Convex Geometry: Cetraro, Italy 2021 (Lecture Notes in Mathematics, 2332) [1st ed. 2023] 3031378822, 9783031378829

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Lecture Notes in Mathematics 2332 CIME Foundation Subseries

Shiri Artstein-Avidan Gabriele Bianchi · Andrea Colesanti Paolo Gronchi · Daniel Hug Monika Ludwig · Fabian Mussnig

Convex Geometry Cetraro, Italy 2021

Andrea Colesanti · Monika Ludwig Editors

Lecture Notes in Mathematics

C.I.M.E. Foundation Subseries Volume 2332

Editors-in-Chief Jean-Michel Morel, Ecole Normale Supérieure Paris-Saclay, Paris, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi Germany

, Institute of Mathematics, Leipzig University, Leipzig,

Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

2023 Paolo Salani

[email protected]

Daniele Angella

[email protected]

Shiri Artstein-Avidan • Gabriele Bianchi • Andrea Colesanti • Paolo Gronchi • Daniel Hug • Monika Ludwig • Fabian Mussnig

Convex Geometry Cetraro, Italy 2021 Andrea Colesanti • Monika Ludwig Editors

Authors Shiri Artstein-Avidan School of Mathematical Sciences Tel Aviv University Tel Aviv, Israel

Gabriele Bianchi Dipartimento di Matematica e Informatica “Ulisse Dini” University of Florence Firenze, Italy

Andrea Colesanti Dipartimento di Matematica e Informatica “Ulisse Dini” University of Florence Firenze, Italy

Paolo Gronchi Dipartimento di Matematica e Informatica “Ulisse Dini” University of Florence Firenze, Italy

Daniel Hug Institut für Stochastik Karlsruher Institut für Technologie (KIT) Karlsruhe, Germany

Monika Ludwig Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wien, Austria

Fabian Mussnig Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wien, Austria Editors Andrea Colesanti Dipartimento di Matematica e Informatica “Ulisse Dini” University of Florence Firenze, Italy

Monika Ludwig Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wien, Austria

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics C.I.M.E. Foundation Subseries ISBN 978-3-031-37882-9 ISBN 978-3-031-37883-6 (eBook) https://doi.org/10.1007/978-3-031-37883-6 Mathematics Subject Classification: 52-02, 49Q22, 52A20, 52A40, 52A41, 52B45 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This volume collects the lecture notes of the Summer School on Convex Geometry, held in Cetraro, Italy, from August 30th to September 3rd, 2021. Convex geometry is a very active area in mathematics with a solid tradition and a promising future. Its main objects of study are convex bodies, that is, compact and convex subsets of n-dimensional Euclidean space. The so-called Brunn–Minkowski theory currently represents the central part of convex geometry. The Summer School aimed to provide an introduction to various aspects of convex geometry: The theory of valuations, including its recent developments concerning valuations on function spaces; geometric and analytic inequalities, including those which come from the .Lp Brunn–Minkowski theory; geometric and analytic notions of duality, along with their interplay with mass transportation and concentration phenomena; symmetrizations, which provide one of the main tools to many variational problems (not only in convex geometry). Each of these parts is represented by one of the courses given during the Summer School and corresponds to one of the chapters of the present volume. The initial chapter contains some basic notions in convex geometry, which form a common background for the subsequent chapters. The material of this book is essentially self-contained and, like the Summer School, is addressed to PhD and post-doctoral students and to all researchers approaching convex geometry for the first time. We are deeply grateful to Fondazione CIME for giving us the opportunity to carry out the Summer School and providing constant support for its organization. Firenze, Italy Wien, Austria January 2023

Andrea Colesanti Monika Ludwig

v

Contents

1

Notation and Introductory Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrea Colesanti

1

2

Valuations on Convex Bodies and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monika Ludwig and Fabian Mussnig

19

3

Geometric and Functional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrea Colesanti and Daniel Hug

79

4

Dualities, Measure Concentration and Transportation . . . . . . . . . . . . . . . . . . 159 Shiri Artstein-Avidan

5

Symmetrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Gabriele Bianchi and Paolo Gronchi

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

vii

Contributors

Shiri Artstein-Avidan School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel Gabriele Bianchi Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy Andrea Colesanti Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy Paolo Gronchi Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy Daniel Hug Karlsruher Institut für Technologie (KIT), Institut für Stochastik, Karlsruhe, Germany Monika Ludwig Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wien, Austria Fabian Mussnig Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wien, Austria

ix

Chapter 1

Notation and Introductory Material Andrea Colesanti

Abstract This is a synthetic presentation of basic notions in convex geometry, with corresponding elementary properties, such as: Minkowski addition; Hausdorff distance; support function; the polar body and the gauge function; the Steiner formula; intrinsic and mixed volumes; surface area measure.

1.1 Introduction This short note collects the preliminary material for the lecture notes of the courses of the C.I.M.E. Summer School entitled “Convex Geometry”, held in Cetraro, Italy, in the summer of 2021. It contains basic notions in convex geometry, such as: the definition of a convex body; the Minkowski addition; Hausdorff distance; support function; polar body, gauge function; the Steiner formula; intrinsic and mixed volumes; the surface area measure. These notions are supplied by examples and basic properties. In general we do not include proofs. The most elementary statements are left as exercises; for the rest we refer to [2] and [1], where the reader can find exhaustive presentations of the material of this note. The author is grateful to the other lecturers of the summer school, Shiri Artstein, Gabriele Bianchi, Paolo Gronchi, Daniel Hug, Monika Ludwig and Fabian Mussnig, for their advice, which contributed to improve this chapter.

A. Colesanti (O) Dipartimento di Matematica e Informatica “U. Dini”, University of Florence, Florence, Italy e-mail: andrea.colesanti@unifi.it © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6_1

1

2

A. Colesanti

1.2 Notation We work in the n-dimensional Euclidean space .Rn , with .n ≥ 1. The standard Euclidean norm and scalar product will be denoted by .| · | and ., respectively. We set Sn−1 = {x ∈ Rn : |x| = 1},

.

and B n = {x ∈ Rn : |x| ≤ 1}.

.

For .k ∈ [0, n], .Hk denotes the k-dimensional Hausdorff measure in .Rn . In particular .Hn is the Lebesgue measure in .Rn . For brevity, integration with respect to the Lebesgue measure will be often denoted by “.dx”. We set κn = Hn (B n ).

.

For reasons that will be clear once that the notion of intrinsic volumes will be introduced, the Lebesgue measure in .Rn will be also denoted by .Vn (and referred to as the volume).

1.3 Convex Bodies A convex body is a non-empty compact and convex subset of .Rn ; the family of convex bodies will be denoted by .Kn : Kn = {K ⊂ Rn : Kis compact and convex, K /= ∅}.

.

Examples . The unit ball .B n . . Let .p ≥ 1; the set Bpn = {x = (x1 , . . . , xn ) : |x1 |p + · · · + |xn |p ≤ 1}

.

is the so-called .Lp -ball, and it is a convex body. The standard Euclidean ball is retrieved for .p = 2. . The unit cube: [0, 1]n = {x = (x1 , . . . , xn ) ∈ Rn : 0 ≤ x1 ≤ 1, . . . , 0 ≤ xn ≤ 1}.

.

1 Notation and Introductory Material

3

Given .A ⊂ Rn , the convex hull of A, denoted by .conv A, is the set formed by the points of the form x=

m W

.

λi xi ,

i=1

where .m ∈ N, .x1 , . . . , xm ∈ A, and (λ1 , . . . , λn ) ∈ [0, 1]n

.

with

m W

λi = 1.

i=1

The convex hull of finitely many points in .Rn belongs to .Kn , and is called a polytope. The family of polytopes will be denoted by .Pn . The unit cube is an example of polytope, as it is the convex hull of the points .x = (x1 , . . . , xn ), where each component .xi is either 0 or 1. We will need the definition of the dimension of a convex body. Definition 1.1 Let .K ∈ Kn ; we say that the dimension of K is j (.j ∈ {0, . . . , n}), if K is contained in an affine subspace of .Rn of dimension j , but it is not contained in any affine subspace of dimension .(j − 1). In this case we write .dim(K) = j . We can clearly construct examples of convex bodies in .Rn of any dimension .j ∈ {0, . . . , n}. In particular, singletons have dimension zero.

1.4 Supporting Hyperplanes and the Metric Projection The aim of this part is to state some results concerning the existence of supporting hyperplanes to a convex bodies. This question is connected to the notion of metric projection. Let .K ∈ Kn and let H be a hyperplane of .Rn . H is called a supporting hyperplane to K if .K ∩H /= ∅, and K is contained in one of the two closed half-spaces bounded by H . More precisely, let .u ∈ Sn−1 , and let H be a hyperplane orthogonal to u. Then H can be written in the form H = {x ∈ Rn : = α}

.

for some .α ∈ R. Given a convex body K, we say that H is a supporting hyperplane to K with outer normal u, if .H ∩ K /= ∅ and K ⊂ H − := {x ∈ Rn : ≤ α}.

.

If .x ∈ H ∩ K, we say that u is an outer normal vector to K at x.

4

A. Colesanti

We recall that given a (non-empty) subset A of .Rn and .x0 ∈ Rn , the distance of .x0 from A is defined as dist(x0 , A) = inf |x0 − a|.

.

a∈A

Exercise 1.2 Let .K ∈ Kn . For every .x ∈ Rn there exists a unique point .pK (x) ∈ ∂K, such that .

dist(x, K) = |x − pK (x)|.

The function .pK : Rn → ∂K is called the metric projection. Exercise 1.3 Let .K ∈ Kn and let .x ∈ Rn \ K. Then the hyperplane passing through .pK (x) and orthogonal to .x − pK (x) is a supporting hyperplane for K. Moreover, u=

.

x − pK (x) |x − pK (x)|

is an outer normal vector to K at .pK (x). Exercise 1.4 Let .K ∈ Kn . Prove the following contraction property of the metric projection .pK of K: |pK (x) − pK (y)| ≤ |x − y|

.

for every .x, y ∈ Rn . Proposition 1.5 Let .K ∈ Kn . For every .x ∈ ∂K there exists .y ∈ Rn \ K such that .x = pK (y). In particular, for every .x ∈ ∂K there exists a supporting hyperplane to K, passing through x. Note that the supporting hyperplane through the boundary point of a convex body may not be unique. The previous result is complemented by the following one. Proposition 1.6 Let .K ∈ Kn ; for every unit vector .u ∈ Sn−1 there exists a (unique) supporting hyperplane H to K, with outer normal u. Based on this result, we may give the following definition. Definition 1.7 Let .K ∈ Kn and .u ∈ Sn−1 . We denote by .K(u) the subset of those points .x ∈ ∂K such that u is an outer normal vector to K at x. Exercise 1.8 Prove that, for every .K ∈ Kn and for every .u ∈ Sn−1 , .K(u) is a non-empty closed convex set (a convex body), with .dim(K(u)) ≤ n − 1. Let .K ∈ Kn and .x ∈ ∂K. We say that x is a regular point if there is only one outer normal vector to K at x.

1 Notation and Introductory Material

5

Proposition 1.9 Let .K ∈ Kn ; then .∂K is .Hn−1 -measurable, and .Hn−1 (∂K) < ∞. Moreover .Hn−1 -a.e. point of .∂K is regular.

1.5 The Minkowski Addition Minkowski, or vector, addition is one of the fundamental constructions in convex geometry, and it is essential for most of the notions that will be introduced in these notes. Let .K1 and .K2 be convex bodies. Their Minkowski or vector sum is defined as K1 + K2 = {x + y : x ∈ K1 , y ∈ K2 }.

.

The multiplication of a convex body K by a number .t ∈ R is t K = {tx : x ∈ K}.

.

We observe that multiplication by non-negative numbers is the usual dilation. The Minkowski sum of .K1 and .K2 can be obtained as the union of translates of one of the bodies by vectors from the other one: K1 + K2 =

||

.

||

(x + K2 ) =

x∈K1

(y + K1 ).

y∈K2

Note that .x + K2 is just the translation of .K2 by x. Hence, to have a picture of K1 + K2 we have to draw a copy of .K2 translated by any point x of .K1 and take the union over x (or vice versa). This may help to figure out how the sum of two convex bodies looks like.

.

Exercise 1.10 Based on the previous definitions: . draw a picture of the sum of a circular disk and a square in the plane; . draw a picture of the sum of an equilateral triangle T and the rotation of T of an angle .π ; . determine the sum of two orthogonal segments in the plane, or of three pairwise orthogonal segments in .R3 ; . determine the sum of a two-dimensional circular disk in .R3 , and a segment orthogonal to it. Here is an important property of addition of convex bodies; the proof is left as an exercise. Proposition 1.11 For convex bodies .K1 , K2 ∈ Kn , and .t1 , t2 ≥ 0, t 1 K1 + t 2 K2

.

is again a convex body in .Rn .

6

A. Colesanti

Proposition 1.11 says that addition and multiplication by non-negative numbers are internal operations of the family of convex bodies in .Rn . In other words, .Kn is a convex cone of sets, with respect to the operations the we have introduced so far. Let K be a convex body in .Rn , and let .ε ≥ 0. The set Kε = K + εB n

.

is called the parallel set of K with distance (parameter) .ε. Exercise 1.12 Prove that for .K ∈ Kn and for .ε ≥ 0, Kε = {x ∈ Rn : dist(x, K) ≤ ε}.

.

1.6 The Hausdorff Distance We have seen in the previous section that .Kn is endowed with two internal operations. We will now equip .Kn with a metric, namely the one induced by the Hausdorff distance. Let .K1 and .K2 be convex bodies. Their Hausdorff distance is: { } δ(K1 , K2 ) = max max dist(x, K2 ), max dist(y, K1 ) .

.

x∈K1

y∈K2

Exercise 1.13 Prove that .δ is a distance. Exercise 1.14 Prove the following characterization of the Hausdorff distance, based on the notions of Minkowski addition and of parallel sets. For .K1 , K2 ∈ Kn : δ(K1 , K2 ) = min{r ≥ 0 : K1 ⊂ K2 + rB n and K2 ⊂ K1 + rB n }.

.

Exercise 1.15 Let .Ki , .i ∈ N, be a sequence of elements of .Kn , and let .K ∈ Kn . Prove that .

lim δ(Ki , K) = 0

i→∞

if and only if the following two conditions hold simultaneously. 1. Each point .x ∈ K is the limit of a sequence .xi , .i ∈ N, with .xi ∈ Ki for every i. 2. If .xi , .i ∈ N, is such that .xi ∈ Ki for every i, then the limit of any convergent subsequence of .xi belongs to K. Exercise 1.16 Compute the distance between a square of side length 2 and a disk of radius 1, both centered at the origin, in .R2 .

1 Notation and Introductory Material

7

With the metric induced by this distance, the space of convex bodies is a complete metric space. Besides completeness, .Kn has the following important compactness property. Theorem 1.17 (Blaschke Selection Theorem) Let .Ki , .i ∈ N, be a bounded sequence of elements of .Kn , i.e. there exists .r > 0 such that, for every i, Ki ⊂ rB n .

.

Then the sequence .Ki admits a subsequence converging to an element of .Kn .

1.6.1 Two Important Dense Subclasses of Kn Every convex body can be approximated by a sequence of convex polytopes. In other words, convex polytopes are dense in the family of convex bodies. Proposition 1.18 For every convex body .K ∈ Kn , there exists a sequence of polytopes .Pj , .j ∈ N, such that .

lim δ(Pj , K) = 0.

j →∞

There is another important dense sub-family of convex bodies. .K ∈ Kn is said to be of class .C 2,+ , if: 1. the boundary .∂K of K is of class .C 2 ; 2. for every .x ∈ ∂K, the Gauss curvature of .∂K at x is strictly positive. Proposition 1.19 For every convex body .K ∈ Kn , there exists a sequence of convex bodies .Kj , .j ∈ N, such that .Kj is of class .C 2,+ for every j , and .

lim δ(Kj , K) = 0.

j →∞

1.7 The Support Function We present here a powerful tool in the study of convex bodies, the support function. Definition 1.20 Let .K ∈ Kn ; the support function of K, .hK : Rn → R, is defined as hK (u) = sup ,

.

x∈K

u ∈ Rn .

8

A. Colesanti

The support function of a convex body K may be also indicated by .h(K, ·). The geometric meaning of the support function can be explained as follows. Let K be a convex body in .Rn and assume that .o ∈ K. Fix a unit vector u and let .Hu be the unique supporting hyperplane to K perpendicular to u and such that u is the outer unit normal. The support function of K evaluated at u is the distance of .Hu from the origin. The support function is then extended from .Sn−1 to .Rn as a 1-homogeneous function. Exercise 1.21 Determine the support function of the following convex bodies. . The ball centered at the origin with radius .r ≥ 0. . A segment. . The rectangle: K = {(x, y) ∈ R2 : |x| ≤ 1, |y| ≤ 2}.

.

1.7.1 Properties of Support Functions Let .K ∈ Kn ; its support function .hK , being the supremum of linear (and then additive) functions, is sub-additive: hK (u + v) ≤ hK (u) + hK (v).

.

Moreover, .hK is 1-homogeneous: if .x ∈ Rn and .λ ≥ 0, then hK (λx) = λhK (x).

.

(We remark that, here and throughout, by homogeneous we mean in fact positively homogeneous). Hence .hK turns to be a convex function. We have thus proved the following statement. Proposition 1.22 For every .K ∈ Kn , .hK is a 1-homogeneous convex function. Let .h : Rn → R be a 1-homogeneous convex function. Consider the set K = {x ∈ Rn : ≤ h(u) for every u ∈ Sn−1 } =

.

n

{x ∈ Rn : ≤ h(u)}.

u∈Sn−1

Exercise 1.23 Prove that K is non-empty. As an intersection of closed half-spaces, K is a closed convex set. Moreover, as h is continuous (by convexity), it is bounded on .Sn−1 . This implies in particular that K is bounded, and then .K ∈ Kn .

1 Notation and Introductory Material

9

Exercise 1.24 In the above notation, prove that hK = h.

.

We then see that to each convex body we can associate a unique 1-homogeneous convex function, its support function; vice versa to each 1-homogeneous convex function h we can associate a convex body having h as support function. {convex bodies} ←→ {support functions} = {1-hom. convex functions}.

.

This one-to-one correspondence is an isometry if the space of support functions, restricted to .Sn−1 , is endowed with the .L∞ distance. Proposition 1.25 For .K, L ∈ Kn δ(K, L) = ||hK − hL ||L∞ (Sn−1 ) .

.

Exercise 1.26 Prove the previous proposition, using Exercise 1.14. We conclude this part with two further properties of support functions, contained in the next exercise. Exercise 1.27 Prove the following statements. . For .K, L ∈ Kn , K⊂L

.

if and only if hK ≤ hL .

. For .K ∈ Kn o∈K

.

if and only if

hK ≥ 0.

1.7.2 Linearity of the Support Function with Respect to the Minkowski Addition One of the most important properties of the support function is its relation with the operations on convex bodies that we have introduced before, as the following result shows. Proposition 1.28 For convex bodies .K, L ∈ Kn , and for .α, β ≥ 0 hαK+βL = αhK + βhL .

.

Exercise 1.29 Prove Proposition 1.28.

10

A. Colesanti

1.8 The Polar Body We now introduce another important notion in convex geometry: the polar body. For simplicity, we restrict ourselves to polarity with respect to the origin. Let Kno = {K ∈ Kn : o ∈ K},

.

Kn(o) = {K ∈ Kn : o is an interior point of K}.

Definition 1.30 Let .K ∈ Kn(o) . The polar body of K (with respect to the origin) is defined as K ◦ = {x ∈ Rn : ≤ 1 for every y ∈ K}.

.

Exercise 1.31 Let .K = rB n , for some .r > 0. Prove that K◦ =

.

1 n B . r

Exercise 1.32 Let .Q ∈ K2 be defined by Q = {(x, y) ∈ R2 : |x| ≤ 1, |y| ≤ 1}.

.

Prove that .Q◦ is the convex hull of the four points .(±1, 0), .(0, ±1). .Q◦ is a cross polytope, i.e. is the convex hull of the elements of an orthogonal basis and their opposites. An analogous fact holds in general dimension .n ≥ 2. Proposition 1.33 Let .K ∈ Kn(o) . Then . .K ◦ ∈ Kn(o) ; . .(K ◦ )◦ = K.

1.8.1 The Gauge Function Let K be a convex body containing the origin in its interior, and assume moreover that K is symmetric with respect to the origin. The function .||·||K : Rn → R defined by: ||x||K = inf{λ ≥ 0 : x ∈ λK},

.

is called the gauge function of K. Exercise 1.34 Let .K ∈ Kn(o) be symmetric with respect to the origin. Then . .||x||K ≥ 0 for every .x ∈ Rn , and .||x||K = 0 if and only if .x = o;

1 Notation and Introductory Material

11

. .||λx||K = |λ| ||x||K for every .x ∈ Rn and for every .λ ∈ R; . .||x + y||K ≤ ||x||K + ||y||K for every .x, y ∈ K. Hence .|| · ||K is a norm on .Rn , and more precisely is the norm for which K is the “unit ball”, or the norm induced by K. .|| · ||K is also called the Minkowski functional of K. Exercise 1.34 tells us also that .|| · ||K is a support function. We have the following relation. Proposition 1.35 For every .K ∈ Kn(o) , symmetric with respect to the origin, we have || · ||K = hK ◦ .

.

