. Int. Statist. Inst. 50, 872-83.
1984, Densities of quermassintegrals for stationary random sets. In: Proc. Conf.
Stochastic Geom., Geom. Statist., Stereology, Oberwolfach 1983 (eds R. V. Ambartzu-
mian & W. Weil), Teubner, Leipzig, pp. 233-47.
1987, Point processes of cylinders, particles and flats. Acta Applicandae Math. 9,
103-36.
1988, Expectation formulas and' isoperimetric properties for non-isotropic Boolean
models. J. Microscopy 151, 235-45.
1989a, Collision probabilities for convex sets. J. Appl. Prob. 26, 649-54.
1989b, Integral geometry, stochastic geometry, and stereology. Acta Stereologica 8,
65-76.
1989c, Translative integral geometry. In: Geobild 89 (eds A. Hiibler et al.). Math.
Research 51, Akademie-Verlag, Berlin 1989.
1990a, Iterations of translative integral formulae and non-isotropic Poisson processes of
particles. Math. Z. 205, 531-49.
1990b. Lectures on translative integral geometry and stochastic geometry of anisotropic
random geometric structures. Rend. Sem. Mat. Messina (Serie 11), 13 @), 79-97.
Weil, W. & Wieacker, J. A. 1984, Densities for stationary random sets and point
processes. Adv. Appl. Prob. 16, 324-46.
1988, A representation theorem for random sets. Prob. Math. Statist. 9, 147-55.
Weiss. V. 1986, Relations between mean values for stationary random hyperplane mosaics
of R''. Forschungserg. FSV N/86/33.
Weiss, V. & Zahle, M. 1988, Geometric measures for random curved mosaics of R''.
Math. Nachr. 13«, 313-26.
Weissbach, B. 1972, Rotoren im reguUren Dreieck. Publ. Math. Debrecen 19, 21-7.
1977a, Zur Inhaltsschatzung von Eibereichen. Beitr. Alg. Geom. 6, 27-35.
1977b, Ausbohrung von Rhomben. Beitr. Alg. Geom. 6, 153-8.
Wets, R. J.-B. 1974, Ober einen Satz von Klee und Strascewicz. Oper. Res. Verf. 19,
185-9.

472 References
Weyl, H. 1917, Uber die Starrheit der Eifl^chen und konvexen Polyeder. Sitzungsber.
Preufi. Akad. Wiss. Berlin, 250-66.
1939, On the volume of tubes. Amer. J. Math. 61, 461-72.
Wieacker, J. A. 1982, Translative stochastische Geometrie der konvexen KOrper.
Dissertation, Univ. Freiburg.
1984, Translative Poincar^ formulae for Hausdorff rectifiable sets. Geom. Dedicata 16,
231-48.
1986, Intersections of random hypersurfaces and visibility. Prob. Th. Rel. Fields 71,
405-33.
1988a, Helly-type decomposition theorems for convex sets. Arch. Math. 50, 59-67.
1988b, The convex hull of a typical convex set. Math. Ann. 2«2, 637-44.
1988c, Decomposition of closed convex sets. Geom. Dedicata 28, 221-8.
1989, Geometric inequalities for random surfaces. Math. Nachr. 142, 73-106.
Wiesler, H. 1964, On convexity for generalized barycentric coordinates (in Russian). Stud.
Cere. Mat. 15, 369-73.
Wijsman. R. A. 1964, Convergence of sequences of convex sets, cones and functions. Bull.
Amer. Math. Soc. 70, 186-8.
1966, Convergence of sequences of convex sets, cones and functions 11. Trans. Amer.
Math. Soc. 123, 32-45.
Wills, J. M. 1970, Zum Verhaltnis von Volumen zu Oberflache bei konvexen Korpern.
Arch. Math. 21, 557-60.
1973, Zur Gitterpunktanzahl konvexer Mengen. Elem. Math. 2S, 57-63.
1990a, Kugellagerungen und Konvexgeometrie. Jahresber. Deutsche Math.-Ver. 92,
21-46.
1990b, Minkowski's successive minima and the zeros of a convexity function. Monatsh.
Math. 109, 157-64.
Willson, S. J. 1980, A semigroup on the space of compact convex bodies. SIAM J. Math.
Anal. 11, 448-57.
Wintgen, P. 1982, Normal cycle and integral curvature for polyhedra in Riemannian
manifolds. In; Differential Geometry, Budapest 1979 (eds G. Soos & J. Szenthe),
Coll. Math. Soc. Jdnos Bolyai 31, North-Holland, Amsterdam, pp. 805-16.
Wintner, A. 1952, On parallel surfaces. Amer. J. Math. 74. 365-76.
Witsenhausen, H. S. 1973, Metric inequalities and the zonoid problem. Proc. Amer. Math.
Soc. 40, 517-20.
1978, A support characterization of zonotopes. Mathematika 25, 13-16.
Wu, H. 1974, The spherical images of convex hypersurfaces. J. Differential Geom. 9,
279-90.
Wunderlich, W. 1939, Uber eine Klasse zwangUufiger hoherer Elementenpaare. Z.
Angew. Math. Mech. 19, 177-81.
Yano, K. & Tani, M. 1969, Integral formulas for closed hypersurfaces. Kddai Math. Sem.
Rep. 21,335-49.
Yost, D. I99I Irreducible convex sets. Mathematika 3«, 134-55.
Zahle, M. 1982, Random processes of Hausdorff rectifiable sets. Math. Nachr. 108, 49-72.
1984a, Curvature measures and random sets, I. Math. Nachr. U9, 327-39.
1984b, Properties of signed curvature measures. In; Proc. Conf Stochastic Geom.,
Geom. Statist., Stereology, Oberwolfach 1983 (eds R. V. Ambartzumian & W. Weil),
Teubner, Leipzig, pp. 256-66.
1986a, Integral and current representation of Federer's curvature measures. Arch. Math.
46, 557-67.
1986b, Curvature measures and random sets 11. Proh. Th. Rel. Fields 71, 37-58.
1987a, Curvatures and currents for unions of sets with positive reach. Geom. Dedicata
23, 155-71.
1987b, Normal cycles and second order rectifiable sets. Forschungsergehnisse FSU
N/87/40.