1.9 The Steiner Formula It is frequently said that convex geometry is based on the study of the relations between volume and Minkowski addition. The Steiner formula is one of the simplest and most significant examples of these relations. The Steiner formula is also the first evidence of the following general phenomenon: the volume has a polynomial behavior with respect to the addition of convex bodies. Let us recall that given a convex body .K ∈ Kn , for a fixed .ε > 0, the parallel set of K is Kε = K + εB n = {x ∈ Rn : dist(x, K) ≤ ε}.

.

Roughly speaking, from a geometric point of view .Kε is the union of K and a “shell” of uniform width .ε surrounding K. The Steiner formula expresses the volume of the parallel set of a convex body. We recall that .Vn denotes the Lebesgue measure on n .R . Theorem 1.36 (Steiner Formula) Let .K ∈ Kn be a convex body in .Rn , .ε > 0 and n .Kε = K + εB be the parallel set of K. The volume of .Kε is a polynomial of degree n in .ε. More precisely, there exist non-negative numbers .V0 (K), . . . , Vn−1 (K), such that Vn (Kε ) =

n W

.

Vi (K)κn−i εn−i ,

i=0

(recall that .κj is the volume of the unit ball in .Rj ). The quantities .V0 (K), . . . , Vn (K) are called the intrinsic volumes of K.

12

A. Colesanti

Remark 1.37 The Steiner formula can be presented in a different form, namely n ( ) W n j ε Wj (K). .Vn (Kε ) = j j =0

The coefficients appearing in this version, .Wi (K), .i = 0, . . . , n, are called the quermassintegrals of K. Some of the intrinsic volumes have a familiar geometric meaning. . .V0 ≡ 1. In other words, this intrinsic volume is a constant independent of K. Sometimes is useful to think about .V0 as the Euler characteristic, which is 1 for every convex body. . .V1 is proportional to the mean width. Given a convex body K, and .u ∈ Sn−1 , the width .wK (u) of K in the direction u is the distance between the two supporting hyperplanes to K, orthogonal to u. This quantity can be expressed in terms of the support function of K, as follows: wK (u) = hK (u) + hK (−u).

.

The mean width .w(K) of K is the mean value of the width function: f 1 wK (u) dHn−1 (u). .w(K) = nκn Sn−1 The following relation holds V1 (K) =

.

nκn w(K). 2κn−1

. .Vn−1 (K) is, up to a factor, the so-called Minkowski content of K, defined as the following limit .

lim

ε→0+

Vn (Kε ) − Vn (K) . ε

In fact the Minkowski content of a convex body K equals .2Vn−1 (K). In particular, if K has interior points, then its Minkowski content coincides with the .(n − 1)-dimensional measure of the boundary of K. Hence, in this case: Vn−1 (K) =

.

1 n−1 H (∂K). 2

In general, given .K ∈ Kn , the quantity .2Vn−1 (K) will be called the surface area of K. . .Vn (K) is the volume of K.

1 Notation and Introductory Material

13

1.9.1 Basic Properties of Intrinsic Volumes Intrinsic volumes are very well studied objects in convex geometry. We list here some of their fundamental properties (see Chap. 3, Theorem 3.5, Theorem 3.14, and the discussion after Theorem 3.16, for more details). For .j ∈ {0, . . . , n}, let us consider .Vj as a functional from .Kn to .R. Proposition 1.38 The following properties hold. . .Vj (K) ≥ 0 for every .K ∈ Kn ; moreover, .Vj is monotone increasing: K, L ∈ Kn , K ⊂ L

.

⇒

Vj (K) ≤ Vj (L).

. .Vj is j -homogeneous: if .K ∈ Kn and .λ ≥ 0, then Vj (λK) = λj Vj (K).

.

. .Vj is continuous with respect to the Hausdorff metric. . .Vj is rigid motion invariant. . If .dim(K) = j , then .Vj (K) coincides with the Lebesgue measure of K as a subset of .Rj (whence the name of intrinsic volumes). . .Vj verifies the following additivity, or valuation, property; let .K, L ∈ Kn be such that .K ∪ L ∈ Kn , then: Vj (K ∪ L) + Vj (K ∩ L) = Vj (K) + Vj (L).

.

1.10 Mixed Volumes The polynomiality of volume with respect to Minkowski addition goes far beyond what the Steiner formula shows. For any natural number m, .K1 , . . . , Km ∈ Kn convex bodies and .t1 , . . . , tm non-negative real numbers, we have Vn (t1 K1 + · · · + tm Km ) is a homogeneous polynomial of degree n, in t1 , . . . , tm .

.

As a rigorous formulation of this fact, we present the following result. Theorem 1.39 There exists a mapping .V : (Kn )n → R+ , such that for every natural number m, .K1 , . . . , Km , convex bodies and .t1 , . . . , tm , non-negative real numbers, Vn (t1 K1 + · · · + tm Km ) =

m W

.

i1 ,...,in =1

V (Ki1 , . . . , Kin )ti1 . . . tin .

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A. Colesanti

Given .K1 , . . . , Kn ∈ Kn , the number V (K1 , . . . , Kn )

.

is called the mixed volume of .K1 , . . . , Kn . Remark 1.40 The Steiner formula for K can be obtained from the previous theorem, setting: .m = 2, .K1 = K, .K2 = B n , .t1 = 1, .t2 = ε. In this way we see that intrinsic volumes and quermassintegrals are special mixed volumes (possibly up to dimensional coefficients). More precisely, in the case of quermassintegrals we have the relation Wn−i (K) = V (K, . . . , K , B n , . . . B n ) =: V (K[n − i], B n [i]). ' '' ' ' '' '

.

i times

n−i times

As in the previous formula, the following notation will be used throughout: for m ∈ {1, . . . , n}, .K, K1 , . . . , Kn−m ∈ Kn :

.

V (K[m], K1 , . . . , Kn−m ) := V (K, . . . , K , K1 , . . . Kn−m ). ' '' '

.

m times

1.10.1 Properties of Mixed Volumes The following proposition gathers the basic properties of mixed volumes (see also Theorem 3.14 in Chap. 3 for a more detailed presentation). Proposition 1.41 The mixed volume functional .V : (Kn )n → R has the following properties. . For every .K ∈ Kn , V (K, . . . , K) = Vn (K).

.

. V is symmetric in its entries, i.e. for every .K1 , . . . , Kn ∈ Kn , .V (K1 , . . . , Kn ) is invariant with respect to permutations of .K1 , . . . , Kn . . V is continuous (with respect to the product topology on .(Kn )n ). . V is translation invariant in each entry. . V is linear in each entry: for every .α, β ≥ 0 and for every .K, L, K2 , . . . , Kn ∈ Kn , V (αK + βL, K2 , . . . , Kn ) = αV (K, K2 , . . . , Kn ) + βV (L, K2 , . . . , Kn ).

.

. V is non-negative, and it is monotone in each entry.

1 Notation and Introductory Material

15

. V has the valuation property in each entry: for every .K, L, K2 , . . . , Kn ∈ Kn such that .K ∪ L ∈ Kn , V (K ∪ L, K2 , . . . , Kn ) + V (K ∩ L, K2 , . . . , Kn )

.

= V (K, K2 , . . . , Kn ) + V (L, K2 , . . . , Kn ).

1.11 The Surface Area Measure To each convex body .K ∈ Kn we can associate a non-negative Borel measure n−1 , called the surface area measure of K; its construction proceeds .Sn−1 (K, ·) on .S as follows. For every regular point .x ∈ ∂K, we denote by .νK (x) the outer unit normal to K at x. The function νK : {x ∈ ∂K : x is regular} −→ Sn−1

.

is the Gauss map of K. For future use, we remark that more generally, given a subset β of .∂K we can define the spherical image of K at .β as the set:

.

{u ∈ Sn−1 : u is an outer unit normal to Kat x, for some x ∈ β}.

.

For .ω ⊂ Sn−1 , we define the reverse spherical image of .ω −1 νK (ω) = {x ∈ ∂K : x is regular and νK (x) ∈ ω}.

.

−1 It can be proved that if .ω is a Borel subset of .Sn−1 , then .νK (ω) is a .Hn−1 measurable subset of .∂K.

Definition 1.42 Let .K ∈ Kn ; the surface area measure .Sn−1 (K, ·) is a non-negative Borel measure defined on .Sn−1 as follows: for every Borel subset .ω of .Sn−1 −1 Sn−1 (K, ω) = Hn−1 (νK (ω)).

.

Remark 1.43 The total mass of the surface area measure of .K ∈ Kn is just the surface area of K Sn−1 (K, Sn−1 ) = 2Vn−1 (K).

.

Example Let P be a polytope. Let .ν1 , . . . , νm , .m ∈ N, be the outer normal vectors to the .(n − 1)-dimensional faces .F1 , . . . , Fm , which form the boundary of P . Then the surface area measure of P is the linear combination of Dirac point masses, concentrated at the unit vectors .νi , .i = 1, . . . , m, where each point mass is

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A. Colesanti

multiplied by the .(n − 1)-dimensional measure of the corresponding face. In other words Sn−1 (P , ·) =

m W

.

Hn−1 (Fi )δνi (·).

i=1

Example Let .K ∈ Kn be of class .C 2,+ . In this case, the Gauss map .νK is a diffeomorphism between .∂K and .Sn−1 . Moreover, the surface area measure of K is absolutely continuous with respect to the .(n − 1)-dimensional Hausdorff measure restricted to .Sn−1 , and f 1 dHn−1 (x) .Sn−1 (K, ω) = −1 ω κK (νK (x)) for every Borel set .ω ⊂ Sn−1 . Here .κK : ∂K → R is the Gauss curvature. The surface area measure has a continuity property with respect to the metric induced on .Kn by the Hausdorff distance. We recall that a sequence of finite Borel measures .μj , .j ∈ N, on .Sn−1 is said to converge weakly to the finite Borel measure n−1 , if .μ on .S f f . lim f (x) dμj (x) = f (x) dμ(x) j →∞ Sn−1

Sn−1

for every function .f ∈ C(Sn−1 ). Proposition 1.44 Let .K ∈ Kn and let .Kj , .j ∈ N, be a sequence in .Kn , such that .

lim Kj = K.

j →∞

Then the sequence of measures .Sn−1 (Kj , ·) converges weakly to the measure Sn−1 (K, ·).

.

1.11.1 The Minkowski Problem We have seen that to each convex body .K ∈ Kn , we can associate its surface area measure .Sn−1 (K, ·), which is a Borel measure on .Sn−1 . The Minkowski problem asks to find a convex body which has a prescribed finite Borel measure on the unit sphere as its surface area measure.

1 Notation and Introductory Material

17

Problem 1.45 Let .σ be a non-negative Borel measure on .Sn−1 ; find a convex body K in .Kn such that Sn−1 (K, ·) = σ (·).

.

This problem is well posed, under natural conditions on .σ , as the following theorem shows. Theorem 1.46 A non-negative Borel measure .σ on .Sn−1 is the surface area measure of a unique, up to translations, convex body .K ∈ Kn , if and only if the following two conditions are satisfied: (i) .σ is centered, that is f .

Sn−1

u dσ (u) = 0;

(ii) the support of .σ is not contained in any great sub-sphere of .Sn−1 . Remark Under the same conditions of the previous result, the solution to the Minkowski problem is a polytope if and only if the given measure .σ is discrete, i.e. is the sum of point masses.

1.11.2 A Formula for the Volume The following theorem contains an important formula, connecting volume, support function and surface area measure of a convex body. Theorem 1.47 For every convex body .K ∈ Kn : Vn (K) =

.

1 n

f Sn−1

hK (x) dSn−1 (K, x).

(1.1)

This formula can be proved in an elementary way in the case of polytopes. The general case can then be obtained approximating a general convex body by polytopes, and using the continuity properties of the volume, the support function and the surface area measure.

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A. Colesanti

1.11.3 The Surface Area Measure as the First Variation of the Volume Let us conclude this section with a formula which shows that the area measure is, roughly speaking, the first variation of the volume with respect to Minkowski addition. Theorem 1.48 Let .K, L ∈ Kn . Then .

lim

ε→0+

Vn (K + εL) − Vn (K) = ε

f Sn−1

hL (x) dSn−1 (K, x).

(1.2)

Remark 1.49 Formula (1.1) can be obtained by (1.2), choosing .L = K and exploiting the homogeneity of volume. Remark 1.50 The left hand side of (1.2) expresses the first variation of the volume functional at K, with respect to the Minkowski addition. The right hand side is a linear integral functional in the support function .hL . Recalling that the Minkowski addition behaves linearly with respect to support functions, we conclude that the surface area measure represents the first variation of the volume with respect to Minkowski addition.

References 1. D. Hug, W. Weil, Lectures on Convex Geometry, Graduate Texts in Mathematics, vol. 286 (Springer, Cham, 2020) 2. R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 151, Second expanded edn. (Cambridge University Press, Cambridge, 2014)

Chapter 2

Valuations on Convex Bodies and Functions Monika Ludwig and Fabian Mussnig

Abstract An introduction to geometric valuation theory is given. The focus is on classification results for .SL(n) invariant and rigid motion invariant valuations on convex bodies and on convex functions.

2.1 Introduction In his Third Problem, Hilbert asked whether, given any two polytopes of equal volume in .R3 , it is always possible to dissect the first into finitely many polytopes which can be reassembled to yield the second. In 1900, it was known that the answer to the corresponding question in .R2 is yes, but the question was open in higher dimensions. Let .Pn be the set of convex polytopes in .Rn . We say that .P ∈ Pn is dissected into .P1 , . . . , Pm ∈ Pn and write .P = P1 u · · · u Pm , if .P = P1 ∪ · · · ∪ Pm and the polytopes .P1 , . . . , Pm have pairwise disjoint interiors. So, Hilbert’s Third Problem asks whether for any .P , Q ∈ Pn of equal volume there are dissections P = P1 u · · · u Pm ,

.

Q = Q1 u · · · u Q m ,

and rigid motions .φ1 , . . . , φm such that Pi = φi Qi

.

for .1 ≤ i ≤ m. We write .P ∼ Q in this case. We call a function .Z : Pn → R a valuation if .

Z(P ) + Z(Q) = Z(P ∪ Q) + Z(P ∩ Q)

M. Ludwig · F. Mussnig (O) Institut für Diskrete Mathematik und Geometrie, TU Wien, Wien, Austria e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6_2

19

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M. Ludwig and F. Mussnig

for all .P , Q ∈ Pn with .P ∪ Q ∈ Pn (and we set .Z(∅) := 0). We call .Z simple if .Z(P ) = 0 for all polytopes that are not full-dimensional. We say that .Z is rigid motion invariant if .

Z(φP ) = Z(P )

for all rigid motions .φ : Rn → Rn and .P ∈ Pn . If .Z : Pn → R is a simple, rigid motion invariant valuation, it is not difficult to see that .P ∼ Q implies that .Z(P ) = Z(Q). Dehn [35] constructed a simple, rigid motion invariant valuation, now called Dehn invariant (see Sect. 2.5), that is not a multiple of volume. He showed that the Dehn invariant of a regular simplex and a cube of the same volume do not coincide. Thereby he solved Hilbert’s Third Problem and showed that the answer to Hilbert’s question is no for .n ≥ 3. Blaschke [14] took the critical next step by asking for classification results for G invariant valuations on .Pn and on the space of convex bodies, .Kn , that is, of nonempty, compact, convex sets in .Rn , where G is any group acting on .Rn . Blaschke’s question is motivated by Felix Klein’s Erlangen Program. We will discuss some of the results obtained in this tradition, in particular, focusing on the special linear group, .SL(n), and the group of rigid motions, .SO(n)xRn , where .SO(n) is the group of (orientation preserving) rotations. Often additional regularity assumptions are required, and we consider continuous and upper semicontinuous valuations, where we equip .Kn and its subspaces with the topology induced by the Hausdorff metric. In addition to classification results and their applications, structural results for spaces of valuations have attracted much attention in recent years. We refer to the books and surveys [4, 6, 11]. Valuations were also considered on various additional spaces, particularly on manifolds (see [3]). Valuations with values in linear spaces and Abelian semigroups, including the space of convex bodies, were also studied (see [54]). We will restrict our attention to real-valued valuations defined on subspaces of .Kn and to recent results on valuations on spaces of realvalued functions. On a space X of (extended) real-valued functions, a functional .Z : X → R is a valuation if .

Z(f ) + Z(g) = Z(f ∨ g) + Z(f ∧ g)

for all .f, g ∈ X such that also their pointwise maximum .f ∨ g and pointwise minimum .f ∧ g belong to X. Since we can embed spaces of convex bodies in various function spaces in such a way that unions and intersections of convex bodies correspond to pointwise minima and maxima of functions, this notion generalizes the classical notion. We will discuss the results on valuations on convex functions.

2.2 Basic Properties Let .S be a class of subsets of .Rn . We say that .Z : S → R is a valuation if .

Z(P ) + Z(Q) = Z(P ∪ Q) + Z(P ∩ Q)

2 Valuations on Convex Bodies and Functions

21

for all .P , Q ∈ S such that .P ∩ Q, P ∪ Q ∈ S, and .Z(∅) = 0. Given a Borel measure on .Rn , its restriction to .Kn is clearly a valuation. So, in particular, ndimensional Lebesgue measure, .Vn , induces a valuation on .Kn . As we will see, there are important valuations that are not induced by measures. Let .S be intersectional, that is, if .P , Q ∈ S, then .P ∩ Q ∈ S. We say that .Z : S → R satisfies the inclusion-exclusion principle on .S if .

Z(P1 ∪ · · · ∪ Pm ) =

E

(−1)|J |−1 Z(PJ )

(2.1)

∅/=J ⊂{1,...,m}

n for .P1 , . . . , Pm ∈ S and .m ≥ 1 whenever .P1 ∪ · · · ∪ Pm ∈ S. Here .PJ := j ∈J Pj and .|J | is the cardinality of the set J . The inclusion-exclusion principle holds for every valuation on .Pn and every continuous valuation on .Kn (see [41, 72]). If .Z : Pn → R is, in addition, simple, we have .

Z(P1 u · · · u Pm ) = Z(P1 ) + · · · + Z(Pm )

(2.2)

for .P1 , . . . , Pm ∈ Pn . For .K, L ∈ Kn , define the Minkowski sum by K + L := {x + y : x ∈ K, y ∈ L}.

.

The following lemma describes a way to obtain new valuations from a given one. Lemma 2.1 Let .Z : Kn → R be a valuation. If .C ∈ Kn is a fixed convex body and .

ZC (K) := Z(K + C),

for .K ∈ Kn , then .ZC is a valuation on .Kn . Proof The following statement is easily seen to hold for subsets .C, K, L ⊂ Rn , K ∪ L + C = (K + C) ∪ (L + C).

.

(2.3)

Now, let .C, K, L ∈ Kn be such that .K ∪ L ∈ Kn . If .x ∈ (K + C) ∩ (L + C), then .x = y + c = z + d with .y ∈ K, z ∈ L and .c, d ∈ C. Since .K ∪ L is convex, there is .t ∈ [0, 1] such that .(1 − t)y + tz ∈ K ∩ L and hence x = (1 − t)(y + c) + t (z + d) = (1 − t)y + tz + (1 − t)c + td.

.

Thus .(K + C) ∩ (L + C) ⊂ (K ∩ L) + C. Since it is easy to see that .(K ∩ L) + C ⊂ (K + C) ∩ (L + C), it follows that (K + C) ∩ (L + C) = (K ∩ L) + C.

.

(2.4)

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M. Ludwig and F. Mussnig

Applying .Z to (2.3) and to (2.4) for convex bodies .C, K, L and adding, we obtain the statement. u n For .p ∈ R, a functional .Z : Kn → R is called homogeneous of degree p (or p-homogeneous), if .

Z(t K) = t p Z(K)

for .t > 0 and .K ∈ Kn . A functional .Z : Kn → R is increasing if .K ⊂ L implies that .Z(K) ≤ Z(L). We will also use corresponding definitions for subsets of .Kn . The n-dimensional volume .Vn : Kn → [0, x) is a valuation. Lemma 2.1 implies that also .K |→ Vn (K + r B n ) is a valuation on .Kn for .r ≥ 0, where .B n is the ndimensional unit ball. Therefore, it follows from the Steiner formula, Vn (K + rB n ) =

n E

.

r n−j κn−j Vj (K),

(2.5)

j =0

where .r ≥ 0 and .κj is the j -dimensional volume of the unit ball in .Rj (with the convention that .κ0 = 1), that all intrinsic volumes .V0 , . . . , Vn are valuations on .Kn . Recall that all intrinsic volumes are continuous and increasing functionals on .Kn and that .V0 is the Euler characteristic and .V0 (K) = 1 for all .K ∈ Kn . Also, recall that .Vj (K) is the j -dimensional volume of K if K is contained in a j -dimensional plane and that .Vj is j -homogeneous. We will use the following notation. Let .e1 , . . . , en be the vectors of the canonical basis of .Rn . For .x, y ∈ Rn , we write . for the inner product and .|x| for the Euclidean norm of x. The convex hull of subsets .A1 , . . . , Am ⊂ Rn is written as n n .[A1 , . . . , Am ] and the convex hull of .x1 , . . . , xm ∈ R as .[x1 , . . . , xm ]. If .E ⊂ R is n an affine plane in .R , then .K(E) and .P(E) are the sets of convex bodies and convex polytopes, respectively, contained in E.

2.3 SL(n) Invariant Valuations Blaschke [14] obtained the first classification theorem of invariant valuations on .Kn . Theorem 2.2 (Blaschke) A functional .Z : Kn → R is a continuous, translation and .SL(n) invariant valuation if and only if there are constants .c0 , cn ∈ R such that .