References 473
1987c, Polyhedron theorems for non-smooth cell complexes. Maih. Nachr. 131, 299-310.
1988, Random cell complexes and generalized sets. Ann. Prob. 16, 1742-66.
1989, Absolute curvature measures. Math. Nachr. 140, 83-90.
1990, Approximation and characterization of generalized Lipschitz-Killing curvatures.
Ann. Global. Anal. Geom. 8, 249-60.
ZajiCek, L. 1979, On the differentiation of convex functions in finite and infinite
dimensional spaces. Czech. Math. J. 29, 340-8.
Zalgaller, V. A. 1967, Mixed volumes and the chance of a hit in a convex region under
multidimensional normal distribution (in Russian). Mat. Zametki 2, 97-104. English
translation: Math. Notes 2, 542-5.
1972, Uber Ar-dimensionale Richtungen, die fiir einen konvexen Ksrper F in R" singular
sind (in Russian). Zapiski nauin. Sent. Leningrad. Otd. mat. ln.it. Steklov 27, 67-72.
English translation: J. Soviet Math. 3, 437-41.
Zamfirescu, T. 1966a, Sur les corps associes h un corps convexe. Rev. Roum. Math. Pures
Appl. 11,727-35.
1966b, Sur les series lin^aires de corps convexes k frontifires non differentiables et
applications k\a reductibility. Rev. Roum. Math. Pures. Appl. U, IOI5-I022.
1966c, Reductibility et series lin^aires de corps convexes. L'Enseignement Math. 12,
57-67.
1967a, Sur la reductibility des corp.s convexes. Math. Z. 95, 20-33.
1967b, Sur quelques que.stions de continuity liees & la reductibility des corps convexes.
Rev. Roum. Math. Pures Appl. 12, 989-98.
1967c, Reducibility of convex bodies. Proc. London Math. Soc. 17, 653-68.
1967d, Conditions necessaires et suffisantes pour la reductibilite des voisinages des corps
convexes. Rev. Roum. Math. Pures Appl. 12, 1523-7.
1973, Two characterizations of the reducible convex bodies. Abh. Math. Sem. Univ.
Hamburg 39, 69-75.
1975, Metric space.s consisting of classes of convex bodies. Rend. 1st. Mat. Univ. Trieste
7, 128-36.
1980a, The curvature of most convex surfaces vanishes almost everywhere. Math. Z.
174, 135-9.
1980b, Nonexistence of curvature in mo.st points of most convex surfaces. Math. Ann.
252,217-19.
1982a, Most convex mirrors are magic. Topology 21, 65-9.
1982b, Many endpoints and few interior points of geodesies. Invent Math. 69, 253-7.
1984a, Intersecting diameters in convex bodies. Ann. Discrete Math. 20, 311-16.
1984b, Points on infinitely many normals to convex surfaces. J. reine angew. Math. 350,
183-7.
1985, Using Baire categories in geometry. Rend. Sem. Mat. Univers. Politecn. Torino
43, 67-88.
1987a, Nearly all convex bodies are smooth and strictly convex. Monatsh. Math. 103,
57-62.
1987b, Typical convex curves on convex surfaces. Monatsh. Math. 103, 241-7.
1988a, Curvature properties of typical convex surfaces. Pacific J. Math. 131, 191-207.
1988b, Too long shadow boundaries. Proc. Amer. Math. Soc. 103, 587-90.
1989+, Long geodesies on convex surfaces. Preprint.
1990, Nondifferentiability properties of the nearest point mapping. J. d'Analyse Math.
54, 90-8.
1991, On two conjectures of Franz Hering about convex surfaces. Discrete Comput.
Geom. 6, 171-80.
Zhang Gaoyong 1991, Restricted chord projection and affine inequalities. Geom. Dedicata
39, 213-22.
Zivaljeviii, R. T. 1989, Extremal Minkowski additive selections of compact convex sets.
Proc. Amer. Math. Soc. 105. 697-700.

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AUTHOR INDEX
Section references are given. Most of the references occur in the section
notes.
Ahrens. I.. .3.1
Aleksandrov, A. D., 15. 1.7, 2.2, 2.4. 2.5,
3.5. 4.2. 4.3, 4.6, 5.1-5.3. 6.2-6.6,
6.8,7.1, 7.2, Appendix
Alexander, R.. 3.5
Alfsen, E. M.. 1.4
Allendoerfer. C. B.. 4.2
Ambartzumian. R. V., 3.5
Anderson, R. D.. 2.2
Anikonov, Ju. E., Appendix
Arnold, R.. 5.4
Arrow, K. J., 3.1
Artstein. Z., 3.1
Ash, R. B..4.1
Asplund. E.. 1.5. 2.1
Assouad, P.. 3.5
Baddeley, A. J., 4.4
Baebler. F..6.2
Bair, J.. 1.3, 1.4. 3.1, 3.2
Ball. K, 7.4
Banchoff. T. F., 4.4
Bandle. C.,6.2
Bandt.Ch., 1.8
Bangert, V., 1.5
Bantegnie, R., 1.8
Baraki. G., 1.8
Bdr^nv, I.. 2.3. 2.6. 7.1
Barthel. W.,6.1.6.2
Batson, R. G., 1.4
Bauer. H.. 1.4,4.1
Baum, D., 1.7
Beer, G. A., 1.8
Bensimon, D.. 2.1
Benson. D. C. 6.2
Berg, Ch.. 3.3, 3.4. 4.3, Appendix
Bernstein. D. N., 6.3
Berwald. L., 4.5
Besicovitch. A. S., 2.2. 2.3
Betke, U.. 3.5. 4.2, 4.5, 5.3. 6.2, 7.4
Bellinger, W., 6.2
Bieberbach, L.. 6.2
Bing, R. H.,2.3
Bj6rck,G..2.1
Blackwell, D.. 3.5
Blaschke, W.. 1.8. 2.5. 3.2. 3.3, 3.5, 4.3.
4.5, 5.3, 5.4, 6.1, 6.2, 6.4. 7.1, 7.4.
Appendix
Blind, R..7.2
Blumenlhal, L. M., 1.8
Boiek, L.. 6.2
Bdhm, J., 3.4
Bokowski. J.,2.4. 4.5, 5.3.6.2
Bol. G.. 3.1, 6.2, 6.5-6.7, Appendix
Bolker. E. D., 3.5
Bolljanski. W. G.. 3.5.6.5
Bonnesen, T.. 1.7, 2.2. 3.1, 3.3, 4.3, 5.1,
5.3. 5.4.6.1,6.2,6.4,6.7,7.1.7.3
Bonnice. W.. 1.3
Boroczky, K., 7.1
Borwein, J. M.. 3.1
Bose, R. C. 5.4
Bolls. T., 1.2
Bourbaki, N., 1.3, 1.4,2.1,2.2
Bourgain. J., 3.3. 3.5. 6.1. 6.2. 7.2. 7.4,
Appendix
Brascamp, H. J., 6.1
Brehm. U.,4.4
Brondsled, A., 1.4, 2.2, 2.4, 3.4
Bronshlein. E. M.. 1.8, 2.1, 3.2. 7.1

Author index
479
Brooks, J. N., 3.2
Brown, A. L., 2.1
Brunn, H., 6.1
Budach, L., 4.4
Buldygin, V. V., 6.1
Bunt, L. N. H., 1.2
Burago, Ju. D., 1.2, 1.3, 6.1-6.3
Burckhardt, J. J., 3.5
Burton, G. R., 2.1, 3.2, 3.5, 4.5
Busemann, H.. 1.5. 2.5, 4.6, 6.1-6.3, 6.8,
7.1,7.2,7.4
Campi, S., 7.2, 7.4, Appendix
CaratWodory, C, 1.1
Cassels, J. W. S.,3.I
Chakerian, G. D,, 3.1, 3.5, 6.2, 6.5,
7.1-7.4
Cheeger, J., 4.4
Cheng, S.-Y., 7.1
Chern, S. S., 2.5, 4.5, 5.3, 7.2
Chernoff, P. R., Appendix
Choquet, G., 1.4,2.1, 3.5
Christoffel, E. B, 4.3
CieSlak, W., Appendix
Clack, R.,7.4
Coifman, R. R., Appendix
Collier, J. B.,2.1
Corson, H. H.,2.I
Coxeter. H. S. M., 3.5
Cressie, N., 3.1
Curtis, D. W., 1.8
Czipszer, J., 6.5
Dalla, L.,2.I
Danzer, L., 1.1, 1.3
Das Gupta, S, 6.1
Davis, C, 1.3
Davy, P. J., 4.5, 5.4
Debrunner, H., 5.3
Debs, G.,2.I
Deicke, A.. 7.4
Delgado, J. A., 3.2
Dierolf, P., 1.8
Dingha.s, A., 1.8, 3.1, 5.3, 6.1, 6.2, 6.5,
6.7, Appendix
Diskant, V. L. 6.1, 6.2, 6.4, 6.5, 6.7, 7.2
Dor, L. E.,3.5
DreSevic, M. 1.8
Dubins, L. E., 1.4
Dubois, C, 3.1
Dudley, R., 1.5. 1.8
Dupin, Ch., 7.4
Duporcq, E., 5.4
Dziechcirtska-Halamoda, Z., 3.1
Eckhoff, J., 1.1,3.4,4.5
Edelstein, M., 2.1, 3.1
Efimov, N. W., 6.6
Eggleston, H. G., 2.1, 6.2, 7.3
Egorychev. G. P., 6.8
Eifler, L. Q..2.1
Ekeland. I.. 3.1
Epelman, M. S., 2.2
Ewald, G., 1.7,2.3.3.1,3.2,6.6
Falconer, K. J., Appendix
Faro Rivas, R., 3.3
FSry, I.,5.I.6.5,7.1
Favard. J., 2.2, 4.3, 6.2, 6.4. 6.6. 6.7
Federer, H., 1.5, 2.2, 4.2, 4.4-4.6. 6.2
Fedotov, V. P.. 2.1, 2.2, 7.1
FejesT6th, L., 6.2
Feller, W., 1.5, 2.5
Fenchel, W., 1.7, 2.2, 3.1, 3.3, 4.2, 4.3,
5.1, 5.3, 5.4, 6.1-6.4, 6.6, 6.7, 7.1-7.3
Ferrers, N. M., 5.4
Fesmire, S., 3.1
Fet, A. I., 7.2
Figiel,T.,3.5
Filliman, P., 3.5
Fillmore, J. R., 4.3, Appendix
Firev, W. J., 1.7,2.1,2.5,3.2,3.3, 3.5,
4.3,4.5,4.6,5.1.5.3,6.1,6.2,6.8,
7.1,7.4
Fisher, J. C, Appendix
Flaherty, F. J., 4.4
Handers, H., 5.4, 6.2
Ratto, L., 2.3
Florian, A., 1.8, 2.4, 6.2
Focke, J., 3.5, Appendix
Folkman, J. H., 3.1
Fourneau, R., 1.3, 3.1, 3.2, 7.3
Franz, G, 6.1
Fu, J. H. G., 4.4, 4.5
Fuglede, B., 1.4, 6.2, Appendix
Fujiwara. M., 2.2, 3.5, Appendix
Gage, M. E., 6.2
Gale, D., 1.7, 3.2
Gallivan, S.,2.I
Gardner, R. J., 2.1
Geivaerts, M., 3.1, 3.2
Gensel, B., 3.5
Geppert, H., 6.7, Appendix
Gericke, H., 5.4, 6.4, Appendix
Giannopoulos, A. A., 7.4
Giertz, M., 7.4
Gin6, E., 1.8,3.1
Gluck, H.,7.1
Godbersen, C, 7.3
Godet-Thobie, Ch., 1.8
Goikhman, D. M., 7.1
Goldberg, M.. 3.5
Goodey, P. R., 3.2, 3.5, 4.2, 4.3, 4.5, 4.6,
5.1, 5.3, 6.6, 7.1, 7.2. 7.4, Appendix
Goodman, A. W., 5,4
Gordon, Y, 3.5
Gortler, H., 5.1, Appendix
G6idi, S., Appendix
Gray, A, 4.2