Z(K) = c0 V0 (K) + cn Vn (K)

for every .K ∈ Kn . In the next section, we will obtain a complete classification of translation invariant valuations in the one-dimensional case, and in the following section, a complete

2 Valuations on Convex Bodies and Functions

23

classification of translation and .SL(n) invariant valuations on convex polytopes. Here, no assumptions on the continuity of the valuation are needed. Theorem 2.2 will be a simple consequence. The situation is different for valuations on convex bodies, where additional (non-continuous) valuations exist that vanish on convex polytopes. We will describe some of these valuations in Sect. 2.3.3.

2.3.1 The One-Dimensional Case We call a function .ζ : [0, x) → R a Cauchy function if it is a solution to the Cauchy functional equation, that is, ζ (x + y) = ζ (x) + ζ (y)

.

for every .x, y ∈ [0, x). Cauchy functions are well understood and can be completely described (if we assume the axiom of choice) by their values on a Hamel basis. Proposition 2.3 A functional .Z : P1 → R is a translation invariant valuation if and only if there are a constant .c0 ∈ R and a Cauchy function .ζ : [0, x) → R such that .

( ) Z(P ) = c0 V0 (P ) + ζ V1 (P )

for every .P ∈ P1 . Proof Set .c0 := Z({0}) and define .Z˜ : P1 → R by ˜ ) := Z(P ) − c0 V0 (P ). Z(P

.

Note that .Z˜ is a simple, translation invariant valuation on .P1 . Define the function .ζ : [0, x) → R by setting ˜ ζ (x) := Z([0, x]).

.

Since .Z˜ is a simple, translation invariant valuation, ˜ ˜ ˜ ζ (x + y) = Z([0, x + y)) = Z([0, x]) + Z([x, x + y]) = ζ (x) + ζ (y)

.

for every .x, y ∈ [0, x). Hence .ζ is a Cauchy function. Using that .Z˜ is translation ˜ ) = ζ (V1 (P )) for .P ∈ P1 , which concludes the proof. invariant, we get .Z(P u n Since every continuous Cauchy function is linear, we obtain the following result.

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M. Ludwig and F. Mussnig

Corollary 2.4 A functional .Z : P1 → R is a continuous and translation invariant valuation if and only if there are constants .c0 , c1 ∈ R such that .

Z(P ) = c0 V0 (P ) + c1 V1 (P )

for every .P ∈ P1 . A corresponding classification result holds for upper semicontinuous and translation invariant valuations on .P1 . Such a result also holds for Borel measurable and translation invariant valuations on .P1 , since every Borel measurable Cauchy function is linear.

2.3.2 SL(n) Invariant Valuations on Convex Polytopes The following result gives a complete classification of translation and .SL(n) invariant valuations on polytopes. Theorem 2.5 A functional .Z : Pn → R is a translation and .SL(n) invariant valuation if and only if there are a constant .c0 ∈ R and a Cauchy function .ζ : [0, x) → R such that .

( ) Z(P ) = c0 V0 (P ) + ζ Vn (P )

for every .P ∈ Pn . Proof Set .c0 := Z({0}) and define .Z˜ : Pn → R by ˜ ) := Z(P ) − c0 V0 (P ). Z(P

.

Note that .Z˜ is a translation invariant valuation on .Pn that vanishes on singletons, that is, sets of the form .{x} with .x ∈ Rn . We show that there is a Cauchy function .ζ : [0, x) → R such that ˜ ) = ζ (Vn (P )) Z(P

.

(2.6)

for every .P ∈ Pn . We use induction on the dimension n. By Proposition 2.3, the statement (2.6) is true for .n = 1. Let .n ≥ 2. Assume that it is true for valuations on .Pn−1 . Hence it is also true for valuations on .P(E) with E any hyperplane in .Rn . The induction assumption implies that there is a Cauchy function .ζ˜ : [0, x) → R such that ˜ ) = ζ˜ (Vn−1 (P )) Z(P

.

2 Valuations on Convex Bodies and Functions

25

for every .P ∈ P(E). Note that the invariance properties of .Z˜ imply that .ζ˜ does not depend on E. Since .Z˜ vanishes on singletons, we have .ζ˜ (0) = 0. Let E be spanned by the first .(n − 1) basis vectors .e1 , . . . , en−1 and define .φ ∈ SL(n) by setting 1 .φe1 = t e1 with .t > 0 and .φej = ej for .1 < j < n and .φen = en . Since .φE = E, t it follows from the .SL(n) invariance of .Z˜ that ˜ ˜ ζ˜ (t) = Z(φ[0, 1]n−1 ) = Z([0, 1]n−1 ) = ζ˜ (1)

.

for every .t > 0. This implies that .ζ˜ ≡ 0 and shows that .Z˜ is simple. Thus it suffices to show that (2.6) holds for every simple, translation and .SL(n) invariant valuation ˜ : Pn → R. .Z Define .ζ : [0, x) → R by setting √ ˜ n s n! [0, e1 , . . . , en ]) ζ (s) := Z(

.

and note that ˜ Z(S) = ζ (Vn (S))

.

for every simplex .S ∈ Pn , as the valuation .Z˜ is simple, translation and .SL(n) invariant and√every n-dimensional simplex is a translate of an .SL(n) image of the simplex . n s n! [0, e1 , . . . , en ] for some .s > 0. For .0 < r < 1, we dissect the n-dimensional simplex with vertices .v0 , . . . , vn ∈ Rn into the n-dimensional simplices .T1 with vertices .v0 , r v0 + (1 − r)v1 , v2 , . . . , vn and .T2 with the vertices ˜ is a simple valuation, .(1 − r)v0 + r v1 , v1 , v2 , . . . , vn (Fig. 2.1). Since .Z ˜ 1 ∪ T2 ) = Z(T ˜ 1 ) + Z(T ˜ 2 ). Z(T

.

Fig. 2.1 Decomposition of .[v0 , . . . , vn ] into .T1 and .T2

(2.7)

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M. Ludwig and F. Mussnig

√ Choosing .v0 := 0 and .vj := n (s + t)n! ej for .j = 1, . . . , n as well as .r := s/(s + t), we obtain from (2.7) that ζ (s + t) = ζ (s) + ζ (t)

.

for every .s, t ∈ (0, x). Hence, .ζ is a Cauchy function. By (2.2) and since we can dissect every polytope into simplices, we conclude that (2.6) holds for every n .P ∈ P . n u Properties of Cauchy functions immediately give the following result, which, in turn, implies Theorem 2.2. Corollary 2.6 A functional .Z : Pn → R is a continuous, translation and .SL(n) invariant valuation if and only if there are constants .c0 , cn ∈ R such that .

Z(P ) = c0 V0 (P ) + cn Vn (P )

for every .P ∈ Pn . Corresponding statements hold for upper semicontinuous valuations and for Borel measurable valuations. We remark that classification results for .SL(n) invariant valuations are also known without assuming translation invariance (see [57]). In particular, the following result holds. Let .Pno be the space of convex polytopes containing the origin. Theorem 2.7 A functional .Z : Pno → R is a continuous, .SL(n) invariant valuation if and only if there are constants .c0 , cn ∈ R such that .

Z(P ) = c0 V0 (P ) + cn Vn (P )

for every .P ∈ Pno . On .Pn , there are additional .SL(n) invariant valuations. In particular, .P |→ Vn ([0, P ]) is such a valuation (see [57] for a complete classification). On .Pn(o) , the space of convex polytopes containing the origin in their interiors, .P |→ Vn (P ◦ ), the functional that associates with P the volume of its polar body, is an .SL(n) variant valuation. A complete classification of .SL(n) invariant valuations on .Pn(o) was established by Haberl and Parapatits [38].

2.3.3 Affine Surface Area While we have established a complete classification of translation and .SL(n) invariant valuations on .Pn , such a result is not known on .Kn , and there are additional valuations on .Kn that vanish on .Pn . The classical affine surface area .o : Kn → R is such a valuation. It is defined by f 1 .o(K) = κ(K, x) n+1 dHn−1 (x), (2.8) ∂K

2 Valuations on Convex Bodies and Functions

27

where .κ(K, x) is the generalized Gaussian curvature of .∂K at x and integration is with respect to the .(n − 1)-dimensional Hausdorff measure .Hn−1 on the boundary, .∂K, of K. By a classical result of Aleksandrov, the boundary of a convex body is twice differentiable almost everywhere and hence .κ(K, x) is defined almost everywhere and it can be shown that .x |→ κ(K, x) is measurable. We remark that the generalized Gaussian curvature is the density of the absolutely continuous part of the curvature measure .C0 (K, ·), where .C0 (K, B) := Hn−1 (νK (B)) for a Borel set .B ⊂ ∂K, and .νK is the spherical image map that assigns to .x ∈ ∂K the set of all unit normal vectors of supporting hyperplanes of K containing x (see [72, Chapter 4]). Hence f . κ(K, x) dHn−1 (x) ≤ Hn−1 (∂B n ) = nκn ∂K

for every .K ∈ Kn , and by Jensen’s inequality, f .

) 1 ( ( 1 κ(K, x) n+1 dHn−1 (x) ≤ nκn n+1

f dHn−1 (x)

)

n n+1

.

(2.9)

∂K

∂K

This implies that the integral in (2.8) is finite for every .K ∈ Kn . The definition of affine surface area for convex bodies with smooth boundary is classical and goes back to Blaschke and Pick [13]. They established that .o is equi-affine invariant, that is, .o is translation and .SL(n) invariant. The extension to general convex bodies is more recent and due to Leichtweiß [45], Lutwak [58] and Schütt and Werner [74]. Lutwak [58] proved that .o is upper semicontinuous on .Kn , that is, for every sequence of convex bodies .Kj converging to a convex body K, we have o(K) ≥ lim sup o(Kj ).

.

j →x

It follows from (2.8) that .o vanishes on polytopes and is therefore not continuous. The valuation property of .o on .Kn follows directly from (2.8). Note that .o is translation invariant and that .Ω(K) = 0 if K is lower dimensional. Hence we may assume in the following that the origin is an interior point of K. Clearly, (2.8) can be rewritten as f o(K) =

1

κ0 (K, x) n+1 dVK (x),

.

∂K

where κ0 (K, x) :=

.

κ(K, x) n+1

(2.10)

28

M. Ludwig and F. Mussnig

and dVK (x) := dHn−1 (x).

.

Here, .nK (x) is the unit outer normal vector of K at x, which is uniquely defined almost everywhere on .∂K, and . is the distance to the origin of the tangent hyperplane to K at such x. In (2.10), it is easy to see that .o is .SL(n) invariant. Indeed, for a Borel set .B ⊂ ∂K, using the fact that the volume of a cone is the product of its height divided by n and the .(n − 1)-dimensional volume of its base, we see that . n1 VK (B) is just the n-dimensional volume of the set .{t B : t ∈ [0, 1]}. Consequently, VφK (φB) = VK (B)

.

for every .φ ∈ SL(n) and every Borel set .B ⊂ ∂K. Moreover, κ0 (φK, φx) = κ0 (K, x)

.

for every .φ ∈ SL(n) and every .x ∈ ∂K where .κ0 (K, x) > 0. This is a simple consequence of the following geometric interpretation of .κ0 (K, x), κ0 (K, x) =

.

κn2 , Vn (EK (x))2

where .EK (x) is the unique centered ellipsoid that osculates K at x. We remark that f K |→

ζ (κ0 (K, x)) dVK (x)

.

(2.11)

∂K

is an .SL(n) invariant valuation on .Kn(o) , the set of convex bodies containing the origin in their interiors when .ζ : [0, x) → [0, x) is a suitable continuous function. The functionals defined in (2.11) are called Orlicz affine surface areas. If .ζ (t) := t p for .t > 0 with .p > −n, the so-called .Lp affine surface area of K is obtained, which was introduced by Lutwak [59]. Classification results for .SL(n) invariant valuations on .Kn(o) were established in [38, 49, 56] and characterizations of .Lp and Orlicz affine surface areas in [56]. The following result from [48, 55] strengthens Theorem 2.2 and establishes a characterization of affine surface area. Theorem 2.8 A functional .Z : Kn → R is an upper semicontinuous, translation and .SL(n) invariant valuation if and only if there are constants .c0 , cn ∈ R and .c ≥ 0 such that .

for every .K ∈ Kn .

Z(K) = c0 V0 (K) + cn Vn (K) + c o(K)

2 Valuations on Convex Bodies and Functions

29

Fig. 2.2 Support triangle of a convex body .K ∈ K2 with endpoints .x, y ∈ ∂K

We present the proof of Theorem 2.8 in the case .n = 2 from [48]. We call a closed triangle .T = T (x, y) a support triangle of .K ∈ K2 with endpoints x and y, if .x, y ∈ ∂K and T is bounded by support lines (that is, 1-dimensional support hyperplanes) to K at x and y and the chord connecting x and y (Fig. 2.2). A cap of a convex body K is the intersection of a closed half-space and K. We set .δs (K, L) := V2 (KAL) for .K, L ∈ K2 , where .KAL := (K ∪ L)\(K ∩ L) is the symmetric difference of K and L. Note that the symmetric difference metric .δs induces on full-dimensional convex bodies the same topology as the Hausdorff metric. We require the following lemma, whose proof is omitted as it is very similar to the proof of Proposition 2.3. Lemma 2.9 If . Z : K2 → R is an upper semicontinuous, rotation invariant valuation that vanishes on polytopes, then .

Z(C) = c o(C)

for every cap C of .B 2 , where .c := Z(B 2 )/ o(B 2 ). Let .E2 be the family of all convex bodies in .R2 which may be dissected into finitely many polytopes and caps of unit ellipses. Here, any equi-affine image of the two-dimensional unit ball .B 2 is called a unit ellipse. Since planar polytopes belong to .E2 , the set .E2 is dense in .K2 . Proposition 2.10 If . Z : K2 → [0, x) is an upper semicontinuous, translation and . SL(2) invariant valuation that vanishes on polytopes, then .

Z(K) = sup{lim sup Z(Ek ) : Ek → K, Ek ∈ E2 } k→x

for every .K ∈ K2 .

30

M. Ludwig and F. Mussnig

Proof Since .Z is upper semicontinuous, we have .

Z(K) ≥ lim sup Z(Ek ) k→x

for every .K ∈ K2 and for every sequence .Ek ∈ E2 such that .Ek → K. To prove the statement of the proposition, assume on the contrary that there is .K ∈ K2 such that .

Z(K) > lim sup Z(Ek )

(2.12)

k→x

for all sequences .Ek with .Ek ∈ E2 and .Ek → K. By (2.9), the affine surface area of E is uniformly bounded for all convex bodies E with .δs (K, E) < 1, say. Therefore, by (2.12), for every .ε > 0 small enough, there is .0 < δ < 1 such that .

Z(K) ≥ Z(E) + ε o(E)

(2.13)

for every .E ∈ E2 with .δs (K, E) < δ. We approximate .B 2 by a sequence of convex bodies built from suitable pieces of K and show that (2.13) leads to a contradiction. Without loss of generality, assume that the origin is an interior point of K. Choose k rays starting at the origin such that k E .

(k)

(2.14)

V2 (Ti ) < δ

i=1 (k) (k) where .Ti(k) = T (xi(k) , xi+1 ) are support triangles and .x1(k) , . . . , xk(k) , xk+1 = x1(k) are (k) the consecutive points where the rays intersect .∂K. For every .Ti with non-empty interior, there is a unique arc of the unit ellipse which touches the two sides of .Ti(k) which are given by the support lines of K. We denote by .Ei(k) the convex body (k) bounded by this arc of an ellipse and the chord connecting .xi(k) and .xi+1 . In the case that .Ti(k) has empty interior, we set .Ei(k) := Ti(k) . We define

Ek :=

k ||

.

(k)

Ei

k ) ( || (k) . ∪ K\ Ti

i=1

i=1

Note that .Ek ∈ E2 and that (2.14) implies that .δs (K, Ek ) < δ. Since .Z and .o vanish on polytopes, (2.13) implies that k E .

i=1

(k)

Z(K∩Ti ) = Z(K) ≥ Z(Ek )+ε o(Ek ) =

k E ( i=1

(k) (k) ) Z(Ek ∩Ti )+ε o(Ek ∩Ti ) .

2 Valuations on Convex Bodies and Functions

31

Consequently, for every k, there exists a support triangle .Ti(k) with non-empty k interior such that .

(k)

(k)

(k)

Z(K ∩ Tik ) ≥ Z(Ek ∩ Tik ) + ε o(Ek ∩ Tik ).

(2.15)

We take an equi-affine transformation .φ (k) which transforms .Ti(k) into a support k (k) 2 (k) (k) (k) ˜ ˜ triangle .T of .B , and denote by .C and .B the images under .φ of the caps (k) (k) .K ∩ T ik and .Ek ∩ Tik , respectively. By (2.15) and the equi-affine invariance of .Z, we have .

Z(C˜ (k) ) ≥ Z(B (k) ) + ε o(B (k) ).

(2.16)

Let .lk be the largest integer such that there are rotations .ψ1 , . . . , ψlk with the property that .ψ1 (T˜ (k) ), . . . , ψlk (T˜ (k) ) are non-overlapping support triangles of .B 2 . Since for a sector of .B 2 with an angle .2 α at the origin, the area of a support triangle to .B 2 is .sin2 α tan α, we have ( ( ) ( ) ) ( ) π π π π 2 tan . tan ≤ V2 (T˜ (k) ) ≤ sin2 (2.17) . sin lk + 1 lk + 1 lk lk We construct convex bodies lk ||

K˜ k :=

.

lk ( ) || ψi (C˜ (k) ) ∪ B 2 \ ψi (T˜ (k) ) .

i=1

i=1

Note that (2.16) implies that .

Z(K˜ k ) ≥ Z(B 2 ) +

ε o(B 2 ) 2

(2.18)

for k sufficiently large. Since .δs (K˜ k , B 2 ) ≤ lk V2 (T˜ (k) ), it follows from (2.17) that K˜ k → B 2

.

(2.19)

as .k → x. Thus by the upper semicontinuity of .Z, by (2.19), (2.18) and (2.21), we obtain that .

ε Z(B 2 ) ≥ lim sup Z(K˜ k ) ≥ Z(B 2 ) + o(B 2 ). 2 k→x

This is a contradiction since .ε > 0 and .o(B 2 ) > 0, which concludes the proof of u n the proposition.

32

M. Ludwig and F. Mussnig

Note that we can apply Proposition 2.10 with .Z = o and obtain that o(K) = sup{lim sup o(Ek ) : Ek → K, Ek ∈ E2 }

.

(2.20)

k→x

for every .K ∈ K2 . Proof of Theorem 2.8 for .n = 2 Let .Z : K2 → R be an upper semicontinuous, translation and .SL(2) invariant valuation. By Theorem 2.5 and since upper semicontinuous Cauchy functions are linear, there are .c0 , c2 ∈ R such that .

Z(P ) = c0 V0 (P ) + c2 V2 (P )

for every .P ∈ P2 . Define .Z˜ : K2 → R by ˜ Z(K) := Z(K) − c0 V0 (K) − c2 V2 (K)

.

and note that .Z˜ is an upper semicontinuous, translation and .SL(2) invariant valuation that vanishes on polytopes. For every .K ∈ K2 , there is a sequence of polytopes .Pk with .Pk → K. Hence, the upper semicontinuity of .Z˜ implies that ˜ ˜ k ) = 0, Z(K) ≥ lim sup Z(P

.

k→x

which shows that .Z˜ is non-negative. Using Lemma 2.9 and the translation and .SL(2) ˜ we see that invariance of .Z, ˜ Z(C) = c o(C)

.

for every cap C of a unit ellipse, where .c = Z(B 2 )/ o(B 2 ). Since .Z vanishes on polytopes, it is a simple valuation, and it follows from (2.2) that ˜ Z(E) = c o(E)

(2.21)

.

for .E ∈ E2 . Proposition 2.10 and (2.20) now complete the proof of the theorem.

u n

For .K ∈ K2 , Blaschke [13] gave the following definition of affine surface area. (k) = x1(k) on .∂K and support triangles Choose subdivision points .x1(k) , . . . , xk(k) , xk+1 (k) (k) (k) (k) (k) .T 1 , . . . , Tk such that .Tj = T (xj , xj +1 ). Define ˜ o(K) := lim

.

k / E 3 j =1

(k)

8 V2 (Tj )

(2.22)

2 Valuations on Convex Bodies and Functions

33

where the limit is taken over a sequence of subdivisions with .

(k)

max V2 (Ti ) → 0

i=1,...,k

as .k → x. For smooth convex bodies in .K2 , Blaschke showed that this limit always ˜ = o(K). exists and that .o(K) If we choose a further subdivision point .y ∈ ∂K in a support triangle .T (x, z) of .K ∈ K2 , we obtain support triangles .T (x, y) and .T (y, z) and the following elementary anti-triangle inequality holds .

/ / / 3 8 V2 (T (x, z)) ≥ 3 8 V2 (T (x, y)) + 3 8 V2 (T (y, z))

/ E (cf. [13, p. 38] or [20]). This implies that . kj =1 3 8 V2 (Tj(k) ) decreases as the subdivision is refined. Consequently, the limit in (2.22) exists and is independent of the sequence of subdivisions chosen and ˜ o(K) = inf

.

k / E 3

(k)

8 V2 (Tj )

j =1

˜ is well defined where the infimum is taken over all subdivisions of .∂K. Thus .o ˜ on .K2 and Leichtweiß [46] proved that .o(K) = o(K) for every .K ∈ K2 . This ˜ : K2 → R is also a simple consequence of Theorem 2.8 for .n = 2. Indeed, .o is equi-affine invariant and vanishes on lower dimensional sets. As an infimum of ˜ is upper semicontinuous. So we have only to show that continuous functionals, .o ˜ is a valuation. Since .maxi=1,...,k V2 (T (k) ) → 0 as .k → x, we have for every line .o i H, ˜ ˜ ˜ o(K) = o(K ∩ H + ) + o(K ∩ H −)

.

where .H + and .H − are the closed halfspaces bounded by H . It is not difficult to see ˜ is a valuation. Thus Theorem 2.8 for .n = 2 shows that that this implies that .o ˜ o(K) = c o(K)

.

with a constant .c ≥ 0 and a simple calculation for .K = B 2 shows that .c = 1.