480
Author index
Green, J. W., 1.7
Grinberg, E. L., 6.8,7.4
Gritzman, P., 3.1, 3.5,6.4
Groemer, H., 3.1, 3.4, 3.5, 4.4, 4.5, 5.1,
5.2, 6.1, 6.2, 6.6, 7.2, 7.4, Appendix
Gromov, M., 6.1, 6.3
Gruber, P. M., 1.8, 2.4, 2.6, 3.2, 7.3
Griinbaum, B., 1.1, 1.3, 1.8, 2.1, 2.4,
3.1-3.4,7.1,7.3
GrzaSlewicz, R., 3.2
Guggenheimer, H., 6.1, 7.2, 7.4
Hadamard, J., 2.5
Hadwiger, H., 1.8, 3.1, 3.3-3.5, 4.2, 4.4,
4.5. 5.3, 5.4, 6.1-6.3, 6.5, 6.7, 6.8, 7.4
Hahn,F. H., 1.8,3.1
Hahn, M. G., 1.8, 3.1
Halmos, P. R., 3.5
Hammer, P. C, 3.1, 3.3
Haralick, R. M.,3.I
Hartmann, P., 2.5
Hausdorff, F., 1.8
Hayashi, T., 5.4, Appendix
Heijenoort, J. van, 2.5
Heil, E., 1.8,5.3,6.2,6.8,7.4
Heine, R., 6.3
Helly, E., 1.1
Henstock, R., 6.1
Herglotz, G.,4.2, 7.1
Hertel, E., 3.4
Hilbert, D.,6.1,6.3
Hildenbrand, W., 1.8, 3.1, 3.3, 3.5
Hille, E., 1.7
Hiriart-Urruty, J.-B., 1.5
Hirose, T., 1.8
Holmes, R. B., 1.3
Holmes, R. D., 6.2,7.4
Hormander, L., 1.8
Hom,B. K. P., 7.1
Howard, R., 3.5
Howe, R., 2.6
Huck, H., 5.3
Hurwitz, A., 4.3, Appendix
Husain, T., 3.1
Inzinger, R., Appendix
Ivanov, B. A., 2.3
Jacobs, K., 1.4
Jaglom, I. M., 3.5
lessen, B., 1.5, 1.7, 2.5, 4.2, 5.1, 6.4, 7.1,
7.2
Jongmans, F., 1.8, 2.2, 3.1, 3.2
Jourlin, M., 3.1
Kahn, J., 6.3
Kakeya, S., 3.5, Appendix
Kallay, M., 1.7,2.4,3.2,4,3
Kalman, J. A., 1.4
Kameneckii, I. M., 3.5, Appendix
Karcher, H., 3.2
Katsurada, Y., 5.3
Kellerer, A. M.,4.5
Kellerer, H. G., 4.5
Kendall, W. S., 3.1, 4.5
Kenderov, P., 2.6
Kharazishvili. A. B, 6.1
Khovanskii, A. G., 6.3, 6.8
Kiener, K., 7.4
Kinczes, J., 3.2
Kirchberger, P., 1.3
Kiselman, C. O., 2.5
Klee, V. L., l.I, 1.3, 1.4, 1.7, 2.1-2.3, 2.6
3.1,3.2,3.4,7.3
Klein, E., 1.8
Klima, V.,2.6
Klotzler, R., 3.5
Kneser, H., 1.7,4.2, 6.1,7.1
Knothe, H., 6.1, 6.2, 6.6, Appendix
Kohlmann, P., 4.4, 5.3, 7.2
Koutroufiotis, D., 3.2, 7.2
Krein, M., 1.4
Kruskal, J. B., 2.1
Kubota, T., 4.3, 5.4, 6.6, Appendix
Kuhnel, W.,4.4
Kuiper, N. H.,4.4
Kuratowski, K., 1.8
Kutateladze, S. S., 7.1
Lafontaine, J., 4.4
Lagarias, J. C, 6.8
Laget, B., 3.1
Larman, D. G., 2.1-2.3, 7.1, 7.4
Lashof, R. K., 2.5
Lebesgue, H., 3.5
Leichtweie, K., 1.7, 5.1, 6.1, 6.3, 6.6, 7.2,
7.4
Leindler, L., 6.1
Lenz, H., 3.4
Letac, G.,4.3
Lettl, G., 1.8
Uvy, P., 3.5
Lewis, J. E., 3.1
Uwy, H.,7.1
Liapounoff, A. A., 3.5
Lieb, E. H.,6.1
Lindelof, L., 6.2
Lindenstrauss, J., 3.3, 3.5, 6.1, 6.2, 7.2,
7.4, Appendix
Undquist, N. F., 3.5
Linhart, J., 3.5
Linke, Ju. E., 1.8
Little, J. J., 7.1
LjaiSenko, N. N., 3.1
Loritz, A., 6.6
Lusternik (Lyusternik), L. A., 6.1
Lutwak, E.,6.I, 6.8, 7.1,7.4
Macbeath, A. M., 1.8, 6.1, 7.4
Maehara, H.,3.1, 3.2

Author index
481
1,2.4, 2.5.
7.1.7.3,7.4
.1
Mahler, K.,7.4
Makai, E., 4.2, 7.1, 7.3
Mani-Levitska (Mani), P., 2.1, 2.4, 3.4, 4.5
Marti, J. T., 1.2, 1.3, 1.5,2.2
Martin, K., 2.1
Martini, H., 3.5,7.3,7.4
Mase, S., 3.1
Matheron, G., 1.8, 3.1, 3.2, 3.5, 4.2, 4.4,
4.5,5.1,5.3,6.5
Matsumura, S., 6.4, Appendix
Mazur, S., 2.2
McKinney, R. L., 1.3
McMinn, T. J., 2.3
McMullen, P., 1.2, 1.7, 1.8, 2
3.2, 3.4, 3.5,4.2, 4.5, 5.1
Mecke, J., 3.1,4.5
Meier, Ch., 5.1, 5.2, 5.4
Meissner, E., 3.5. 5.4, Appendix
Melzak, Z. A.. 2.5
Menger, K., 1.8
Meiichkowski, H., 3
Meyer, M., 2.4, 3.2, 3.4, 4.2, 7.4
Meyer, P., 4.2
Meyer, W., 3.2
Michael, E., 1.8
Miles, R. E.,4.5
Milka, A. D., 3.2
Milman, D., 1.4
Milman, V. D., 3.3, 3.5, 6.1, 7.4
Appendix
Minkowski, H., 1.4
6.1,6.2,6.6,7.1
Minoda, T., Appendix
Miranda, C, 7.1
Mityagin, B. S., 6.1
Molter, U. M., 4.5
Moore, J. D.,7.2
Motzkin, T. S., 1.2
MiJller, C., 4.4, 5.4, Appendix
MUller, H. R., 5.4
Muller, W.,4.4
MUmer, P., 3.5
2.2,3.1,3.3,5.1,5.4,
7.2, Appendix
N^deni'k,
Nadler, S
Nakajima
Netuka, I
Neveu, J.
Newman,
Neyman,
Nicliborc,
Z., 6.2, 7.1
. B., 1.8
, S.. 5.4, Appendix
.,2.6
,4.5
D. J.,2.3
A., 3.3
W.,5.4
Nirenberg, L., 7.1
O'Brien,
Oda, T., 1
Ohmann,
Ohshio, S
Oishi, K.,
Oliker, V
R. C, 3.1
6.2
D.,4.2, 6.1,
.,6.5
, Appendix
. I., 7.1, 7.2
6.2,6
Osserman. R., 6.2
Overhagen, T.. 5.3
Pach, J.,7.1
Pajor, A, 6.1, 7.4
Panina, G. Yu., 3.5
Panov, A. A., 6.8
Papaderou-Vogiatzaki, L,4.5
Papadopoulou, S., 2.1
Pays, R.. 3.2
Perles, M. A., 3.4,4.4
Petermann, E., 6.2
Petty, C. M., 3.5, 7.4, Appendix
Phelps, R. R., 1.4,3.3
Phillips. R. S., 1.7
Pisier, G.,6.I,7.4
Pogorelov, A. V., 4.3, 7.1, 7.2
P61ya, G.,6.2
Pontrjagin. L. S., 3.3
Positsel'skii, E. D., 3.4
Pranger, W., 1.4
Prekopa, A., 6.1
Przeslawski, K., 1.8, 3.4
Quinn, J., 1.8
Rademacher, H., 1.7
Radon, J.. 1.1,3.3
RIdstrom, H., 1.8
Ratschek, H.. 1.8
Rauch, J., 2.5, 3.2
Reay, J. R., 1.1, 1.3
Reidemeister, K., 2.2
Reisner, S., 7.4
Reiter, H. B, 2.1
R^nyi, A., 6.5
ReSetnjak, Ju. G.
Reuleaux, F., 3.5
Rham, G. de, 2.6
Ricker, W., 3.3
Rickert, N. W., 3.5
Riesz, F., 1.7
Rivin, I., 7.4
Roberts, A. W., 1.5
Rockafellar, R. T., I.I
Rogers, C. A., 1.8,2.1
7.4
Rota, G.-C, 3.4
Rother, W., 4.5
Roy, A. K., 1.4,3.1,5.4
Roy, N. M., 1.4
Roy, S. N., 5.4
Rubinov, A. M., 7.1
3.5
1.5, 7.2, Appendix
I.4-I.7
2.3,3.2,7.1,7.3,
Sacksteder. R., 2.5
Saint Pierre, J., 1.8, 3.3. 3.4
Saint Raymond, J., 7.4