2.4 Translation Invariant Valuations For translation invariant valuations on convex polytopes and on convex bodies, Hadwiger developed the basic theory. Many of the results are even valid in the setting of rational polytopes in .Qn and polytopes with integer coordinates (see, for

34

M. Ludwig and F. Mussnig

example, [17]). Nevertheless, we will restrict our attention to convex polytopes and convex bodies in .Rn .

2.4.1 The Canonical Simplex Decomposition For .0 ≤ k ≤ n, a k-dimensional simplex S in .Rn is the convex hull of .(k + 1) affinely independent points .p0 , . . . , pk ∈ Rn . We set .xi := pi − pi−1 for .1 ≤ i ≤ k and .x0 := p0 and write .S = . For .k = 0, we set .S := {x0 }. Lemma 2.11 A set S is an n-dimensional simplex with vertices .p0 , . . . , pn ∈ Rn if and only if n { } E S = x0 + ri xi : 1 ≥ r1 ≥ . . . ≥ rn ≥ 0 .

.

(2.23)

i=1

Conversely, for .x0 , . . . , xn ∈ Rn , the set defined in (2.23) is an n-dimensional simplex if .x1 , . . . , xn are linearly independent. Proof Every point .x ∈ S is a convex combination of .p0 , . . . , pn , that is, x=

n E

.

t i pi

i=0

E E with .ti ≥ 0 and . ni=0 ti = 1. Setting .ri = nj=i tj , we have x = x0 +

n E

.

ri xi .

i=1

Hence, every point contained in the right side of (2.23) is in S. E Conversely, if S is the set defined in (2.23), then, setting .pk = kj =0 xj , we have .S = [p0 , . . . , pn ]. u n The following result is called the Hadwiger canonical simplex decomposition [39, Section 1.2.6] (see Fig. 2.3). Theorem 2.12 Let .S := be an n-dimensional simplex. Defining S 0 := {x0 }, .S n−k := {x0 + · · · + xn },

.

> < S k := x0 ; x1 , . . . , xk ,

.

and

k E > < S n−k := x0 + xi ; xk+1 , . . . , xn , i=1

2 Valuations on Convex Bodies and Functions

35

Fig. 2.3 Canonical simplex decomposition

for .1 ≤ k ≤ n − 1, we have S=

.

n || ( ) (1 − t) S k + t S n−k k=0

for .0 < t < 1. Proof Setting Qk (t) := (1 − t) S k + t S n−k ,

.

we obtain by Lemma 2.23 that k k n { E E E ) ( ) ( Qk (t) = (1 − t) x0 + ri xi + t x0 + xi + si xi :

.

i=1

i=1

i=k+1

} 1 ≥ r1 ≥ · · · ≥ rk ≥ 0, 1 ≥ sk+1 ≥ · · · ≥ sn ≥ 0

n } { E ti xi : 1 ≥ t1 ≥ · · · ≥ tk ≥ t ≥ tk+1 ≥ · · · ≥ tn ≥ 0 . = x0 + i=1

For .x ∈ S, this implies that .x ∈ Qk (t) for a suitable k. We have to show that the sets Qk (t) for .1 ≤ k ≤ n − 1 have pairwise disjoint interiors. If .x ∈ Qi (t) ∩ Qk (t) for .i < k, then .ti+1 = · · · = tk = t and therefore .rj = 0 and .sj = 1 for .i+1 ≤ j ≤ k. It .

36

M. Ludwig and F. Mussnig

follows that .x ∈ ∂Qi (t) and .x ∈ ∂Qj (t). This completes the proof of the statement. u n We say that a simplex . is orthogonal if the vectors .x1 , . . . , xn are pairwise orthogonal. The following result is due to Hadwiger [39, Section 1.3.4]. Lemma 2.13 Let .z ∈ Rn be given. If .P ∈ Pn is n-dimensional, then there are ' , each with a vertex at z, such that orthogonal simplices .S1 , . . . , Sm , .S1' , . . . , Sm ' Pu

m ||

.

'

Si ∼

m ||

Sj' .

j =1

i=1

Proof The statement is easy to prove for .n = 1. Assume that it is true in .P(E) for every .(n − 1)-dimensional hyperplane E and every .zE ∈ E. It suffices to prove the statement for an n-dimensional simplex S. Let F be one of its facets whose affine hull E does not contain z. Let .zE be the closest point to z in E. We use the induction assumption for polytopes in E with .zE and obtain that there are .(n − 1)-dimensional simplices .F1 , . . . , Fk , F1' , . . . , Fk '' , each with a vertex at .zE such that Fu

k ||

.

i=1

'

Fi ∼

k ||

Fj' .

j =1

Setting .Si := [z, Fi ] for .1 ≤ i ≤ k and .Sj' := [z, Fj' ] for .1 ≤ j ≤ k ' , we obtain the statement for S. u n The question of whether every polytope in .Pn can be dissected into finitely many orthogonal simplices is open. Hadwiger conjectured that it is possible, and his conjecture has been proved for .n ≤ 5 (see, for example, [19]).

2.4.2 Valuations Vanishing on Orthogonal Cylinders We say that .P ∈ Pn is a convex orthogonal cylinder if there are orthogonal, complementary subspaces E and F with .dim E, dim F ≥ 1 and polytopes .PE ⊂ E and .PF ⊂ F such that .P = PE + PF . Note that this class includes all polytopes that are not full-dimensional. Proposition 2.14 (Hadwiger) If . Z : Kn → R is a continuous, translation invariant valuation that vanishes on convex orthogonal cylinders, then .Z is homogeneous of degree 1.

2 Valuations on Convex Bodies and Functions

37

Proof Let S be an n-dimensional orthogonal simplex in .Rn and .0 < t < 1. In the canonical simplex decomposition, n || ( ) (1 − t)S k + t S n−k ,

S=

.

(2.24)

k=0

the simplices .S k and .S n−k are orthogonal and lie in orthogonal subspaces for each 1 ≤ k ≤ n−1. Hence .(1−t)S k +t S n−k is an orthogonal cylinder for .1 ≤ k ≤ n−1. Since .Z vanishes on convex orthogonal cylinders, we obtain

.

.

Z((1 − t)S k + t S n−k ) = 0

for .1 ≤ k ≤ n − 1 and we also see that .Z is simple. By (2.2), it now follows from (2.24) that .

Z(S) = Z((1 − t)S) + Z(t S).

˜ with .S = (r + s)S˜ and .t = r/(r + s), we obtain Let .r, s > 0. Setting .α(r) := Z(r S) α(r + s) = α(r) + α(s)

.

for all .r, s > 0. Since .Z is continuous, so is .α : (0, x) → R. It follows that .α is a continuous Cauchy function. Hence .α is linear. Thus, .

Z(t S) = t Z(S)

(2.25)

for every .t > 0 and every orthogonal simplex S. For .P ∈ Pn , by Lemma 2.13 there are orthogonal simplices such that Pu

m ||

.

'

Si ∼

m ||

Sj' .

j =1

i=1

Therefore, for every .t > 0, using that .Z is simple, we obtain

.

Z(t P ) + Z

m ( ||

)

t Si = Z

m' ( ||

t Sj'

)

j =1

i=1

and '

'

m m m m E E E ( ') E ( ) ' . Z(t P ) = Z t Sj − Z t Si = t Z(Sj ) − t Z(Si ) = t Z(P ), j =1

i=1

j =1

i=1

38

M. Ludwig and F. Mussnig

where (2.2) and (2.25) are used. Hence, .Z is homogeneous of degree 1 on polytopes. Since .Z is continuous, this concludes the proof. u n For .1 ≤ l ≤ n, we say that .P ∈ Pn is a convex .l-cylinder if there are subspaces n .E1 , . . . , El of .R which are pairwise orthogonal and at least one-dimensional and convex polytopes .P1 ⊂ E1 , . . . , .Pl ⊂ El such that .P = P1 + · · · + Pl . We say that n .C ⊂ R is an .l-cylinder if it can be dissected into finitely many convex .l-cylinders. The following result was established by Hadwiger [39, Section 1.3.7]. Theorem 2.15 For .P ∈ Pn and each .1 ≤ l ≤ n, there is an .l-cylinder .Cl such that mP =

n || ||

τ (Cl )

.

l=1 τ ∈Tl,m

( ) for every integer .m ≥ 1, where . Tl,m is a set of at most . ml translations. Proof It suffices to prove the statement for P an n-dimensional simplex S. Let S = . For .i < j , define the simplices .Sij := . For .0 < t < 1 and .1 ≤ k < n, set .

Qk (t) := (1 − t) S0k + t

k−1 (E

.

) xi + Skn .

i=0

The canonical dissection into simplices for .m S and .t =

1 m

from Theorem 2.12 gives

mS = m Q0 ( m1 ) u · · · u m Qn ( m1 ).

.

We have .m Q0 ( m1 ) = S and .m Qn ( m1 ) = (m − 1) S while m Ql ( m1 ) = (m − 1) S0l + Sln

.

for .1 ≤ l ≤ n − 1, where .= stands for equal up to translation. Applying the canonical simplex decomposition from Theorem 2.12 to .(m − 1)S0l , we obtain a decomposition into .l-cylinders of the form Tl (j1 , . . . , jl ) := S0j1 + Sj1 n + · · · + Sjl−1 n

.

for .1 ≤ (j1) < · · · < jl = n. We use induction to show that each .Tl (j1 , . . . , jl ) appears . ml times in the decomposition. The statement is trivial for .m = 1. So, let .m > 1. The polytope .Tl (j1 , . . . , jl ) appears when decomposing .(m − 1)S and when decomposing .(m − 1)S0jl−1 + Sjl−1 n . By the induction assumption, it appears (m−1) ( ) . times in the first case and . m−1 l l−1 times in the second case, which proves the

2 Valuations on Convex Bodies and Functions

39

claim. The .l-cylinder .Cl is obtained as union of translates (with pairwise disjoint interiors) of the convex .l-cylinders .Tl (j1 , . . . , jl ) for .1 ≤ j1 < · · · < jl = n. n u

2.4.3 The Homogeneous Decomposition Theorem The following result is fundamental in the theory of translation invariant valuations on convex bodies. Theorem 2.16 (McMullen) If . Z : Kn → R is a continuous, translation invariant valuation, then .

Z = Z0 + · · · + Zn

where . Zj : Kn → R is a continuous, translation invariant and j -homogeneous valuation. We will prove the result under the additional assumption that .Z is simple. This version is due to Hadwiger. The general case was stated without proof by Hadwiger and proved by McMullen [61]. We require the following proposition. Proposition 2.17 If . Z : Pn → R is a simple, translation invariant valuation, then .

Z = Z0 + · · · + Zn

where .Zj : Pn → R for .0 ≤ j ≤ n is a simple, translation invariant valuation that is homogeneous of degree j with respect to multiplication by positive integers. Proof Let .P ∈ Pn . By Theorem 2.15, for .1 ≤ j ≤ n, there are j -cylinders .Cj such that .

Z(m P ) =

n ( ) E m j =1

j

Z(Cj )

(2.26)

for every .m ≥ 1. Note that .m |→ Z(m P ) is a polynomial in m of degree at most n. We define .Zj (P ) as the coefficient of .mj of this polynomial. For .k, m ≥ 1, we obtain n E .

Zj (k P ) mj = Z(k m P ) =

j =1

n E

Zj (P ) (k m)j .

j =1

Therefore, .

Zj (k P ) = k j Zj (P ),

40

M. Ludwig and F. Mussnig

that is, .Zj is homogeneous of degree j with respect to multiplication with positive integers. To show that .Zj is a valuation, it suffices to show that .

Zj (P u Q) = Zj (P ) + Zj (Q).

This follows using (2.26) for .P uQ, P and Q and comparing coefficients of .mj . n u Proof of Theorem 2.16 for Simple Valuations First, we show that for non-negative λ ∈ Q,

.

.

Z(λP ) =

n E

Zj (P ) λj .

j =1

Indeed, let .λ = p/q with .p, q ∈ N. We have .q j Zj ( q1 P ) = Zj (P ) and

.

n E

Z( pq P ) =

Zj ( q1 P ) pj =

j =1

n E

Zj (P )

( p )j q

.

j =1

So far, the valuations .Zj are only defined on .Pn . Note that the system of equations,

.

Z(m P ) =

n E

Zj (P ) mj

j =1

for .m = 1, . . . , n with unknowns .Z1 (P ), . . . , Zn (P ) has a unique solution, as the matrix is just the Vandermonde matrix. This gives us explicit representations,

.

Zj (P ) =

n E

αij Z(i P )

i=1

with suitable .αij ∈ R independent of P , which we use as definition of .Zj on .Kn . It is easy to see that the resulting functionals are continuous, translation invariant valuations that are homogeneous of degree j . u n For fixed .K¯ ∈ Kn and a given continuous, translation invariant valuation .Z, ¯ defines a continuous, translation invariant, Lemma 2.1 shows that .K |→ Zj (K + K) j -homogeneous valuation on .Kn . We may use this argument repeatedly and obtain the following theorem, where we call a function .Z¯ : (Kn )m → R symmetric if it is not changed when its arguments are permutated.

2 Valuations on Convex Bodies and Functions

41

Theorem 2.18 Let .1 ≤ m ≤ n. If . Z : Kn → R is a continuous, translation invariant, m-homogeneous valuation, then there is a symmetric function .Z¯ : (Kn )m → R such that .

E

Z(λ1 K1 + · · · + λk Kk ) =

i1 ,...,ik ∈{0,...,m} i1 +···+ik =m

(

) m ¯ 1 [i1 ], . . . , Kk [ik ]) λi1 · · · λikk Z(K i 1 · · · ik 1

for every .k ≥ 1, every .K1 , . . . , Kk ∈ Kn and every .λ1 , . . . , λk ≥ 0. Moreover, .Z¯ is Minkowski additive in each variable and the map ¯ K |→ Z(K[j ], K1 , . . . , Km−j )

.

is a continuous, translation invariant, j -homogeneous valuation for .1 ≤ j ≤ m and every .K1 , . . . , Km−j ∈ Kn . Here, we write .K[j ] if K appears j times as an argument in .Z¯ while a function n .Y : K → R is called Minkowski additive if .

Y(K + L) = Y(K) + Y(L)

for every .K, L ∈ Kn . The special case .m = 1 in Theorem 2.18 leads to the following result. Corollary 2.19 If . Z : Kn → R is a continuous, translation invariant valuation that is homogeneous of degree 1, then .Z is Minkowski additive. Theorem 2.16 allows to reduce questions on continuous and translation invariant valuations to questions on such valuations with a given degree of homogeneity .j ∈ {0, . . . , n}. It is easy to see that every continuous, translation invariant, and 0-homogeneous valuation is a multiple of the Euler characteristic. For the degrees of homogeneity .j = n and .j = n − 1, we mention (without proofs) the following results by Hadwiger [39] and McMullen [62]. Theorem 2.20 (Hadwiger) A functional .Z : Pn → R is a translation invariant and n-homogeneous valuation if and only if there is a constant .c ∈ R such that .

for every .P ∈ Pn .

Z(P ) = c Vn (P )

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M. Ludwig and F. Mussnig

Theorem 2.21 (McMullen) A functional .Z : Kn → R is a continuous, translation invariant, .(n − 1)-homogeneous valuation if and only if there is .ζ ∈ C(Sn−1 ) such that f . Z(K) = ζ (y) dSn−1 (K, y) Sn−1

for every .K ∈ Kn . The function .ζ is uniquely determined up to addition of the restriction of a linear function. Here, .Sn−1 (K, ·) is the surface area measure of K. Continuous, translation invariant, 1-homogeneous valuations were classified by Goodey and Weil [37]. While a complete classification of continuous, translation invariant valuations on n .K is out of reach, Alesker [2] proved the following result. Theorem 2.22 (Alesker) For . 0 ≤ j ≤ n, the space of linear combinations of the valuations { .

K |→ V (K[j ], K1 , . . . , Kn−j ) : K1 , . . . , Kn−j ∈ Kn

}

is dense in the space of continuous, translation invariant, j -homogeneous valuations. Here, .V (K[j ], K1 , . . . , Kn−j ) is the mixed volume of .K ∈ Kn taken j times and n .K1 , . . . , Kn−j ∈ K while the topology on the space of continuous, translation invariant valuations is induced by the norm || Z || := sup{| Z(K)| : K ∈ Kn , K ⊆ B n }.

.

Alesker’s result confirms a conjecture by McMullen [62] and is based on Alesker’s so-called irreducibility theorem [2], which has further far-reaching consequences. For simple valuations, the following complete classification was established by Klain [40] and Schneider [71]. Theorem 2.23 (Klain and Schneider) A functional .Z : Kn → R is a continuous, translation invariant, simple valuation if and only if there are .c ∈ R and an odd function .ζ ∈ C(Sn−1 ) such that f .

Z(K) =

Sn−1

ζ (y) dSn−1 (K, y) + c Vn (K)

for every .K ∈ Kn . The function .ζ is uniquely determined up to addition of the restriction of a linear function. Klain [40] used his classification of simple valuations in his proof of the Hadwiger theorem. For an alternate proof of Theorem 2.23, see [44].

2 Valuations on Convex Bodies and Functions

43

A classification of weakly continuous, translation invariant valuations on .Pn was obtained by McMullen [63]. Here, a valuation is weakly continuous if it is continuous under parallel displacements of the facets of polytopes.

2.5 Rigid Motion Invariant Valuations The following rigid motion invariant, simple valuations are called Dehn invariants. For .P ∈ P3 and .ζ : [0, x) → [0, x) a Cauchy function with .ζ (π ) = 0, set .

Dζ (P ) :=

E

V1 (E) ζ (αP (E))

where the sum is taken over all edges E of P and .αP (E) is the dihedral angle of P at E. It is not difficult to see that .Dζ is a rigid motion invariant, simple valuation on 3 3 .P and that .Dζ vanishes on cubes. The regular tetrahedron T in .R has the dihedral angle .α := arccos(1/3) at every edge, and the ratio .α/π is irrational. Hence there are Cauchy functions with .ζ (α) /= 0. Hence .Dζ (T ) /= 0 for every regular simplex T . Since .Dζ is a rigid motion invariant, simple valuation, it follows from (2.2) that T is not equi-dissectable to any cube. This shows that Hilbert’s Third Problem has a negative answer (for an introduction to Hilbert’s Third Problem and the dissection theory of polytopes, see [16]). In general, .Dζ is far from being continuous. A complete classification of rigid motion invariant and continuous valuations on n .K was obtained by Hadwiger [39, Section 6.1.10] in his celebrated classification theorem. Theorem 2.24 (Hadwiger) A functional .Z : Kn → R is a continuous, translation and rotation invariant valuation if and only if there are constants .c0 , . . . , cn ∈ R such that .

Z(K) = c0 V0 (K) + · · · + cn Vn (K)

for every .K ∈ Kn . Hadwiger [39, Section 6.1.10] also obtained a complete classification of monotone increasing, translation and rotation invariant valuations by showing that any such valuation is a linear combination with non-negative coefficients of intrinsic volumes. McMullen [64] showed that every monotone increasing, translation invariant valuation is continuous. Hence the monotone version of Hadwiger’s theorem is a simple consequence of Theorem 2.24. We present a variation of Hadwiger’s original proof, which we got to know through lecture notes by Ulrich Betke. The main step is to prove the following result for simple valuations. Proposition 2.25 A functional .Z : Kn → R is a continuous, translation and rotation invariant, simple valuation if and only if there is a constant .c ∈ R such

44

M. Ludwig and F. Mussnig

that .

Z(K) = c Vn (K)

for every .K ∈ Kn . We first show how to deduce the Hadwiger theorem from this proposition and then describe its proof. An alternate proof of the Hadwiger theorem is due to Dan Klain [40]. It can also be found in [41] and [72]. Klain also uses the simple argument in the following subsection. Proof of the Hadwiger Theorem Using Proposition 2.25 We use induction on the dimension n and note that the statement is true for .n = 1 by Proposition 2.3. Assume that the statement is true in dimension .n − 1 and let E be an .(n − 1)-dimensional linear subspace of .Rn . The restriction of .Z to .K(E) is a continuous, translation and rotation invariant valuation on .K(E). By the induction assumption, there are constants .c0 , . . . , cn−1 ∈ R such that .

Z=

n−1 E

cj Vj

j =0

for every .K ∈ K(E). Define .Z˜ : Kn → R by Z˜ := Z −

n−1 E

.

cj Vj

j =0

and note that .Z˜ is a continuous, translation and rotation invariant valuation on .Kn . Moreover, .Z˜ is simple, as .Z˜ vanishes on .K(E) and hence, because of its translation and rotation invariance, on all convex bodies contained in an affine hyperplane. Using Proposition 2.25, we obtain that there is a constant .cn ∈ R such that ˜ Z(K) = cn Vn (K)

.

for every .K ∈ Kn . This concludes the proof.

u n

2.5.1 A Characterization of the Mean Width For .K ∈ Kn and .u ∈ Sn−1 , the support function of K in the direction u is hK (u) = sup

.

x∈K

and the width of K in direction u is .hK (u) + hK (−u). For the intrinsic volume .V1 , which is defined in (2.5), we have

2 Valuations on Convex Bodies and Functions

V1 (K) =

.