482
Author index
Saks, M..6.3
Salinetti, G., 1.8
Sallee, G. T., 2.5,3.1, 3.2, 3.4,4.4
Sandgren. L., 1.7
Sangwine-Yager. J. R., 6.2, 6.5, 6.6
Santal6, L. A., 4.2, 4.5, 5.3, 6.2, 6.4,
Sas. E., Appendix
§a5kin, Ju. A., 1.4
Schaal, H., 3.5
Schmidt, E.,6.1
Schmidt, K. D., 1.8
Schmttt, K.-A.,3.1. 3.4
Schneider, R.. 1.7, 1.8. 2.1. 2.3-2.6,
3.1-3.5. 4.2-4.6, 5.1-5.4, 6.2, 6.3
6.5-6.8, 7.1-7.4. Appendix
Schort, R. M., 1.8
7.4
Schrader, R.,
Schroder, G..
4.4
1.8
Schiirger, K., 3.1
Schiitt, C, 7.4
Schwarz. T.,
2.6
Seeger, A., 1.5
Seeley, R. T.. 2.5,
Sen'kiti, E. P., 4.3
Serra. J., 3.1
Shahin, J. K.
Shapley. L. S
Shephard, G-
3.1-3.5,
. 5.3
..3.1
C.
4.4,:
, Appendix
1.2,
5.4,
1.3,
6.3,
1.8,
7.3,
2.4,
7.4,
Appendix
Silverman, R., 3.2
Simon. U., 5.3
Smilansky, Z.. 3.2
Soltan,P. S.,6.5
Sorger. H., 2.6
Sorokin, V. A., 3.1
Spiegel. W., 1.8, 3.4, 4.2, 5.1
Stacho, L. L. 4.4
Stanley, R. P., 6.3
Starr. R, 3.1
Stavrakas, N. M., 1.8,2.1
Steenaerts, P., 4.5
Steiner, J.,4.2, 5.4
Steinitz, E., 1.3
Stepanov, V. N.. Appendix
Sternberg, S. R., 3.1
Stoka, M. L.4.5
Stoker, J. J., 2.5,7.2
Stoyan, D., 3.1,4.5
Strantzen, J. B., 3.2
Straszewicz. S., 1.4
Streit. F.,4.5
Sturmfels, B., 3.1
Su, B., 5.4, Appendix
Sung. C.H., 2.2
Suss, W., 4.3, 5.3, 6.1, 6.2, 6.6, 7.1
Sz. Nagy (SzSkefalvi-Nagy), B. v., 4.2, 6.5
Sz. Nagy (Sz6ketalvi-Nagy), G. v., 5.4
Szego, G., 6.2
Szwiec, W., 3.1
Tam. B. S., 2.2
Tani, M., 5.3
Tanno, S., 3.3,7.4
Teissier, B., 6.2, 6.3
Temam, R.. 3.1
Tennison, R. L-. 3.2
The Lai, Ph., 1.8
Thomas, C, 3.5
Thompson, A. C. 1.8, 6.2, 7.4
Tichy. R., 1.8
Tolstonogov, A. A.. 1.8
Treibergs, A., 7.1. 7.2
Tverberg, H., I.I
Tweddle, I., 3.1
Uhrin, B., 6.1
Urbariski, R., 1.8
Valentine. F. A., 1.2
Valette.G., 1.7.3.1,3.4
Vanhecke, L.. 4.2
Varberg, D. E.. 1.5
Varga, 0.,4.5
Vidal Abaseal, E.. 4.2
Vincensini, P., 3.1
Vitale, R. A., 1.8,3.1,6.1,6.6
Vlasov, L. P., 1.2
Vodop'yanov. S. K., 7.2
Voiculescu, D., 3.1
Volkov, Ju. A.. 6.2, 7.2
Volland, W., 3.4
Wak.sman, Z., 2.2
Wallen, L. J., 6.2
Webster, R. J., 1.8
Wegmann. R., 3.1
Wegner, B., 3.3
Weil.W., 1.7,2.5. 3.1-3.3, 3.5,4.2,4.3,
4.5, 4.6.5.1-5.3,
Weiss, G., Appendix
Weiss, v., 4.4
Weissbach, B., 3.5
Werner, E.. 7.4
Wets, R. J.-B.. 1.4.
Weyl, H.,4.2, 6.6
Wieacker. J. A., 2.6, 3.2, 3.5, 4.5, 7.1, 7.4
1.4
A., 1.8
.1-3.3, 3.5,4.2,
6.2,7.1.7.4
1.8
Wiesler. H.,
Wijsman, R
Wills, J. M., 4.5, 5.3, 6.2, 6.4
Willson, S-. J., 6.5
Wintgen, P., 4.4
Wintner, A., 2.5
Witsenhausen, H. S., 3.5
Wrase, D., 6.4
Wu, H.. 2.5
Wunderlich, W., 3.5

Author index 483
Yano, K., 5.3 Zalgaller, V. A., 1.2, 1.3, 2.3, 2.5, 6.1-6.3
Yau, S.-T., 7.1 Zamfirescu, T., 2.6, 3.1
Yo,st, D., 3.2 Zhang. G.. 7.4
Zhuang, X.. 3.1
zahle, M., 4.2, 4.4.4.5 Zinn,J.,3.I
Zaji'cek, L., 2.2 2;ivaljevitS R. T., 1.8

SUBJECT INDEX
abstract
convex cone, 42
Steiner point. 178
adapted, one convex body to another, 374
additive function, 173
affine
combination, x
hull, X
isoperimetric inequality, 419
quermassintegral, 387
surface area, 419
affinely independent points, x
Aleksandrov-Fenchel inequality, 327
Aleksandrov-Fenchel-Jessen theorem,
400,407
approximable convex body, 162
area measure, 203
of order m. 207
mixed, 274
associated bodies, 140
asymmetry class, 156
barycentre function, 21
Baire space, 119
Bieberbach inequality, 318
Blaschke
diagram, 322
-Hausdorff metric, 58
-Santald inequality, 387, 421, 425
selection theorem, 50
sum, 394
symmetrization, 165
Blaschke's rolling theorem, 150
body of constant width, 128
Bonnesen-style inequalities, 323
Boolean model, 256
Brunn-Minkowski inequality, 309
extended, 383
Brunn-Minkowski theorem, 309
general, 339
Busemann
intersection inequality, 417
-Petty centroid inequality, 418
Busemann's theorem, 316
canal class, 377
cap, 81
cap-body, 76
cap-covering theorem, 81, 84
Carathtodory's theorem, 3
Cauchy's surface area formula, 295
cc-hyperspace, 57
centred convex body, 383
centroid body, 417
Chebyshev
set, U, 67
subspace, 94
Choquet theory, 20
Christoffel's problem, 216, 218
circumball, 129
circumradius, 129
class
asymmetry, 156
canal, 377
universal approximating, 162
closed
convex function, 35
convex hull, 6
halfspace, xi
segment, x
closing of set A by set B, 138
combination, I
affine, x
convex, 1
linear, x
positive, 2
concave function, 21
conditionally positive definite function, 194
cone, 80
convex, I
abstract, 42

Subject index
485
normal, 70
projection, 70
recession, 17
touching, 74
conjugate function, 36
contact measure, 260
convex body,8
adapted, 347
approximable by a class, 162
centred, 383
decomposable, 150
determined by closed set, 321, 343
equivalent by dilatation, 383
freely rolling, 150
freely sliding, 143
indecomposable, 150, 157, 172
irreducible, 141, 156
Ar-curved, 121
normalized, 150
reducible, 141, 156
stable, 67
strictly convex, 77
typical. 119
convex cone, 1
convex function, 21
convex hull, 2
convex ring, 175
convex set, 1
continuous, 46
convexity number, 225
critical set, 141
Crofton's intersection formula, 235
curvature, 104
Gauss-Kronecker, 106
integral of mean, 216
lessen radius of, 117
lower, 104
lower radius of, 104
mean, 106
principal, 105
principal radii of, 108
specific, 269
upper, 104
upper radius of, 104
curvature centroid, 305
curvature function, 419
curvature image, 420
curvature measure, 203
generalized, 202
curved convex body, 121
decomposable convex body, 150
defined locally, measure, 206
determined by closed set, convex body.
321, 343
diameter, xi
difference body, 127, 409
difference body inequality, 409
dilatation, xii, 138
dimension, 7
direct
sum, xi
summand, 142
directly indecomposable, 142
distance, xi
dual
affine quermassintegral, 387
cone, 34
mixed volume, 385
quermassintegral, 386
Dupin indicatrix, 116
edge, 96
effective domain, 22
endomorphism, 170
envelope, 67
epigraph, 22
equivalent by dilatation, convex bodies, 383
equivalent by telescoping, convex bodies,
380
equivariant map, 166
equivariant under rigid motions, Steiner
point. 43
erosion, 138
Euler
characteristic, 175
point, 116
relation, 179
Euler-type relation, 180
Euler's theorem, 106
even measure, 183
exponent of entropy, 60
exposed, 65
face, 63, 65
normal vector, 74
point, 19, 65
r-skeleton, 65
support plane, 74
extended
Brunn-Minkowski inequality, 383
convex ring, 256
exterior normal vector, 11, 70
external angle, 100
extreme
normal vector, 74, 76
point, 18, 65
ray, 18
support plane, 74, 76
face, 18, 62
exposed, 63, 65
isolated, 79
perfect, 73
proper, 62
strongly connected family of faces, 1.52
face-function, 66
facet, 62, 96
first
category, 119
fundamental form, 105

486
Subject index
nat.xi, 121
form body, 321
freely rolling convex body, 150
freely sliding convex body, 143
full set of inequalities, 333
fully additive function, 174
function
additive, 173
barycentre. 21
closed convex, 35
concave, 21
conditionally positive definite, 194
conjugate, 36
convex, 21
curvature. 419
fully additive, 174
gauge, 43
indicator, 22
Minkowski additive, 41
mixed curvature, 115
positive definite, 194
positively homogeneous, 26
proper, 21
radial, 44, 416
rigid motion invariant, 42
semiaxis, 46
subadditive, 26
sublinear, 26
support, 37, 176
Funk-Hecke theorem, 431
gauge function, 43
Gauss
map, 104
Kronecker curvature, 106
general
Brunn-Minkowski theorem, 339
relative position, 239
generalized
curvature measure, 202
normal bundle, 198
zonoid, 187
generating measure. 184
generic property. 119
geominimal surface area, 425
grain distribution, 256
//•convex set, 350
half-flat, xi
half-open segment, xi
Hammer body, 178
harmonic
isepiphanic inequality, 387
quermassintegral, 386
Urysohn inequality, 387
Hausdorff
di.stance, 48
metric, 48
Helly's theorem, 4
homothet, xii
homothetic sets, xii
homothety, xii
homothety invariant M-class, 164
hull
affine, x
convex. 2
linear, x
positive, 2
hypermetric vector space, 194
hyperplane, xi
improper rigid motion, xii
inclusion-exclusion principle, 174
indecomposable
convex body, 150
endomorphism, 172
set, 157
index, 219, 223
indicator function, 22
inequality
Aleksandrov-Fenchel, 327
Bieberbach, 318
Blaschke-Santal6, 387, 425
Brunn-Minkowski, 309
extended, 383
Busemann, 417
Busemann-Petty, 418
difference body, 409
harmonic isepiphanic, 387
harmonic Urysohn, 387
isodiametric, 318
isoperimetric, 318, 325
affine, 419
Jensen, 22
Minkowski, 317, 321
Petty projection, 387, 416
Urysohn, 318
infimal convolution, 37
inner
parallel body, 1.34, 350
radias, 130
inradius, 135
relative, 135
integral geometry, 253
integral of mean curvature, 210
intensity, 256
internal point, 7
intersection body, 416
intersectional family of sets, 173
intrinsic
volume, 210
moment, 304
irreducible convex body
Hammer's definition, 141
Shephard's definition, 156
isodiametric inequality, 318
isolated face, 79
isoperimetric
inequality. 318, 325
affine, 419

Subject index
487
ratio. 321
isoperimetrix, 416
isotropic random closed set, 256
Jensen's inequality, 22
lessen radius of curvature, 117
A--curved convex body, 121
(Ki A^„_i)-extreme vector, 76
Kirchberger's theorem, 15
Krein-Milman theorem, 20. 162
Kubota's integral recursion, 295
Lp metric. 59
length measure, 215
limit section, 118
line, xi
line-free set, 17
linear
combination, x
extension, 181
hull, X
linearity direction, 26
local parallel set, 198
locally defined measure, 206
locally similar (strongly isomorphic)
polytope, 101
log-concave
measure, 315
sequence, 334
lower
curvature, 104
radius of curvature, 104
M-class, 164, 196
Macbeath region, 81
map
Gauss, 104
Minkow.ski additive, 166
radial, 79
spherical image, 78, 104
reverse, 78. 107
Weingarten, 105
reverse, 107
meagre subset. 119
mean
curvature. 106
width. 42
measure, 183
contact, 260
curvature, 203
generalized, 202
even, 183
length. 215
locally defined, 206
log-concave, 315
of symmetry, Minkowski. 141
signed, 183
metric
entropy, 60
projection. 9
minimal width. 42
Minkowski
addition, 41
additive
function. 41
map, 166
difference, 133
functional, 209
linear map, 166
subtraction, 126, 133
symmetrization, 165
Minkowskian integral formulae, 291, 300
Minkowski's
existence theorem, 390
inequalities, 317, 321
measure of symmetry, 141
theorem, 19
mixed
affine surface area, 424
area, 321
area measure, 274
body, 396
brightness. 422
curvature function, 115
discriminant, 115, 383,388
moment vector, 303
projection body, 421
volume, 272
width integral, 385
moment vector, 303
monotypic polytope, 103, 153
■most' convex bodies, 119
nearest-point map. 9
'nearly all' elements of Baire .space, 124
negative type, 194
normal cone, 70
normal point. 116
normal vector, xi, 70, 96
exposed, 74
exterior, II. 70
extreme, 74, 76
outer, 11,70,96
outward, 70
r-exposed, 75
r-extreme, 74
regular, 77
singular, 77
normalized convex body, 150
opening of set A by set B. 138
outer
normal vector, II, 70. 96
parallel body, 134, 197
outward normal vector. 70
p-quermassintegral, 384
p-sum, 383