1 2 κn−1

45

f Sn−1

(hK (u) + hK (−u)) dHn−1 (u),

so .V1 (K) is proportional to the mean width of K. We say that a convex body M is a rotational Minkowski mean of .K ∈ Kn if there are rotations .ϑ1 , . . . , ϑm ∈ SO(n) such that M=

.

) 1( ϑ1 K + · · · + ϑm K . m

We require the following result due to Hadwiger [39, Section 4.5.3]. Theorem 2.26 For each .K ∈ Kn , there exists a sequence of rotational Minkowski means of K that converges to a centered ball. We remark that f y |→

hϑK (y) dϑ

.

SO(n)

is the support function of a centered ball associated with K, where integration is with respect to the Haar probability measure on .SO(n). Hence the sequence from Theorem 2.26 can be obtained by a suitable discretization. For a complete proof, see, for example, [72, Theorem 3.3.5]. Theorem 2.27 (Hadwiger) A functional .Z : Kn → R is continuous, translation and rotation invariant, and Minkowski additive if and only if there is a constant .c ∈ R such that .

Z(K) = c V1 (K)

for every .K ∈ Kn . Proof For .K ∈ Kn , Theorem 2.26 implies that there exists a sequence .(Kj ) of rotational Minkowski means of K with .Kj → rB n , where .rB n is a centered ball of radius r, where r depends on K. As every .Kj is of the form Kj =

.

) 1( ϑ1 K + · · · + ϑm K m

with suitable rotations .ϑ1 , . . . , ϑm , we have .Z(Kj ) = Z(K), as .Z is rotation invariant, Minkowski additive and homogeneous of degree 1. The continuity of .Z implies that .

Z(K) = lim Z(Kj ) = Z(rB n ) = r Z(B n ). j →x

46

M. Ludwig and F. Mussnig

The first intrinsic volume, .V1 , is continuous, translation and rotation invariant, and Minkowski additive. Hence we also have .V1 (K) = r V1 (B n ). Combined this gives n n .Z(K) = c V1 (K) with .c = Z(B )/V1 (B ). u n

2.5.2 Proof of Proposition 2.25 We use induction on the dimension n. The statement is true for .n = 1 by Proposition 2.3. Let .n ≥ 2 and assume that the statement is true for valuations defined on .Kk for .1 ≤ k ≤ n − 1. Let E and F be orthogonal and complementary subspaces with .dim E = k and .dim F = n−k. If we fix a convex body .KF in F , then the functional KE |→ Z(KE + KF )

.

is a valuation on .K(E) by Lemma 2.1 which is easily seen to be simple and continuous. Moreover, it is invariant with respect to translations and rotations in E. Hence, by the induction assumption, there is .c(KF ) ∈ R such that .

Z(KE + KF ) = c(KF ) Vk (KE )

for every .KE ∈ K(E). It follows from Lemma 2.1 that .c : K(F ) → R is a valuation, which is easily seen to be simple, continuous, translation and rotation invariant. Hence, by the induction assumption, there is .ck ∈ R such that .

Z(KE + KF ) = ck Vn−k (KF ) Vk (KE )

(2.27)

for every .KE ∈ K(E) and .KF ∈ K(F ). Since .Z is translation and rotation invariant, (2.27) holds for convex bodies .KE and .KF in any orthogonal and complementary subspaces E and F with .dim E = k. Evaluating on the unit cube, we obtain that .c1 = · · · = cn−1 =: c. Define .Z˜ : Kn → R by ˜ Z(K) = Z(K) − c Vn (K).

.

Note that .Z˜ is a simple, continuous, translation and rotation invariant valuation that vanishes on orthogonal cylinders. By Proposition 2.14, it is homogeneous of degree 1, and by Corollary 2.19, it is Minkowski additive. Using Theorem 2.27, we obtain that there is a constant .d ∈ R such that ˜ Z(K) = d V1 (K)

.

for every .K ∈ Kn . Since .Z˜ is simple and .n ≥ 2, we obtain that .d = 0 which concludes the proof of the theorem. .O

2 Valuations on Convex Bodies and Functions

47

2.5.3 Valuations Invariant Under Subgroups of O(n) For valuations invariant under the action of subgroups of the orthogonal group, O(n), Alesker [1, 3] obtained the following result.

.

Theorem 2.28 (Alesker) For a compact subgroup G of .O(n), the linear space of continuous, translation and G invariant valuations on .Kn is finite dimensional if and only if G acts transitively on .Sn−1 . As the classification of such subgroups G is known, it is a natural task (which was already proposed in [1]) to find bases for spaces of continuous, translation and G invariant valuations for all such subgroups (see [2, 10, 12] for some of the contributions).

2.5.4 An Application of the Hadwiger Theorem The following result is a special case of the principal kinematic formula, which is due to Blaschke, Chern, Federer and Santaló (see [41, 73]). We use integration with respect to the Haar measure on .SO(n) x Rn , and the normalization is chosen so that on .SO(n) we have the Haar probability measure, and translations are identified with n .R with the standard Lebesgue measure. Theorem 2.29 For .K, L ∈ Kn , f .

φ∈SO(n)xRn

V0 (K ∩ φL) dφ =

n E κi κn−i (n) Vi (K) Vn−i (L). κ n i i=0

Proof For .K, L ∈ Kn , set f .

Z(K, L) :=

φ∈SO(n)xRn

V0 (K ∩ φL) dφ.

For .L ∈ Kn , it is easy to see that .K |→ Z(K, L) is a continuous, translation and rotation invariant valuation on .Kn . By Theorem 2.24, there are .c0 (L), . . . , cn (L) ∈ R such that .

Z(K, L) =

n E

cj (L)Vj (K)

j =0

for every .K, L ∈ Kn . For given .K ∈ Kn , the functional .L |→ Z(K, L) is also a continuous, translation and rotation invariant valuation on .Kn . Combined with the homogeneity of intrinsic volumes, it follows that also each of the maps .L |→ cj (L)

48

M. Ludwig and F. Mussnig

for .0 ≤ j ≤ n is a continuous, translation and rotation invariant valuation. By Theorem 2.24, there are .c0j , . . . , cnj ∈ R such that

.

n E

Z(K, L) =

cij Vi (K) Vj (L)

i,j =0

for every .K, L ∈ Kn . The constants .cij can be determined by evaluating this formula for suitable convex bodies K and L. u n

2.6 Valuations on Function Spaces We will extend and generalize valuations from (subsets of) the space of convex bodies to function spaces. Let .F (Rn ; R) denote the space of all real-valued functions on .Rn .

2.6.1 Definition One way to represent the set of convex bodies, .Kn , within .F (Rn ; R) is to assign to each body .K ∈ Kn its characteristic function .χK ∈ F (Rn ; R), which is given by χK (x) =

{ 1

.

0

for x ∈ K for x ∈ / K.

Using this embedding we assign to each functional .Z : F (Rn ; R) → R the functional .Z˜ : Kn → R by setting ˜ Z(K) := Z(χK )

.

for every .K ∈ Kn . We now ask which conditions .Z needs to satisfy so that .Z˜ is a valuation. By the definition of .Z˜ we have .

˜ ˜ Z(χK ) + Z(χL ) = Z(K) + Z(L) ˜ ˜ = Z(K ∩ L) + Z(K ∪ L) = Z(χK∩L ) + Z(χK∪L ) = Z(χK ∧ χL ) + Z(χK ∨ χL )

2 Valuations on Convex Bodies and Functions

49

for every .K, L ∈ Kn such that also .K ∪ L ∈ Kn . Here, .f ∨ g and .f ∧ g denote the pointwise maximum and minimum of .f, g ∈ F (Rn ; R), respectively. This motivates the following definition. Definition 2.30 Let .X ⊆ F (Rn ; R). A map .Z : X → R is a valuation if .

Z(f ) + Z(g) = Z(f ∧ g) + Z(f ∨ g)

for every .f, g ∈ X such that also .f ∧ g, f ∨ g ∈ X. Similarly, one may define valuations with values in any Abelian semigroup. Examples include vector-valued, matrix-valued, measure-valued valuations, and even Minkowski valuations, which are valuations with values in the space of convex bodies equipped with Minkowski addition. We remark that there are various ways to represent convex bodies within the space of real-valued functions on .Rn . For many such representations, pointwise maxima and minima of functions correspond to unions and intersections of bodies. Repeating the above steps for other embeddings of .Kn into .F (Rn ; R) leads to the same definition of valuations on (subsets of) .F (Rn ; R).

2.6.2 First Examples As for valuations on convex bodies, the simplest valuation .Z : X → R for any X ⊆ F (Rn ; R) is of the form .Z(f ) = c with .c ∈ R. It is straightforward to check that this defines a valuation, although it may f not be the most interesting one. Next, let .X ⊆ {f ∈ F (Rn ; R) : | Rn f (x) dx| < x}, where we consider Lebesgue integrals. Define .Z : X → R as

.

f .

Z(f ) :=

Rn

f (x) dx.

We claim that .Z is a valuation. Let .f, g ∈ X. Since .Rn can be represented as the disjoint union Rn = {f ≥ g} u {f < g},

.

50

M. Ludwig and F. Mussnig

where .{f ≥ g} := {x ∈ Rn : f (x) ≥ g(x)} and .{f < g} is defined accordingly, we have f f Z(f ) + Z(g). = f (x) dx + g(x) dx f =

Rn

Rn

f

{f ≥g}

f (x) dx +

f

+

f

{f ≥g}

f =

{f ≥g}

g(x) dx +

{f 0. Note that this implies {u ≤ t} ⊆ {x : a|x| ≤ t} = at B n

.

2 Valuations on Convex Bodies and Functions

57

for every .t ≥ 0. Combined with (2.34), this implies that f 0≤

.

Rn

e−u(x) dx =

f f

+x

−x +x

≤

Vn 0

= =

Vn ({u ≤ t})e−t dt

κn an

f

(t

+x

aB

n

) −t e dt

t n e−t dt

0

κn n! . an

In the general case there exist .a > 0 and .b ∈ R such that .u(x) ≥ a|x| + b for x ∈ Rn and thus, since .u(x) − b ≥ a|x| for .x ∈ Rn , we have

.

f 0≤

.

Rn

e−u(x) dx = e−b

f Rn

e−(u(x)−b) dx ≤ κn e−b

n! . an

(2.35)

The fact that (2.33) defines a valuation follows from (2.34) and (2.29) together with the fact that the n-dimensional volume, .Vn , is a valuation on convex bodies. Similarly, using (2.31), it is easy to obtain .SL(n) and translation invariance. The valuation property can be proved analogous to (2.28). It remains to show continuity. Let .uk be a sequence in .Convcoe (Rn ) that epiconverges to .u ∈ Convcoe (Rn ). By Lemma 2.35, there exist .a > 0 and .b ∈ R such that (2.30) holds. Thus, similar to (2.35), we obtain f 0≤

.

Rn

e−uk (x) dx ≤ κn e−b

n! an

for every .k ∈ N. Since the n-dimensional volume is continuous with respect to Hausdorff convergence, we have .Vn ({uk ≤ t}) → Vn ({u ≤ t}) as .k → x for almost every .t ∈ R. Therefore, by the dominated convergence theorem, we obtain that f +x f f . lim e−uk (x) dx = Vn ({u ≤ t}) e−t dt = e−u(x) dx, k→x Rn

Rn

−x

and thus (2.33) is continuous. In order to establish the properties of (2.32), observe that − minx∈Rn u(x)

e

.

f =

+x

−t

e minx∈Rn u(x)

f dt =

+x −x

V0 ({u ≤ t}) e−t dt

for every .u ∈ Convcoe (Rn ). Thus, it is easy to see that the same arguments as above can be applied. u n

58

M. Ludwig and F. Mussnig

The proof above demonstrates the simple strategy for finding a functional analog of an operator .Z : Kn → R by considering the map f u |→

+x

.

−x

Z({u ≤ t}) e−t dt

on .Convcoe (Rn ), where we set .Z(∅) := 0. Indeed, in many cases, this will define an operator on . Convcoe (Rn ) with similar properties as the original operator .Z. More generally, one can often replace .e−t dt by a suitable measure .dμ(t). For some examples, see [15, 65].

2.7.3 A Functional Analog of Blaschke’s Result Similar to Blaschke’s characterization of the Euler characteristic and the volume, Theorem 2.2, we can characterize their functional analogs, which we introduced in the last section. We will prove the following result. Theorem 2.37 For .n ≥ 2, a map .Z : Convcoe (Rn ) → R is a continuous, .SL(n) and translation invariant valuation such that .

Z(u + t) = e−t Z(u)

(2.36)

for every .u ∈ Convcoe (Rn ) and .t ∈ R, if and only if there are constants .c0 , cn ∈ R such that f − minx∈Rn u(x) . Z(u) = c0 e + cn e−u(x) dx (2.37) Rn

for every .u ∈ Convcoe (Rn ). We remark that if we omit condition (2.36), additional valuations will appear in a classification result [25, 66, 67]. However, all known proofs of such results are considerably more involved than the proof of Theorem 2.37, which we present here. Observe that (2.36) becomes more natural if we consider the corresponding result on the space of log-concave functions, LCcoe (Rn ) = {e−u : u ∈ Convcoe (Rn )}.

.

The properties of valuations on .LCcoe (Rn ) are defined analogously to the corresponding properties for valuations on .Convcoe (Rn ). The following result, which is equivalent to Theorem 2.37, is a consequence of [68, Theorem 4].

2 Valuations on Convex Bodies and Functions

59

Theorem 2.38 For .n ≥ 2, a map .Z : LCcoe (Rn ) → R is a continuous, .SL(n) and translation invariant valuation such that Z(sf ) = s Z(f )

.

for every .f ∈ LCcoe (Rn ) and .s > 0, if and only if there are constants .c0 , cn ∈ R such that f . Z(f ) = c0 max f (x) + cn f (x) dx n Rn

x∈R

for every .f ∈ LCcoe (Rn ). To prove Theorem 2.37, we will start with the following simple observation. For K ∈ Kno , where .Kno = {K ∈ Kn : 0 ∈ K}, and .x ∈ Rn , set

.

gK (x) := min{λ > 0 : x ∈ λK}.

.

Then .gK is the gauge function or Minkowski functional of K. Since .0 ∈ K, it follows that .gK ∈ Convcoe (Rn ) with {gK ≤ t} = tK

.

for every .t ≥ 0 and .{gK ≤ t} = ∅ for every .t < 0. Lemma 2.39 Let .n ≥ 2. If .Z : Convcoe (Rn ) → R is a continuous and .SL(n) invariant valuation that satisfies (2.36), then there are constants .c0 , cn ∈ R such that f − minx∈Rn (gK (x)+t) . Z(gK + t) = c0 e + cn e−(gK (x)+t) dx Rn

for every .K ∈ Kno and .t ∈ R. Proof Since gK ∨ gL = gK∩L

.

and

gK ∧ gL = gK∪L

for every .K, L ∈ Kno such that .K ∪ L ∈ Kno and convergence of .Kj to K on .Kno implies gKj → gK ,

.

it is easy to see that ˜ Z(K) := Z(gK )

.

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M. Ludwig and F. Mussnig

defines a continuous valuation on .Kno . Furthermore, since .Z is .SL(n) invariant, also ˜ has this property. Thus it follows from Theorem 2.7 that there exist .c˜0 , c˜n ∈ R .Z such that ˜ Z(K) = c˜0 V0 (K) + c˜n Vn (K)

.

for every .K ∈ Kno . Note that this also implies that .Z˜ is translation invariant, which is not evident from its definition. Next, observe that e− minx∈Rn (gK (x)+t) = e−t = e−t V0 (K)

.

and .

1 n!

f Rn

e−(gK (x)+t) dx =

e−t n!

f

x

Vn (sK)e−s ds = e−t Vn (K)

0

for every .K ∈ Kno and .t ∈ R, where we used a similar computation as in (2.34). The result now follows by setting .c0 := c˜0 and .cn := c˜n /n!, since .

Z(gK + t) = e−t Z(gK ) ˜ = e−t Z(K) = c˜0 e−t V0 (K) + c˜n e−t Vn (K) f = c0 e− minx∈Rn (gK (x)+t) + cn

Rn

for every .K ∈ Kno and .t ∈ R.

e−(gK (x)+t) dx u n

We will use the following reduction principle, which was first established for valuations on Sobolev spaces in [52]. For simplicity, we will present the proof in dimension one and remark that its extension to higher dimensions is straightforward. See, for example, [66, Lemma 5.1]. Lemma 2.40 Let .Z1 , Z2 : Convcoe (Rn ) → R be continuous, translation invariant valuations. If .

Z1 (gP + t) = Z2 (gP + t)

for every polytope .P ∈ Pno and .t ∈ R, then .

for every .u ∈ Convcoe (Rn ).

Z1 (u) = Z2 (u)

(2.38)

2 Valuations on Convex Bodies and Functions

61

Proof (for .n = 1) Since .Z1 and .Z2 are continuous, it is enough to consider the case where .u ∈ Convcoe (Rn ) is such that u=

m A

.

wi

i=1

with affine functions .wi : Rn → R. In addition, we may also assume that the graph of u has no edges parallel to the coordinate axis. We will use induction on the number k of vertices of the graph of u. If .k = 1, then u must be of the form u(x) = gP (x − x0 ) + t

.

for some polytope .P ∈ Pno , .t ∈ R and .x0 ∈ R. Thus, it follows from translation invariance and (2.38) that .Z1 (u) = Z2 (u). Assume now that the statement is true for .k − 1 and let the graph of u have k vertices. Denote by .(x, ¯ t¯) ∈ R2 one of the highest vertices. It is easy to see that we can find a polytope .P¯ ∈ Pno such that the graph of u ∧ u¯ has k − 1 vertices,

.

where .u¯ ∈ Convcoe (Rn ) is given by .u(x) ¯ := gP¯ (x − x) ¯ + t¯ for .x ∈ Rn . See Fig. 2.4.

Fig. 2.4 Illustration of u and .u¯ for the case .k = 3

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Since the graph of .u ∨ u¯ has only one vertex, it follows from our induction assumption and the valuation property that .

Z1 (u) = Z1 (u ∨ u) ¯ + Z1 (u ∧ u) ¯ − Z1 (u) ¯ = Z2 (u ∨ u) ¯ + Z2 (u ∧ u) ¯ − Z2 (u) ¯ = Z2 (u), u n

which completes the proof. We can now proceed with the proof of our classification result.

Proof of Theorem 2.37 If .Z : Convcoe (Rn ) → R is as in (2.37), then it follows from Lemma 2.36 that .Z is a continuous, .SL(n) and translation invariant valuation. Furthermore, it is easy to see that it satisfies (2.36). Conversely, let .Z : Convcoe (Rn ) → R be a continuous, translation and .SL(n) invariant valuation that satisfies (2.36). By Lemma 2.39, there exist .c0 , cn ∈ R such that f − minx∈Rn (gK (x)+t) . Z(gK + t) = c0 e + cn e−(gK (x)+t) dx Rn

for every .K ∈ Kno and .t ∈ R. Define now .Z¯ : Convcoe (Rn ) → R as f − minx∈Rn u(x) ¯ .Z(u) := c0 e + cn e−u(x) dx. Rn

By the first part of the proof, this is a continuous and translation invariant valuation ¯ P + t) = Z(gP + t) for every .P ∈ Pno and .t ∈ R. The result now such that .Z(g follows from Lemma 2.40. u n

2.7.4 Homogeneity For .u ∈ Convcoe (Rn ) and .λ > 0, define .λ o u ∈ Convcoe (Rn ) by (x ) .λ o u(x) := u λ for .x ∈ Rn . This definition is motivated by the fact that for .t ∈ R, {λ o u ≤ t} = λ{u ≤ t}.

.

For .p ∈ R, an operator .Z : Convcoe (Rn ) → R is horizontally p-homogeneous if .

Z(λ o u) = λp Z(u)

for every .u ∈ Convcoe (Rn ) and .λ > 0.

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It is straightforward to see .u |→ e− minx∈Rn u(x) is horizontally 0f that −u(x) homogeneous and that .u |→ Rn e dx is horizontally n-homogeneous. One might hope that similar to McMullen’s decomposition theorem for valuations on .Kn , Theorem 2.16, every continuous and translation invariant valuation on .Convcoe (Rn ) can be written as a sum of horizontally homogeneous valuations. However, such a result fails, and counterexamples were constructed in [27, Theorem 1.2], where the following classes of functionals were studied. For .ζ ∈ Cc (R × Rn ), the space of continuous functions with compact support on .R × Rn , consider f u |→

ζ (u(x), ∇u(x)) dx,

.

dom u

which is well-defined since convex functions are differentiable almost everywhere on the interior of their domains. More general examples can be written as f u |→

.

Rn

ζ (u(x), ∇u(x)) [D2 u(x)]i dx,

(2.39)

if in addition .u ∈ C 2 (Rn ). Here .D2 u(x) denotes the Hessian matrix of u at x ∈ Rn and .[D2 u(x)]i the ith elementary symmetric function of its eigenvalues for n .0 ≤ i ≤ n. We remark that (2.39) can be extended to general .u ∈ Convcoe (R ) where essentially .[D2 u(x)]i dx is replaced by the so-called Hessian measures. The examples above are continuous and translation invariant. Still, due to their dependence on the gradient of the convex function, they cannot be decomposed into homogeneous terms for all .ζ ∈ Cc (R × Rn ) (see [27] for details). Nevertheless, we will establish a functional analog of Theorem 2.16 in Sect. 2.8.2. .

2.8 Valuations on Super-Coercive Convex Functions To obtain a homogeneous decomposition theorem, we will restrict to the smaller space of super-coercive convex functions and shift our attention from sublevel sets to epi-graphs.

2.8.1 Definitions and First Examples We consider the space of super-coercive convex functions, { Convsc (Rn ) := u ∈ Conv(Rn ) :

.

} u(x) = +x . |x|→+x |x| lim

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Obviously, .Convsc (Rn ) is a subspace of .Convcoe (Rn ). Note that for differentiable n .u ∈ Convsc (R ), the property .

u(x) = +x |x|→+x |x| lim

implies that also .

lim

|x|→+x

|∇u(x)| = +x.

(2.40)

The space of super-coercive convex functions is, in the following way, closely connected to the space of finite-valued convex functions, Conv(Rn ; R) := {v : Rn → R : v is convex}.

.

Recall that the Legendre transform is defined for .u ∈ Conv(Rn ) by ( ) u∗ (x) := sup − u(y)

.

y∈Rn

for .x ∈ Rn . By standard properties of the Legendre transform, we now have {u∗ : u ∈ Convsc (Rn )} = Conv(Rn ; R).

.

(2.41)

This relation allows us to translate results for valuations on .Convsc (Rn ) easily to results on .Conv(Rn ; R) and vice versa. A valuation .Z : Convsc (Rn ) → R is epi-translation invariant if it is vertically translation invariant in addition to having the usual translation invariance, that is, if .

Z(u ◦ τ −1 + α) = Z(u)

for every .u ∈ Convsc (Rn ), translation .τ on .Rn and .α ∈ R. Note that this means that Z is invariant under translations of the epi-graph of u in .Rn+1 , where the epi-graph of u is given by

.

.

epi u := {(x, t) ∈ Rn × R : u(x) ≤ t}

and is a closed, convex subset of .Rn+1 for every .u ∈ Conv(Rn ). For .u, v ∈ Convsc (Rn ), let ( ) (u O v)(x) := infn u(x − y) + v(y)

.

y∈R

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denote their infimal convolution at .x ∈ Rn . Note that also .u O v ∈ Convsc (Rn ) and that .

epi(u O v) = epi u + epi v,

where the addition on the right side is Minkowski addition of closed, convex sets in Rn+1 . The infimal convolution is also called epi-addition. It naturally induces the following operation. For .λ > 0 and .u ∈ Convsc (Rn ), define .λ u ∈ Convsc (Rn ) as

.

(λ u)(x) := λ u

.

(x ) λ

for .x ∈ Rn . In addition, set .0 u := I{0} . It is easy to see that epi(λ u) = λ epi u

.

for every .λ > 0 and .u ∈ Convsc (Rn ) and that k u = 'u O ·'' · · O u'

.

k times

for every .k ∈ N and .u ∈ Convsc (Rn ). For .p ∈ R, a valuation .Z : Convsc (Rn ) → R is epi-homogeneous of degree p if .

Z(λ u) = λp Z(u)

for every .u ∈ Convsc (Rn ) and .λ > 0. We will present two examples of continuous, epi-homogeneous, and epi-translation invariant valuations on .Convsc (Rn ). First, it is easy to see that the constant map .u |→ c for .c ∈ R defines a continuous, epi-translation invariant valuation that is epi-homogeneous of degree 0. In fact, it is the only valuation with these properties (see [28, Theorem 25]). Next, let .ζ ∈ Cc (Rn ), that is, .ζ is continuous with compact support. Consider the map f .

Z(u) =

ζ (∇u(x)) dx

(2.42)

dom u

for .u ∈ Convsc (Rn ). Because of (2.40), it is at least plausible that .Z(u) is welldefined and finite for every .u ∈ Convsc (Rn ). On the other hand, the example .u(x) := |x| shows that .Z defined by (2.42) is not a well-defined (finite) map on the larger space . Convcoe (Rn ). We will state the following result from [28, Proposition 20] without proof.

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Lemma 2.41 For .ζ ∈ Cc (Rn ), the map f u |→

ζ (∇u(x)) dx

.

dom u

defines a continuous and epi-translation invariant valuation on .Convsc (Rn ), which is epi-homogeneous of degree n. We remark that in the proof of this lemma, the map f v |→

.

dom v ∗

ζ (∇v ∗ (x)) dx

on .Conv(Rn ; R) is considered, which is well-defined by (2.41). This map can be written as f .v |→ ζ (x) dMA(v; x) Rn

for .v ∈ Conv(Rn ; R). Here, .MA(v; ·), the Monge–Ampère measure of v, is a Radon measure on .Rn , which can be defined as a continuous extension of the measure 2 2 n n .det(D v(x)) dx from .C (R ) to . Conv(R ; R). The valuation .Z defined in (2.42) can be seen as a further generalization of ndimensional volume on convex bodies to convex functions. Indeed, we have f . Z(IK ) = ζ (0) dx = ζ (0)Vn (K) (2.43) K

for every .K ∈ Kn . See also Sect. 2.8.4.

2.8.2 A Homogeneous Decomposition Theorem We will prove a functional analog from [28] of the homogeneous decomposition theorem, Theorem 2.16. Theorem 2.42 If .Z : Convsc (Rn ) → R is a continuous, epi-translation invariant valuation, then .

Z = Z0 + · · · + Zn

where .Zj : Convsc (Rn ) → R is a continuous, epi-translation invariant valuation that is epi-homogeneous of degree j .

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Similar to the proof of Theorem 2.37, we first show that the result is true on a restricted set of functions and then use a reduction argument. For .y ∈ Rn , define n .ly : R → R by .ly (x) := . Lemma 2.43 Let . Z : Convsc (Rn ) → R be a continuous, epi-translation invariant valuation. For every .y ∈ Rn , the map .Z˜ y : Kn → R, defined by Z˜ y (K) := Z(ly + IK ),

.

is a continuous and translation invariant valuation. Proof Continuity and the valuation property are easy to obtain. For translation invariance, observe that ly (x) + IK+x0 (x) = + IK (x − x0 )

.

= + IK (x − x0 ) + = ly (x − x0 ) + IK (x − x0 ) + for every .K ∈ Kn and .x, x0 , y ∈ Rn . In other words, the epi-graph of .ly + IK+x0 is a translate of the epi-graph of .ly + IK . Thus, by the epi-translation invariance of .Z, we now obtain that Z˜ y (K + x0 ) = Z(ly + IK+x0 ) = Z(ly + IK ) = Z˜ y (K)

.

for every .K ∈ Kn and .x0 , y ∈ Rn .

u n

Next, we use Theorem 2.16 to show that Theorem 2.42 holds on functions of the form .ly + IK with .y ∈ Rn and .K ∈ Kn . Lemma 2.44 If . Z : Convsc (Rn ) → R is a continuous and epi-translation invariant valuation, then for every .u = ly + IK with .y ∈ Rn and .K ∈ Kn , .

Z(u) = Z0 (u) + · · · + Zn (u)

where .Zi : Convsc (Rn ) → R is a continuous and epi-translation invariant valuation such that .

Zi (λ u) = λi Z(u)

for every .λ ≥ 0 and .u = ly + IK with .y ∈ Rn and .K ∈ Kn . Proof For .y ∈ Rn , define .Z˜ y : Kn → R by .Z˜ y (K) := Z(ly + IK ). It follows from Lemma 2.43 and Theorem 2.16 that for every .y ∈ Rn there exist continuous,

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translation invariant and i-homogeneous valuations .Z˜ y,i : Kn → R for .0 ≤ i ≤ n such that Z˜ y (K) =

n E

.

Z˜ y,i (K)

i=0

for every .K ∈ Kn . Thus, ˜ y (λK) = . Z(λ (ly + IK )) = Z(ly + IλK ) = Z

n E

λi Z˜ y,i (K)

i=0

for every .K ∈ Kn , .y ∈ Rn and .λ ≥ 0, where .00 := 1. Setting .λ := j for every .0 ≤ j ≤ n, we therefore obtain ⎞ ⎛ 0 0 ··· Z(0 (ly + IK )) ⎟ ⎜ ⎜ . . .. .⎝ ⎠ = ⎝ .. . . . n0 · · · Z(n (ly + IK )) ⎛

⎞ ⎞⎛ 0n Z˜ y,0 (K) ⎟ .. ⎟ ⎜ .. ⎠ . ⎠⎝ . nn Z˜ y,n (K)

for every .K ∈ Kn and .y ∈ Rn . The matrix in the equation is invertible since it is a Vandermonde matrix. Denoting its inverse by .(αij )0≤i,j ≤n , we now have ˜ y,i (K) = .Z

n E

αij Z(j (ly + IK ))

j =0

for .0 ≤ i ≤ n and every .K ∈ Kn and .y ∈ Rn . Since the coefficients .αij are independent of .K ∈ Kn and .y ∈ Rn , we may now define .Zi : Convsc (Rn ) → R as .

Zi (u) :=

n E

αij Z(j u)

j =0

for every .0 ≤ i ≤ n. It easily follows from the properties of .Z that also the functionals .Zi for .0 ≤ i ≤ n are continuous and epi-translation invariant valuations. We now have .

Zi (ly + IK ) =

n E

αij Z(j (ly + IK )) = Z˜ y,i (K)

j =0

and thus .

Zi (λ (ly + IK )) = Zi (ly + IλK ) = Z˜ y,i (λK) = λi Z˜ y,i (K) = λi Zi (ly + IK )

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for every .0 ≤ i ≤ n, .K ∈ Kn , .y ∈ Rn and .λ > 0. Furthermore, .

Z(ly + IK ) = Z˜ y (K) =

n E

Z˜ y,i (K) =

i=0

n E

Zi (ly + IK )

i=0

for every .K ∈ Kn and .y ∈ Rn , which completes the proof.

u n

We will now show that every continuous, epi-translation invariant valuation on Convsc (Rn ) is already determined by its values on a particular small set of functions.

.

Lemma 2.45 Let . Z : Convsc (Rn ) → R be a continuous, epi-translation invariant valuation. If .

Z(ly + IP ) = 0

(2.44)

for every .y ∈ Rn and .P ∈ Pn , then .Z(u) = 0 for every .u ∈ Convsc (Rn ). We present two approaches to prove this result. Similar to the proof of Lemma 2.40, it follows from the continuity of .Z that we may reduce to the case that .u ∈ Convsc (Rn ) is piecewise affine. Here, this means that there exist polytopes .P1 , . . . , Pm ∈ Pn with pairwise disjoint interiors and affine functions n .w1 , . . . , wm : R → R such that u=

m A

.

(wi + IPi ),

(2.45)

i=1

that is, u is a piecewise minimum of affine functions restricted to disjoint polytopes. Proof (First Approach) We prove the result by induction on the number m in (2.45). We start with the case .m = 1. Since there exist .y1 ∈ Rn and .t ∈ R such that .w1 = ly + t, we have .u = ly1 + IP1 + t and thus, by the epi-translation invariance of .Z, it follows from (2.44) that .Z(u) = 0. Assume that the statement is true for .m − 1 and let u have m components in the representation (2.45). Without loss of generality, we may assume that there exist disjoint index sets .I1 , I2 ⊂ {1, . . . , m} such that .0 < |I1 |, |I2 | < m and such that the sets || || . Pi and Pi i∈I1

i∈I2

are convex (for a more detailed discussion of such a partition, we refer to [21, Section 7.1]). Let .u1 , u2 ∈ Convsc (Rn ) be defined as u1 :=

A

.

i∈I1

(wi + IPi )

and

u2 :=

A i∈I2

(wi + IPi ).

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Clearly, .u = u1 ∧ u2 and, by our induction assumption, .Z(u1 ) = Z(u2 ) = 0. Furthermore, it is easy to see that if .u¯ := u1 ∨ u2 , then there exist polytopes n .P¯1 , . . . , P¯k ∈ P (which all have to be at most .(n − 1)-dimensional) with .k ≤ m − 1 and affine functions .w¯ 1 , . . . , w¯ k such that u¯ =

k A

.

(w¯ i + IP¯i ).

i=1

Using the induction assumption again, we see that also .Z(u) ¯ = 0. Thus, by the valuation property of .Z, .

Z(u) = Z(u1 ) + Z(u2 ) − Z(u) ¯ = 0, u n

which completes the proof.

Proof (Second Approach) Similar to (2.1), for every continuous valuation .Z¯ on n n n .Convsc (R ) and .u1 , . . . um ∈ Convsc (R ) such that .u1 ∧ . . . ∧ um ∈ Convsc (R ), we have E ¯ 1 ∧ · · · ∧ um ) = ¯ I ), .Z(u (−1)|I |−1 Z(u ∅/=I ⊂{1,...,m}

V where .uI := i∈I ui . Thus, in order to show that .Z vanishes on functions of the form (2.45), it suffices to show that .

Z

(v ) (wi + IPi ) = 0 i∈I

V for every .∅ /= I ⊂ {1, . . . , m}. Since every such function . i∈I (wi + IPi ) is again an affine function restricted to a polytope, the statement follows from (2.44) and the vertical translation invariance of .Z. u n We now have all ingredients to prove the homogeneous decomposition theorem. Proof of Theorem 2.42 Let the valuations .Z0 , . . . , Zn : Convsc (Rn ) → R be given by Lemma 2.44 and define .Z¯ : Convsc (Rn ) → R as ¯ Z(u) := Z(u) −

n E

.

Zi (u).

i=0

Clearly, .Z¯ is a continuous and epi-translation invariant valuation. Furthermore, by the properties of the valuations .Zi for .0 ≤ i ≤ n, we have ¯ y + IP ) = 0 Z(l

.

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for every polytope .P ∈ Pn and .y ∈ Rn . Thus, by Lemma 2.45, .

Z(u) =

n E

Zi (u)

i=0

for every .u ∈ Convsc (Rn ). It remains to show that the valuation .Zi is epi-homogeneous of degree i for each .0 ≤ i ≤ n. For .λ ≥ 0 and .0 ≤ i ≤ n, set Z¯ λ,i (u) = Zi (λ u) − λi Zi (u)

.

for .u ∈ Convsc (Rn ). Note that .Z¯ λ,i is a valuation on .Convsc (Rn ). Using the same arguments as above, we obtain that .Z¯ λ,i ≡ 0, which shows that for each .0 ≤ i ≤ n, u n the valuation .Zi is epi-homogeneous of degree i. With an approach similar to that for Theorem 2.18, we obtain the following result by considering .u |→ Zi (u O u) ¯ for fixed .u¯ ∈ Convsc (Rn ). Theorem 2.46 Let .1 ≤ m ≤ n. If .Z : Convsc (Rn ) → R is a continuous, epitranslation invariant valuation that is epi-homogeneous of degree m, then there is a symmetric function .Z¯ : (Convsc (Rn ))m → R such that .

E

Z(λ1 u1 O · · · O λk uk ) =

i1 ,...,ik ∈{0,...,m} i1 +···+ik =m

(

) m ¯ 1 [i1 ], . . . , uk [ik ]) λi1 · · · λikk Z(u i 1 · · · ik 1

for every .k ≥ 1, every .u1 , . . . , uk ∈ Convsc (Rn ) and every .λ1 , . . . , λk ≥ 0. Moreover, .Z¯ is epi-additive in each variable and the map ¯ u |→ Z(u[j ], u1 , . . . , um−j )

.

is a continuous, epi-translation invariant valuation on .Convsc (Rn ) that is epihomogeneous of degree j for .1 ≤ j ≤ m and every .u1 , . . . , um−j ∈ Convsc (Rn ). Here, a function .Y : Convsc (Rn ) → R is called epi-additive if .

Y(u O v) = Y(u) + Y(v)

for every .u, v ∈ Convsc (Rn ). The special case .m = 1 in the previous theorem leads to the following result, which is a functional version of Corollary 2.19. Corollary 2.47 If . Z : Convsc (Rn ) → R is a continuous, epi-translation invariant valuation that is epi-homogeneous of degree 1, then .Z is epi-additive.

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2.8.3 A Classification Result In this section, we will show that the valuations described in Lemma 2.41 are indeed the only continuous and epi-translation invariant valuations on .Convsc (Rn ) which are epi-homogeneous of degree n. The following result was established in [28, Theorem 2]. Theorem 2.48 A map .Z : Convsc (Rn ) → R is a continuous and epi-translation invariant valuation that is epi-homogeneous of degree n, if and only if there exists n .ζ ∈ Cc (R ) such that f . Z(u) = ζ (∇u(x)) dx (2.46) dom u

for every .u ∈ Convsc (Rn ). Proof For given .ζ ∈ Cc (Rn ), it follows from Lemma 2.41 that (2.46) has the desired properties. Conversely, let .Z : Convsc (Rn ) → R be a continuous, epi-translation invariant valuation that is epi-homogeneous of degree n. For .y ∈ Rn , let .Z˜ y : Kn → R be defined by .Z˜ y (K) := Z(ly + IK ). By Lemma 2.43, the functional .Z˜ y is a continuous and translation invariant valuation. In addition, it is easy to see that .Z˜ y is homogeneous of degree n. Thus, it follows from Theorem 2.20 that for every n .y ∈ R there exists a constant .ζ (y) ∈ R such that .

Z(ly + IK ) = Z˜ y (K) = ζ (y)Vn (K)

(2.47)

for every .K ∈ Kn . Furthermore, it follows from the continuity of .Z that .ζ (y) continuously depends on .y ∈ Rn . This defines a continuous function .ζ : Rn → R. Next, we show that .ζ has compact support. Assume on the contrary that there exists a sequence .yk ∈ Rn with lim |yk | = +x

.

k→x

(2.48)

but .ζ (yk ) /= 0 for every .k ∈ N. By possibly restricting to a subsequence, we may assume without loss of generality that .ζ (yk ) is positive for every .k ∈ N and that there exists a vector .e ∈ Sn−1 such that .

yk = e. k→x |yk | lim

Let .Bk , Bx ∈ Kn be given by Bk = {x ∈ yk⊥ : |x| ≤ 1},

.

Bx = {x ∈ e⊥ : |x| ≤ 1}

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73

and let .Ck ∈ Kn be defined as [ { yk 1 ]} .Ck = x + t : x ∈ Bk , t ∈ 0, |yk | ζ (yk )

(2.49)

for .k ∈ N. Observe that .Ck is an orthogonal cylinder and that Vn (Ck ) = Vn−1 (Bk )

.

1 κn−1 = ζ (yk ) ζ (yk )

(2.50)

for .k ∈ N. Next, set .uk := lyk + ICk for .k ∈ N and note that .uk ∈ Convsc (Rn ). It follows from (2.48) and (2.49) that .uk → IBx as .k → x. Thus, the continuity of .Z combined with (2.47) implies that 0 = Z(IBx ) = lim Z(uk ).

.

k→x

On the other hand, it follows from (2.47) and (2.50) that .

Z(uk ) = ζ (yk )Vn (Ck ) = κn−1 > 0

for every .k ∈ N, which is a contradiction. Hence, we conclude that .ζ has compact support. It remains to show that (2.46) holds. Define .Z¯ : Convsc (Rn ) → R as f ¯ := Z(u) − .Z(u) ζ (∇u(x)) dx. dom u

By Lemma 2.41 and our assumptions on .Z, the operator .Z¯ is a continuous and epitranslation invariant valuation. Furthermore, it follows from (2.47) that f ¯ y + IK ) = Zy (K) − .Z(l ζ (y) dx = 0 K

for every .y ∈ Rn and .K ∈ Kn . Thus, Lemma 2.45 implies that ¯ Z(u) =0

.

for every .u ∈ Convsc (Rn ), which completes the proof.

2.8.4 A Glimpse at the Current State of Research As pointed out in (2.43), for any .ζ ∈ Cc (Rn ), the operator f u |→

ζ (∇u(x)) dx

.

dom u

u n

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can be seen as a functional analog of the n-dimensional volume on .Convsc (Rn ). This interpretation is further supported by Theorem 2.48, which (up to the assumption of continuity) is a functional version of Theorem 2.20. In the following, we restrict to the rotation invariant case, where we say that a valuation .Z : Convsc (Rn ) → R is rotation invariant if .

Z(u ◦ ϑ −1 ) = Z(u)

for every .u ∈ Convsc (Rn ) and .ϑ ∈ SO(n). Define f Vn,α (u) :=

α(|∇u(x)|) dx

.

dom u

for .u ∈ Convsc (Rn ) and .α ∈ Cc ([0, x)). It is a consequence of Theorem 2.43 that, for each .0 ≤ j ≤ n − 1, there exists a continuous, epi-translation invariant valuation .Vj,α : Convsc (Rn ) → R that is epi-homogeneous of degree j such that Vn,α (u O r IB n ) =

n E

.

r n−j κn−j Vj,α (u)

(2.51)

j =0

for every .u ∈ Convsc (Rn ) and .r ≥ 0. Observe that (2.51) corresponds to the classical Steiner formula (2.5) where we have replaced the n-dimensional volume with .Vn,α and where now .IB n plays the role of the unit ball. In many ways, the functionals .Vj,α behave like the classical intrinsic volumes. First, it follows from the rotation invariance of .Vn,α and the radial symmetry of .IB n that also .Vj,α is rotation invariant for every .0 ≤ j ≤ n − 1. Next, since IK O r IB n = IK+rB n ,

.

it follows from (2.5), (2.43) and (2.51) that Vj,α (IK ) = α(0)Vj (K)

.

for every .0 ≤ j ≤ n and .K ∈ Kn . Last but not least, the functionals .Vj,α are characterized by a Hadwiger-type theorem. The version that is stated here follows from [26, Theorem 1.3] and [30, Theorem 1.4]. Theorem 2.49 Let .n ≥ 2. A functional .Z : Convsc (Rn ) → R is a continuous, epi-translation and rotation invariant valuation if and only if there are functions .α0 , . . . , αn ∈ Cc ([0, x)) such that .