488
Subject index
pair of constant width, 139
parallel body, 134
relative, 134
perfect face, 73
Petty projection inequality, 387, 416
point
affinely independent, x
Euler. 116
exposed. 19, 65
extreme. 18, 65
internal, 7
normal, 116
r-expo.sed, 65
r-singular, 73
regular. 73
Samalo. 419
singular. 73
smooth. 73, 153
polar
body, 33
projection body, 414
polytope, 3. 49
locally similar. 101
monotypic, 103, 153
simple, 99
simplicial, 99
space-filling, 192
strongly combinatorially equivalent, 101
strongly isomorphic. 101
porous set. 124
positive
basis, 17
combination, 2
definite function, 194
hull, 2
reach. 212
positively homogeneous function, 26
positively homothetic set. xii
principal
curvatures, 105
kinematic formula, 229
radii of Curvature, 108
projection
body, 296, 414
cone,70
prolongation, 138
proper
face, 62
function, 21
rigid motion, xii
separation, 12
r-exposed
normal vector, 75
point, 65
support plane, 75
r-extreme
normal vector, 74
point, 65
support plane, 74
r-singular point, 73
r-skeleton, 65
exposed, 65
radial
function, 44, 416
map, 79
radius of curvature, 104
lessen, 117
lower, 104
principal, 108
tangential, 119
upper, 104
Radon's theorem, 4
random closed set, 255
ray, xi
extreme, 18
recession cone, 17
reducible convex body, 141, 156
regular
normal vector, 77
point, 73
support plane, 77
relative
boundary, xi
indecomposability, 157
inradius, 135
interior, xi
parallel body, 134
relatively indecomposable set, 157
residual set, 119
reverse
second fundamental form. 107
spherical image, 78
spherical image map, 78, 107
Weingarten map, 107
rigid motion, xii
improper, xii
proper, xi
rigid motion invariant function, 42
rolling theorem, 155
rotation, xii
rotation mean, 161
rotor, 190, 196
quermassintegral, 209
affine, 387
dual, 386
dual affine, 387
harmonic, 386
P-, 384
quermassvector. 304
Santald point, 419
second
category, 119
fundamental form, 105
semiaxis function, 46
separation, 12
proper, 12

Subject index
489
set
isomorphic polytopes, 100
convex, I
continuous, 46
critical, 141
//-convex, 350
homothetic, xii
indecomposable convex, 157
indecomposable non-convex, 157
line-free, 17
local parallel, 198
porous, 124
positively homothetic, xii
random closed, 255
isotropic, 256
standard, 256
stationary, 256
relatively indecomposable, 157
residual
strictly separated sets, 12
strongly separated sets, 12
sublevel, 22
support, 37
Shapley-Folkman theorem, 128
Shapley-Folkman-Starr theorem, 130, 139
signed measure, 183
similarity, xii
simple n-polytope, 99
simplex, 3
simplicial polytope, 99
singular
normal vector, 77
point, 73
relative position, 242, 250
skeleton, 65
smooth point, 73, 153
space-filling polytope, 192
special position, 227, 231
specific curvature, 269
spherical
harmonic, 428
image, 77
image map, 78, 104
stability
estimate, 314, 325
problem, 322
stable convex body, 67
standard random closed set, 256
star body, 416
stationary random closed set, 256
Steiner ball, 353
Steiner compact, 195
Steiner formula, 197, 210
Steiner point, 42, 306
abstract, 178
Steinitz's theorem, 15
Straszewicz's theorem, 20
strictly separated sets, 12
strongly
combinatorially equivalent polytopes, 101
connected family of faces, 152
separated sets, 12
subadditive function, 26
subdifferential, 30
subgradient, 30
subgradient choice, 32
sublevel set, 22
sublinear function, 26
summand, 134, 142
support, 11, 261
support cone, 70
support element, 46, 198
support function, 37, 176
support number, 271
support plane, 11, 37
exposed, 74
extreme, 74, 76
r-exposed, 75
r-extreme, 74
regular, 77
support set, 37
support vector, 288
supporting halfspace, 11, 37
surface area
affine, 419
mixed, 424
geominimal, 425
surface-area centroid, 305
symmetric difference metric, 58
tangent space, 103
tangential
body, 75, 76
radius of curvature, 119
telescoping, 380
touching cone, 74
translate, xii
translation, xii
translation vector, xii
theorem of
Aleksandrov-Fenchel-Jessen, 400,
Brunn-Minkowski, 309
Busemann, 316
Carath^odory, 3
Euler, 106
Funk-Hecke, 431
Helly, 4
Kirchberger, 15
Krein-Milman, 20, 162
Minkowski, 19
Radon,4
Shapley-Folkman-Starr, 130, 139
Steinitz, 15
Straszewicz. 20
Tverberg, 8
typical convex body, 119
unit
ball, xii
sphere, xii
407

490
Subject index
universal approximating class, 162
upper
curvature, 104
radius of curvature, 104
Urysohn inequality, 318
valuation, 173
weak, 179
vertical halfspace, 35
vertex. 3, 96
volume
functional, 55
product, 421
Weingarten map, 105
width, 42
Wirtinger's lemma, 352
Wulff shape, 350
zonoid, 182, 183
generalized, 187
zonotope, 182