Z(u) = V0,α0 (u) + · · · + Vn,αn (u)

for every .u ∈ Convsc (Rn ).

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75

Theorem 2.49 is a functional analog of the Hadwiger theorem, Theorem 2.24, and shows that the valuations .Vj,α clearly play the role of the intrinsic volumes on n .Convsc (R ). In [26], a different approach and notation are used. The functionals there take the form f .u |→ ζ (|∇u(x)|)[D2 u(x)]n−j dx (2.52) Rn

if in addition .u ∈ C 2 (Rn ), where .ζ : (0, x) → R has bounded support and might have a certain singularity at .0+ . It was later shown in [30, Theorem 1.4] that the continuous extensions of (2.52) to .Convsc (Rn ) coincide with the functionals .Vj,α , that are considered here, where .ζ and .α are connected via an integral transform. There are many open questions concerning functional intrinsic volumes and related functionals. Current research topics include characterization results, particularly for further groups of transformations, and the program to obtain results in the integral geometry of function spaces and to establish inequalities for the newly defined functionals. We refer to [5, 29–31, 42, 43] for some recent results. Acknowledgments M. Ludwig was supported, in part, by the Austrian Science Fund (FWF): P 34446, and F. Mussnig was supported by the Austrian Science Fund (FWF): J 4490.

References 1. S. Alesker, On P. McMullen’s conjecture on translation invariant valuations. Adv. Math. 155, 239–263 (2000) 2. S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11, 244–272 (2001) 3. S. Alesker, Theory of valuations on manifolds: a survey. Geom. Funct. Anal. 17, 1321–1341 (2007) 4. S. Alesker, Introduction to the Theory of Valuations. CBMS Regional Conference Series in Mathematics, vol. 126. Published for the Conference Board of the Mathematical Sciences, Washington (American Mathematical Society, Providence, 2018) 5. S. Alesker, Valuations on convex functions and convex sets and Monge-Ampère operators. Adv. Geom. 19, 313–322 (2019) 6. S. Alesker, J.H.G. Fu, Integral Geometry and Valuations. Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser/Springer, Basel, 2014) 7. S. Artstein-Avidan, A. Giannopoulos, V.D. Milman, Asymptotic Geometric Analysis. Part II. Mathematical Surveys and Monographs, vol. 261 (American Mathematical Society, Providence, 2021) 8. Y. Baryshnikov, R. Ghrist, Target enumeration via Euler characteristic integrals. SIAM J. Appl. Math. 70, 825–844 (2009) 9. Y. Baryshnikov, R. Ghrist, M. Wright, Hadwiger’s Theorem for definable functions. Adv. Math. 245, 573–586 (2013) 10. A. Bernig, A Hadwiger-type theorem for the special unitary group. Geom. Funct. Anal. 19, 356–372 (2009)

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11. A. Bernig, Algebraic integral geometry, in Global Differential Geometry. Springer Proceedings in Mathematics, vol. 17, pp. 107–145 (Springer, Heidelberg, 2012) 12. A. Bernig, J.H.G. Fu, Hermitian integral geometry. Ann. Math. (2) 173, 907–945 (2011) 13. W. Blaschke, Differentialgeometrie II (Springer, Berlin, 1923) 14. W. Blaschke, Vorlesungen über Integralgeometrie. H. 2 (Teubner, Berlin, 1937) 15. S.G. Bobkov, A. Colesanti, I. Fragalà, Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities. Manuscripta Math. 143, 131–169 (2014) 16. V. Boltianskii, Hilbert’s Third Problem (Wiley, New York, 1978) 17. K.J. Böröczky, M. Ludwig, Valuations on lattice polytopes, in Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177 (Springer, Cham, 2017), pp. 213–234 18. A. Braides, r-Convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22 (Oxford University Press, Oxford, 2002) 19. J. Brandts, S. Korotov, M. Kˇrížek, J. Šolc, On nonobtuse simplicial partitions. SIAM Rev. 51, 317–335 (2009) 20. E. Calabi, P. Olver, A. Tannenbaum, Affine geometry, curve flows, and invariant numerical approximation. Adv. Math. 124, 154–196 (1996) 21. L. Cavallina, A. Colesanti, Monotone valuations on the space of convex functions. Anal. Geom. Metr. Spaces 3, 167–211 (2015) 22. A. Colesanti, N. Lombardi, Valuations on the space of quasi-concave functions, in Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 2169 (Springer, Cham, 2017), pp. 71–105 23. A. Colesanti, N. Lombardi, L. Parapatits, Translation invariant valuations on quasi-concave functions. Studia Math. 243, 79–99 (2018) 24. A. Colesanti, M. Ludwig, F. Mussnig, Minkowski valuations on convex functions. Calc. Var. Partial Differ. Equ. 56, Paper No. 162 (2017) 25. A. Colesanti, M. Ludwig, F. Mussnig, Valuations on convex functions. Int. Math. Res. Not. IMRN, 2384–2410 (2019) 26. A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions, I (2020). arXiv:2009.03702 27. A. Colesanti, M. Ludwig, F. Mussnig, Hessian valuations. Indiana Univ. Math. J. 69, 1275– 1315 (2020) 28. A. Colesanti, M. Ludwig, F. Mussnig, A homogeneous decomposition theorem for valuations on convex functions. J. Funct. Anal. 279, 108573 (2020) 29. A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions, II: Cauchy– Kubota formulas (2021). Amer. J. Math. (in press) 30. A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge–Ampère measures. Calc. Var. Partial Differ. Equ. 61, Paper No. 181 (2022) 31. A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions, IV: the Klain approach (2022). Adv. Math. 413, Paper No. 108832 (2023) 32. A. Colesanti, D. Pagnini, P. Tradacete, I. Villanueva, A class of invariant valuations on Lip(S n−1 ). Adv. Math. 366, Paper No. 107069 (2020) 33. A. Colesanti, D. Pagnini, P. Tradacete, I. Villanueva, Continuous valuations on the space of Lipschitz functions on the sphere. J. Funct. Anal. 280, Paper No. 108873 (2021) 34. G. Dal Maso, An Introduction to r-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8 (Birkhäuser Boston, Inc., Boston, 1993) 35. M. Dehn, Ueber den Rauminhalt. Math. Ann. 55, 465–478 (1901) 36. R.J. Gardner, The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39, 355–405 (2002) 37. P. Goodey, W. Weil, Distributions and valuations. Proc. Lond. Math. Soc. (3) 49, 504–516 (1984) 38. C. Haberl, L. Parapatits, The centro-affine Hadwiger theorem. J. Amer. Math. Soc. 27, 685–705 (2014)

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39. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (German) (Springer, Berlin, 1957) 40. D.A. Klain, A short proof of Hadwiger’s characterization theorem. Mathematika 42, 329–339 (1995) 41. D.A. Klain, G.C. Rota, Introduction to Geometric Probability (Cambridge University Press, Cambridge, 1997) 42. J. Knoerr, The support of dually epi-translation invariant valuations on convex functions. J. Funct. Anal. 281, Paper No. 109059 (2021) 43. J. Knoerr, Smooth valuations on convex functions. J. Differential Geom. (in press) 44. K. Kusejko, L. Parapatits, A valuation-theoretic approach to translative-equidecomposability. Adv. Math. 297, 174–195 (2016) 45. K. Leichtweiß, Zur Affinoberfläche konvexer Körper. Manuscripta Math. 56, 429–464 (1986) 46. K. Leichtweiß, On the affine rectification of convex curves. Beiträge Algebra Geom. 40, 185– 193 (1999) 47. J. Li, D. Ma, Laplace transforms and valuations. J. Funct. Anal. 272, 738–758 (2017) 48. M. Ludwig, A characterization of affine length and asymptotic approximation of convex discs. Abh. Math. Semin. Univ. Hamb. 69, 75–88 (1999) 49. M. Ludwig, Valuations on polytopes containing the origin in their interiors. Adv. Math. 170, 239–256 (2002) 50. M. Ludwig, Fisher information and matrix-valued valuations. Adv. Math. 226, 2700–2711 (2011) 51. M. Ludwig, Valuations on function spaces. Adv. Geom. 11, 745–756 (2011) 52. M. Ludwig, Valuations on Sobolev spaces. Amer. J. Math. 134, 827–842 (2012) 53. M. Ludwig, Covariance matrices and valuations. Adv. Appl. Math. 51, 359–366 (2013) 54. M. Ludwig, Geometric valuation theory, in European Congress of Mathematics, ed. by A. Hujdurovi´c, K. Kutnar, D. Marušiˇc, Š. Miklaviˇc, T. Pisanski, P. Šparl (ESM Press, 2023), pp. 93–123 55. M. Ludwig, M. Reitzner, A characterization of affine surface area. Adv. Math. 147, 138–172 (1999) 56. M. Ludwig, M. Reitzner, A classification of SL(n) invariant valuations. Ann. Math. (2) 172, 1219–1267 (2010) 57. M. Ludwig, M. Reitzner, SL(n) invariant valuations on polytopes. Discrete Comput. Geom. 57, 571–581 (2017) 58. E. Lutwak, Extended affine surface area. Adv. Math. 85, 39–68 (1991) 59. E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996) 60. D. Ma, Real-valued valuations on Sobolev spaces. Sci. China Math. 59, 921–934 (2016) 61. P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes. Proc. Lond. Math. Soc. (3) 35, 113–135 (1977) 62. P. McMullen, Continuous translation-invariant valuations on the space of compact convex sets. Arch. Math. 55, 377–384 (1980) 63. P. McMullen, Weakly continuous valuations on convex polytopes. Arch. Math. (Basel) 41, 555–564 (1983) 64. P. McMullen, Monotone translation invariant valuations on convex bodies. Arch. Math. (Basel) 55, 595–598 (1990) 65. V. Milman, L. Rotem, Mixed integrals and related inequalities. J. Funct. Anal. 264, 570–604 (2013) 66. F. Mussnig, Volume, polar volume and Euler characteristic for convex functions. Adv. Math. 344, 340–373 (2019) 67. F. Mussnig, SL(n) invariant valuations on super-coercive convex functions. Canad. J. Math. 73, 108–130 (2021) 68. F. Mussnig, Valuations on log-concave functions. J. Geom. Anal. 31, 6427–6451 (2021) 69. M. Ober, Lp -Minkowski valuations on Lq -spaces. J. Math. Anal. Appl. 414, 68–87 (2014)

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70. R.T. Rockafellar, R.J.B. Wets, Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317 (Springer, Berlin, 1998) 71. R. Schneider, Simple valuations on convex bodies. Mathematika 43, 32–39 (1996) 72. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, second expanded edn. (Cambridge University Press, Cambridge, 2014) 73. R. Schneider, W. Weil, Stochastic and Integral Geometry. Probability and Its Applications (New York) (Springer, Berlin, 2008) 74. C. Schütt, E. Werner, The convex floating body. Math. Scand. 66, 275–290 (1990) 75. P. Tradacete, I. Villanueva, Radial continuous valuations on star bodies. J. Math. Anal. Appl. 454, 995–1018 (2017) 76. P. Tradacete, I. Villanueva, Continuity and representation of valuations on star bodies. Adv. Math. 329, 361–391 (2018) 77. A. Tsang, Valuations on Lp spaces. Int. Math. Res. Not. 20, 3993–4023 (2010) 78. A. Tsang, Minkowski valuations on Lp -spaces. Trans. Amer. Math. Soc. 364, 6159–6186 (2012) 79. I. Villanueva, Radial continuous rotation invariant valuations on star bodies. Adv. Math. 291, 961–981 (2016) 80. T. Wang, Semi-valuations on BV(Rn ). Indiana Univ. Math. J. 63, 1447–1465 (2014)

Chapter 3

Geometric and Functional Inequalities Andrea Colesanti and Daniel Hug

Abstract In this chapter, some of the fundamental inequalities for the basic geometric functionals on convex bodies are described. For some of these, functional analogs and extensions have been obtained. Some of these are motivated by the calculus of variations

3.1 Introduction Inequalities involving geometric functionals are one of the principal directions of research in classic and modern convex geometry. Well-known examples of such inequalities are the Brunn–Minkowski inequality, the Aleksandrov–Fenchel inequalities, the Blaschke–Santaló inequality and its converse, the isoperimetric inequality and its reverse form. Many connections exist among these inequalities. For instance, the Brunn– Minkowski inequality for convex bodies, which has great importance by itself, is a special case of the Aleksandrov–Fenchel inequalities. Analogously, the isoperimetric inequality for convex bodies can be easily deduced from the Brunn–Minkowski inequality. The research activity about these inequalities is currently very intense. We start by mentioning the recent progress concerning the classification of equality cases in the Aleksandrov–Fenchel inequality, by Shenfeld and van Handel (see [60] and [61]). Further developments follow different ramifications. First, various extensions of these inequalities have been proposed. On the one hand, new frameworks like the

A. Colesanti Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Firenze, Italy e-mail: andrea.colesanti@unifi.it D. Hug (O) Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6_3

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Lp Brunn–Minkowski theory provided the ground for new formulations of existing inequalities. On the other hand, as it has been found out, some functionals coming from different areas satisfy inequalities very similar to those valid e.g. for the volume, in convex geometry. Examples are provided by the Brunn–Minkowski type inequalities that have been established for some classic functionals in the calculus of variations. Moreover, in recent times, for several of the main inequalities in convex geometry, an analytic counterpart has been found. For instance, the Prékopa–Leindler inequality can be seen as a functional form of the Brunn–Minkowski inequality. As a second example, we mention the functional forms of the Blaschke–Santaló inequality and its converse (we refer to Chap. 4 for this topic). The goal of this chapter is to provide a general overview of this area, selecting some specific and representative examples from the wide landscape of geometric and functional inequalities in convex geometry. Our starting point will be an elementary construction of volume, surface area and mixed volumes of convex bodies, as well as of the class of mixed area measures, contained in the first section. Then we present the two most emblematic inequalities in convex geometry: the Brunn–Minkowski and the Aleksandrov–Fenchel inequalities. This part includes various proofs of these inequalities, applications, and the general Brunn–Minkowski and Minkowski inequalities. In the subsequent section we illustrate the basics of the so-called .Lp Brunn– Minkowski theory, and we present a selection of results coming from this area. In particular, we focus on the .Lp version of the Brunn–Minkowski inequality, for .p ≥ 1, and on the conjecture concerning the log-Brunn–Minkowski inequality, corresponding to the case .p = 0. In Sect. 3.5 we present the Brunn–Minkowski inequalities for three functionals in the calculus of variations: the electrostatic capacity, the torsion and the first Dirichlet eigenvalue of the Laplace operator. We focus in particular on the first one, and in general we provide additional evidence of the analogies of the behavior of these functionals with that of the volume. The final section is devoted to some important functional inequalities, the Brascamp–Lieb and the Barthe inequalities, and their geometric applications. In particular we present the volume ratio inequality and the consequent solution of the reverse isoperimetric problem, including some recent stability results. Throughout this chapter we will use the notation and several notions which have been introduced in Chap. 1. .

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3.2 Functionals and Measures in the Brunn–Minkowski Theory In this section, we give a more detailed introduction to some of the basic functionals and measures which have proved to be key objects in convex geometry. We start by constructing volume and surface area of convex bodies by using induction over the dimension of the space. This can be done in an elementary way for polytopes, the extension to general convex bodies is achieved by approximation of general convex bodies by polytopes (from inside and outside). The approach to mixed volumes and mixed area measures follows a similar path (in principle) but requires more refined arguments.

3.2.1 Polytopal Approximation, Volume and Surface Area We recall from the introductory chapter that by a convex body in .Rn we mean a non-empty compact convex subset of .Rn . The class of all convex bodies in .Rn is denoted by .Kn . The topology on .Kn , induced by the Hausdorff metric .δ(·, ·), allows us to introduce and study geometric functionals on convex bodies by first defining them for a special subclass, for example the class .Pn of polytopes or the class of convex bodies with sufficiently smooth boundaries. Such an approach requires that the geometric functionals under consideration have a continuity or monotonicity property and also that the class .Pn of polytopes is dense in .Kn . The following result ensures that the latter property is indeed available. Theorem 3.1 Let .K ∈ Kn and .ε > 0. (a) There exists a polytope .P ∈ Pn with .P ⊂ K and .δ(K, P ) ≤ ε. (b) There exists a polytope .P ∈ Pn with .K ⊂ P and .δ(K, P ) ≤ ε. (c) If . o ∈ relint K, then there exists a convex polytope .P ∈ Pn which satisfies - ∈ Pn with .P - ⊂ relint K .P ⊂ K ⊂ (1 + ε)P . There is even a convex polytope .P and which satisfies .K ⊂ relint((1 + ε)P ). For the proof of (a), one can use a simple compactness argument to see that .∂K (the boundary of K) can be covered by finitely many Euclidean balls of radius .ε with center in .∂K. The convex hull of these centers is a suitable choice of a convex polytope P , as required in (a). Proofs of the remaining assertions can be found in [43, Theorem 3.5]. Sometimes better or more specific approximation results are required. Examples of such cases are: . Approximation by combinatorially equivalent (simple and strongly isomorphic) polytopes (see [59, Theorem 2.4.15]). . Approximation by smooth bodies (see [59, Section 3.4]).

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. Economic approximation (by polytopes having few vertices or few facets; see, e.g., [13, 26, 56, 57, 63]). . Simultaneous approximation with additional features (for instance, preserving the convexity of the union; see [43, Exercise 3.1.15]).

Volume and Surface Area Volume and surface area of convex bodies can be defined via Lebesgue measure and Hausdorff measure of the appropriate dimension, respectively. Here we indicate a more elementary approach, using polytopal approximation and induction over the dimension. This also serves as a preparation for the introduction of mixed volumes. We start with some preparatory remarks: . A support set of a convex body K is a non-empty intersection of K with a supporting hyperplane, i.e. a hyperplane H which has non-empty intersection with K and is such that K is contained in one of the two half-spaces determined by H . . The support set .K(u) := {x ∈ K : = hK (u)}, .u ∈ Sn−1 , of a non-empty convex body K with (exterior) normal u lies in a hyperplane parallel to .u⊥ , the orthogonal complement of u. . The orthogonal projection .K(u)|u⊥ of the support set .K(u) to .u⊥ is a uniquely determined translate of .K(u), and we can consider .K(u)|u⊥ as a convex body in n−1 if we identify .u⊥ with .Rn−1 . .R . Assuming that the volume is already defined in .(n − 1)-dimensional linear subspaces, we then denote by .V (n−1) (K(u)|u⊥ ) the .(n − 1)-dimensional volume of this projection. . As soon as translation invariance is available, there is no need to mention the projection explicitly. Since our definition is by induction on the dimension and translation invariance of the volume in lower-dimensional subspaces is available by an induction argument, we will not indicate the projection in the following definition. . Furthermore, after Definition 3.2 we will not make explicit by our notation the dimension of the ambient Euclidean space in which volume and surface area are calculated, as long as the dimension is clear from the context. Let .P ∈ Pn be a polytope with .o ∈ int P . Let .u1 , . . . , um ∈ Sn−1 denote the exterior unit normals of the facets (.(n − 1)-dimensional support sets) of P . Then P is the union of the simplices .conv({o} ∪ P (ui )) with base .P (ui ) and common apex o, .i = 1, . . . , m, any two of which do not have common interior points, that is, P =

m U

.

i=1

conv({o} ∪ P (ui )).

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The volume of P equals the sum of the volumes of the simplices and the volume of the simplex .conv({o} ∪ P (ui )) is .1/n times the .(n − 1)-dimensional volume of the base .P (ui ) times the height, which is the distance of o from the affine subspace spanned by .P (ui ). The latter is equal to the value of the support function of P evaluated at .ui . Some argument is required to see that the volume thus defined is indeed translation invariant and that it still works properly if o is not an interior point of P . After these explanations, the following definitions are natural. Definition 3.2 Let .P ∈ Pn be a polytope. If .n = 1, then .P = [a, b] with .a ≤ b and V (1) (P ) := b − a

.

and

S (1) (P ) := 2.

For .n ≥ 2, let ⎧ E 1 ⎪ ⎨ hP (u)V (n−1) (P (u)), (n) n .V (P ) : = (∗) ⎪ ⎩ 0,

if dim P ≥ n − 1, if dim P ≤ n − 2,

and S (n) (P ) : =

.

⎧E ⎪ V (n−1) (P (u)), ⎨

if dim P ≥ n − 1,

(∗)

⎪ ⎩0,

if dim P ≤ n − 2,

where the summation .(∗) is over all .u ∈ Sn−1 for which .P (u) is a facet of P ; here, in .Rn , by a facet of a polytope we mean a support set of dimension .n − 1. In .Rn , we shortly write .V (P ) for .V (n) (P ) and call this the volume of P . Similarly, we write (n) (P ) and call this the surface area of P . .S(P ) instead of .S Note that the definition implies that .V (n) (P ) = 0 if .dim P = n − 1. Moreover, volume and surface area of the empty set are defined as zero. Next we summarize basic properties of volume and surface area. Proposition 3.3 The volume V and the surface area F of polytopes .P , Q ∈ Pn have the following properties: (a) (b) (c) (d) (e)

V (P ) = Hn (P ), V and S are invariant with respect to rigid motions, n n−1 S(P ) for .α ≥ 0, .V (αP ) = α V (P ) and .S(αP ) = α .V (P ) = 0 if and only if .dim P ≤ n − 1, if .P ⊂ Q, then .V (P ) ≤ V (Q) and .S(P ) ≤ S(Q). .

Most of the required arguments easily follow by induction on the dimension, which is natural in view of the inductive nature of the definition of volume and

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surface area of polytopes. The connection to and properties of the n-dimensional Hausdorff measure (which equals Lebesgue measure for the sets under consideration in .Rn ) can also be exploited. See [43, Proposition 3.1] for the details. The monotonicity of volume and surface area hold for general convex bodies, as stated in Theorem 3.5. For the volume this is not surprising in view of the monotonicity of Lebesgue (n-dimensional Hausdorff) measure with respect to inclusion. It should be clear, however, that the monotonicity of the surface area with respect to inclusion of the enclosed sets does not extend to arbitrary compact sets in .Rn . Definition 3.4 Let .K ∈ Kn be a convex body. Then V+ (K) := inf V (P ),

V− (K) := sup V (P ),

where P ∈ Pn ,

S+ (K) := inf S(P ),

S− (K) := sup S(P ),

where P ∈ Pn .

.

P ⊃K

P ⊂K

and .

P ⊃K

P ⊂K

If .V+ (K) = V− (K) =: V (K), we call this value the volume of K. If .S+ (K) = S− (K) =: S(K), we call this value the surface area of K. The following result shows that volume and surface area of general convex bodies exist, and it summarizes important properties of the functionals thus defined. It is not surprising that these properties mirror corresponding properties of the functionals on convex polytopes. Theorem 3.5 Let .K, L ∈ Kn . (a) Then V+ (K) = V− (K) = V (K)

.

and S+ (K) = S− (K) = S(K).

.

(b) Volume and surface area have the following properties: (b1) (b2) (b3) (b4) (b5) (b6)

V (K) = Hn (K), V and S are invariant with respect to rigid motions, n n−1 S(K) for .α ≥ 0, .V (αK) = α V (K) and .S(αK) = α .V (K) = 0 if and only if .dim K ≤ n − 1, if .K ⊂ L, then .V (K) ≤ V (L) and .S(K) ≤ S(L), .K |→ V (K) is continuous. .

Most of the properties are again easy to derive from the corresponding ones stated in Theorem 3.3. However, part (a) and the continuity property require additional arguments (see [43, Theorem 3.6] for the details). The surface area functional is also

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continuous with respect to the Hausdorff metric. In the special case of a sequence of convex bodies .Ki , .i ∈ N, converging to a convex body .K ∈ Kn with non-empty interior, we can provide a direct argument: Since S is translation invariant, we can assume that .K ∈ Kn(o) . Then we also have .Ki ∈ Kn(o) , if i is large enough. Let .ε > 0 be given. By Theorem 3.1 (c) there is a polytope .P ∈ Pn(o) such that P ⊂ int(K) and

.

K ⊂ int((1 + ε)P ).

Then there is some .r > 0 such that .P + rB n ⊂ K and .K + rB n ⊂ (1 + ε)P . Since .Ki → K it follows that Ki ⊂ K + rB n ⊂ (1 + ε)P

.

and

P + rB n ⊂ K ⊂ Ki + rB n

for .i ≥ i0 ; in particular, .P ⊂ Ki . Using these inclusions and the facts that S is increasing under set inclusion and homogeneous of degree .n − 1, we get S(P ) − S((1 + ε)P ) ≤ S(Ki ) − S(K) ≤ S((1 + ε)P ) − S(P ),

.

hence ( ) |S(Ki ) − S(K)| ≤ (1 + ε)n−1 − 1 S(K).

.

This proves the assertion under the assumption that K has non-empty interior. The general case will follow later via the connection to mixed volumes (see Theorem 3.14 (a) and (f)).

3.2.2 Mixed Volumes The classical Steiner formula has already been discussed in Sect. 1.9 of Chap. 1. An elementary proof for the polynomial expansion of the volume of the parallel set of a general convex body can be given by first showing this expansion for the parallel sets of a convex polytope. In this case, the parallel set decomposes into (essentially disjoint) wedges over the faces of the polytope whose volumes can be calculated by means of Fubini’s theorem. The situation is easily illustrated for a convex polygon in dimension two (see Fig. 3.1). Now we consider the more general problem of calculating the volume of Minkowski combinations of finitely many (general) convex bodies in .Rn . More specifically, we are considering the following question. Let .Ki ∈ Kn and .αi ≥ 0 for .i = 1, . . . , m. How does the volume V (α1 K1 + · · · + αm Km )

.

depend on the variables .α1 , . . . , αm ? To answer the question, we provide several auxiliary results.

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Fig. 3.1 Polynomial volume growth of .V (P +qB 2 ) = V (P )+S(P )·q +V (B 2 )·q2 as a function of .q ≥ 0, where .S(P ) is the perimeter and .V (P ) is the area of P

Proposition 3.6 Let .m ∈ N, let .α1 , . . . , αm > 0, let .K1 , . . . , Km ∈ Kn be convex bodies, and let .u, v ∈ Sn−1 . Then (a) .(α1 K1 + · · · + αm Km )(u) = α1 K1 (u) + · · · + αm Km (u), (b) .dim(α1 K1 + · · · + αm Km )(u) = dim(K1 + · · · + Km )(u), (c) if .(K1 + · · · + Km )(u) ∩ (K1 + · · · + Km )(v) /= ∅, then (K1 + · · · + Km )(u) ∩ (K1 + · · · + Km )(v)

.

= (K1 (u) ∩ K1 (v)) + · · · + (Km (u) ∩ Km (v)). Proof The argument in [43, Proposition 3.2] works without changes for general convex bodies, but will be needed only for polytopes. u n Also the next lemma is taken from [43], see Lemma 3.2 there. It identifies the intersection of two given adjacent support sets as a support set of one of the given support sets. Lemma 3.7 Let .K ∈ Kn , let .u, v ∈ Sn−1 be linearly independent unit vectors, and let .w = λu + μv with some .λ ∈ R and .μ > 0. Then .K(u) ∩ K(v) /= ∅ implies that .K(u) ∩ K(v) = K(u)(w). While Lemma 3.7 holds for general convex bodies, the restriction to polytopes is crucial in the following lemma. Lemma 3.8 Let .P ∈ Pn with .dim(P ) = n, let F be a facet of P , and let Q be a facet of F (with respect to the .(n − 1)-dimensional affine subspace spanned by F , as the ambient space). Then there is a (unique) facet G of P such that .Q = F ∩ G.

3 Geometric and Functional Inequalities

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The assertion follows from Exercise 1.4.6 or from Exercise 1.5.5 (d) in [43]. Here we provide another argument. A minor modification of this argument shows that support sets of support sets of a polytope are support sets (and consequently the proper faces of a polytope are support sets). In the argument, we use the important notion of an extreme point. A point x of a polytope P (or of a general compact convex set) is said to be an extreme point of P if .P \ {x} is convex. It is a wellknown fact that a polytope P has finitely many extreme points and P equals the convex hull of its extreme points (which form the minimal subset of P having this property). Proof We may denote the extreme points (vertices) .x0 , . . . , xt of P in such a way that .x0 , . . . , xr are the vertices of Q and .x0 , . . . , xs are the vertices (extreme points) of F , where .r < s < t. Let .F = P (u) for some .u ∈ Sn−1 and .Q = P (u)(w) for some vector .w ∈ Sn−1 orthogonal to u. Then { = 0 for i = 1, . . . , s, . < 0 for i = s + 1, . . . , t, { = 0 for i = 1, . . . , r,

< 0 for i = r + 1, . . . , s. Define }

: i = s + 1, . . . , t . .λ0 := max − {

Then ⎧ ⎪ ⎪ ⎨= 0 . = λ0 + 0, . . . , αm > 0. By the definition of volume and by Proposition 3.6, 1 .V (α1 P1 + · · · + αm Pm ) = n =

E

hEm

i=1 αi Pi

(u) v

(( m E

in =1

1 αin n

E

hPin (u) v

u∈N (P1 ,...,Pm )

( m E

)

αi Pi (u)

i=1

u∈N (P1 ,...,Pm )

m E

)

) αi Pi (u) ,

i=1

where we write v to denote the volume functional in a subspace of codimension 1. The induction hypothesis implies that v

( m E

.

) αi Pi (u) =

m E i1 =1

i=1

···

m E

αi1 · · · αin−1 V (Pi1 (u), . . . , Pin−1 (u)).

in−1 =1

Thus we obtain (d). The most delicate point in the proof is to show the symmetry property (a). Here we apply the recursive definition twice. For unit vectors u, v with u /= ±v we write γ (u, v) ∈ (0, π ) for the angle enclosed by u and v, that is cos γ (u, v) = . Then we obtain, using Proposition 3.6 and Lemmas 3.7 and 3.8, V (P1 , . . . , Pn−2 , Pn−1 , Pn ) E 1 = hPn (u) V (P1 (u), . . . , Pn−1 (u)) n

.

u∈N (P1 ,...,Pn )

=

1 n(n − 1) ×

[

E u,v∈N (P1 ,...,Pn ),v/=±u

] 1 1 hPn (u)hPn−1 (v) − hPn (u)hPn−1 (u) sin γ (u, v) tan γ (u, v)

× V (P1 (u) ∩ P1 (v), . . . , Pn−2 (u) ∩ Pn−2 (v)) = V (P1 , . . . , Pn−2 , Pn , Pn−1 ), which implies the symmetry property (by induction). The remaining assertions and further details can be found in the argument for [43, Theorem 3.7]. n u The next result is a key property of mixed volumes (and other multilinear functionals). It can be used to transfer properties of the volume functional to mixed volumes. It can be easily verified for .n = 2 or .n = 3. A proof can be found in [43, Theorem 3.8] (see also [59, Lemma 5.1.4]), a different (combinatorial) argument is given in [39, Theorem 6.7].

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Theorem 3.13 (Inversion Formula) If .P1 , . . . , Pn ∈ Pn , then V (P1 , . . . , Pn ) =

.

1 E (−1)n+k n! n

E

k=1

1≤r1 1 and suppose that the assertion has already been proved in lower dimensions. We use the identification .Rn = Rn−1 × R and define, for .s ∈ R and .z ∈ Rn−1 , hs (z) := h(z, s),

fs (z) := f (z, s),

.

gs (z) := g(z, s).

Let .z1 , z2 ∈ Rn−1 , .a, b ∈ R and .c := (1 − λ)a + λb. By assumption, we have hc ((1 − λ)z1 + λz2 ) = h((1 − λ)z1 + λz2 , (1 − λ)a + λb)

.

= h((1 − λ)(z1 , a) + λ(z2 , b)) ≥ f (z1 , a)1−λ g(z2 , b)λ = fa (z1 )1−λ gb (z2 )λ . The induction hypothesis yields )1−λ ( f

(f

f .

hc (z) dz ≥ n−1 R '' ' ' =:H (c)

fa (z) dz n−1 R '' ' ' =:F (a)

)λ

gb (z) dz n−1 R '' ' ' =:G(b)

that is, H ((1 − λ)a + λb) ≥ F (a)1−λ G(b)λ

.

for .a, b ∈ R.

,

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The case .n = 1 has already been proved, hence by Fubini’s theorem we get f f f f H (s) ds . h(x) dx = h (z) dz ds = s R Rn−1 R Rn (f )1−λ (f )λ ≥ F (a) da G(b) db R R (f )1−λ (f )λ = f (x) dx g(x) dx . Rn Rn This completes the induction argument and thus the proof.

u n

The classical proof of the BMI also has its benefits. For instance, it has been extended to obtain stability improvements of the BMI. In particular, it clarifies the equality cases of the BMI for convex sets. Proof (Classical Proof of the BMI) The main case to be considered deals with the situation where .V (K) = V (L) = 1. Volume is translation invariant. Hence we can assume that K and L have their center of mass at the origin o. Recall that the center of mass of an n-dimensional convex body M is the point .c = c(M) ∈ Rn for which f 1 dx holds for u ∈ Sn−1 . . = V (M) M Since .V (K) = V (L) = 1, assuming that .c(K) = c(L) = o means that f f . dx = dx = 0 for u ∈ Sn−1 . K

L

Under these assumptions, the equality case then reduces to the claim that .K = L. We now prove the Brunn–Minkowski theorem by induction on n. For .n = 1, the assertion is already clear. The argument employs a method that can be described as using sections of K and L by coordinated parallel hyperplanes. For the induction step we assume that .n ≥ 2 and that the assertion of the Brunn– Minkowski theorem is true in dimension .n − 1. We fix an arbitrary unit vector n−1 .u ∈ S and denote by .Eη := H (u, η) = {x ∈ Rn : = η}, .η ∈ R, the hyperplane in direction u with (signed) distance .η from the origin. Recall that n − .H (u, η) = {x ∈ R : ≤ η} is the halfspace bounded by .H (u, η) having u as an outer normal vector. The function f : [−hK (−u), hK (u)] → [0, 1],

.

β |→ V (K ∩ H − (u, β)),

is strictly increasing, onto, and continuous. This follows since V (K ∩ H − (u, β)) =

f

β

.

−hK (−u)

v(K ∩ Eη ) dη

3 Geometric and Functional Inequalities

107

by Fubini’s theorem (here .v(·) denotes the volume functional in a hyperplane) and since .η |→ v(K ∩ Eη ) is positive on .(−hK (−u), hK (u)) and continuous on .[−hK (−u), hK (u)]. Hence the function f is differentiable on .[−hK (−u), hK (u)] and .f ' (β) = v(K ∩ Eβ ). Since f is invertible, the inverse function .β : [0, 1] → [−hK (−u), hK (u)], which is also strictly increasing and continuous, satisfies .β(0) = −hK (−u), .β(1) = hK (u), and β ' (τ ) =

.

1 f ' (β(τ ))

=

1 , v(K ∩ Eβ(τ ) )

τ ∈ (0, 1).

By the same argument, for the convex body L we obtain a function .γ : [0, 1] → [−hL (−u), hL (u)] with γ ' (τ ) =

.

1 , v(L ∩ Eγ (τ ) )

τ ∈ (0, 1).

Because of α(K ∩ Eβ(τ ) ) + (1 − α)(L ∩ Eγ (τ ) ) ⊂ (αK + (1 − α)L) ∩ Eαβ(τ )+(1−α)γ (τ ) ,

.

for .α, τ ∈ [0, 1] and using a substitution by the map [0, 1] → [α(−hK (−u)) + (1 − α)(−hL (−u)), αhK (u) + (1 − α)hL (u)],

.

τ |→ αβ(τ ) + (1 − α)γ (τ ), we obtain from the induction assumption f ∞ .V (αK + (1 − α)L) = v((αK + (1 − α)L) ∩ Eη ) dη −∞

f

1

=

v((αK + (1 − α)L) ∩ Eαβ(τ )+(1−α)γ (τ ) )(αβ ' (τ ) + (1 − α)γ ' (τ )) dτ

0

f ≥ 0

1

( ) v α(K ∩ Eβ(τ ) ) + (1 − α)(L ∩ Eγ (τ ) )

(

× f ≥

1−α α + v(K ∩ Eβ(τ ) ) v(L ∩ Eγ (τ ) )

1[

α 0

× ≥ 1,

(

) dτ

/ / ]n−1 v(K ∩ Eβ(τ ) ) + (1 − α) n−1 v(L ∩ Eγ (τ ) )

n−1

1−α α + v(K ∩ Eβ(τ ) ) v(L ∩ Eγ (τ ) )

) dτ .

(3.15) (3.16)

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since the integrand is .≥ 1. To see this, choose .r := v(K ∩Eβ(τ ) ), .s := v(L∩Eγ (τ ) ), 1 . Then the assertion follows from the inequality and .t := n−1 ( .

α 1−α + r s

)

[ t ]1 αr + (1 − α)s t t ≥ 1,

α ∈ (0, 1), r, s, t > 0,

with equality if and only if .r = s. (Note that the function .x |→ log x is strictly concave.) Now we discuss the equality case and assume that V (αK + (1 − α)L) = 1.

.

Then we must have equality in (3.16), which implies that the integrand in (3.15) equals 1, for all .τ (since the integrand is a continuous function of .τ ). In turn this yields that v(K ∩ Eβ(τ ) ) = v(L ∩ Eγ (τ ) ),

for τ ∈ [0, 1].

.

Therefore .β ' = γ ' on .(0, 1), hence the function .β−γ is a constant on .[0, 1]. Because the center of gravity of K is at the origin, we obtain f

f

0=

dx =

.

K

β(1)

f

β(1)

ηv(K ∩ Eη ) dη =

β(0)

ηf ' (η) dη =

β(0)

f

1

β(τ ) dτ, 0

where the change of variables .η = β(τ ) was used. In an analogous way, f

1

0=

γ (τ ) dτ.

.

0

Consequently, f

1

.

(β(τ ) − γ (τ )) dτ = 0

0

and therefore .β = γ . In particular, we obtain hK (u) = β(1) = γ (1) = hL (u).

.

Since .u ∈ Sn−1 was arbitrary, .V (αK + (1 − α)L) = 1 implies that .hK = hL , and hence .K = L. u n Conversely, it is clear that .K = L implies that .V (αK + (1 − α)L) = 1.

3 Geometric and Functional Inequalities

109

The Brunn–Minkowski inequality implies that the function defined by 1

t ∈ R,

f (t) := V (tK + (1 − t)L) n ,

.

is concave on .[0, 1]. If .x, y, α ∈ [0, 1], then 1

f (αx + (1 − α)y) = V ([αx + (1 − α)y]K + [1 − αx − (1 − α)y]L) n

.

1

= V (α[xK + (1 − x)L] + (1 − α)[yK + (1 − y)L]) n 1

1

≥ αV (xK + (1 − x)L) n + (1 − α)V (yK + (1 − y)L) n = αf (x) + (1 − α)f (y). 1

The same is true for the function .t |→ V (K + tL) n with .t ∈ [0, 1]. As an important consequence, we obtain an inequality for mixed volumes which was first proved by Hermann Minkowski. Theorem 3.28 (Minkowski’s Inequality, MI) Let .K, L ∈ Kn . Then V (K[n − 1], L)n ≥ V (K)n−1 V (L)

.

with equality if and only if .dim K ≤ n − 2 or K and L lie in parallel hyperplanes or K and L are homothetic. Proof For .dim K ≤ n − 1, the inequality holds since the right-hand side is zero. Moreover, we then have equality, if and only if either .dim K ≤ n − 2 or K and L lie in parallel hyperplanes. Hence, we now assume .dim K = n. By the BMI the function 1

f (t) := V (K + tL) n ,

.

t ∈ [0, 1],

is concave. We write .f + for the right derivative of f . Since f is concave on its domain, 1

1

f + (0) ≥ f (1) − f (0) = V (K + L) n − V (K) n .

.

On the other hand, by the polynomial expansion of .t |→ V (K + tL) and the chain rule, f + (0) =

.

1 1 V (K) n −1 n V (K[n − 1], L). n

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Hence we arrive at 1

1

1

1

V (K) n −1 V (K[n − 1], L) ≥ V (K + L) n − V (K) n ≥ V (L) n ,

.

where we used the BMI for the second inequality (with .t = 12 ). This yields the required inequality. If equality holds, then equality holds in the BMI, which implies that K and L are homothetic. Conversely, if K and L are homothetic, then equality holds. u n The isoperimetric inequality is one of the fundamental classical results in mathematics. It states that among all (convex) bodies of given volume, precisely the Euclidean balls minimize the surface area F . Corollary 3.29 (Isoperimetric Inequality) If .K ∈ Kn is n-dimensional, then ( ) ) ( S(K) n V (K) n−1 . ≥ . S(B n ) V (B n ) Equality holds if and only if K is a ball. Proof We put .L := B n in the Minkowski inequality and get V (K[n − 1], B n )n ≥ V (K)n−1 V (B n )

.

or, equivalently, .

nn V (K[n − 1], B n )n V (K)n−1 ≥ , nn V (B n [n − 1], B n )n V (B n )n−1 u n

which is precisely what we had to show.

Note that the inequality is scaling and rigid motion invariant. It can be expressed by saying that the isoperimetric ratio .

S(K)n V (K)n−1

is minimized precisely by Euclidean balls. Using .V (B n ) = κn and .S(B n ) = nκn , we can rewrite the inequality in the form V (K)n−1 ≤

.

1 nn κ

S(K)n . n

For .n = 2 and using the common terminology .A(K) for the area (the “volume” in R2 ) and .L(K) for the boundary length (the “surface area” in .R2 ), we obtain

.

A(K) ≤

.

1 L(K)2 , 4π

3 Geometric and Functional Inequalities

111

and, for .n = 3, V (K)2 ≤

.

1 S(K)3 . 36π

An exchange of K and .B n in the proof above leads to a similar inequality for the mixed volume .V (B n [n − 1], K), whence we obtain the following corollary for the mean width .w(K). Corollary 3.30 Let .K ∈ Kn be a convex body. Then, ( .

w(K) w(B n )

)n ≥

V (K) . V (B n )

Equality holds if and only if K is a ball. Since .w(K) is not greater than the diameter of K, the corollary yields an inequality for the diameter. The resulting inequality is known as the isodiametric inequality. Using the BMI and second derivatives, we obtain an inequality of quadratic type. Theorem 3.31 (Minkowski’s Second Inequality, MIq) For .K, L ∈ Kn , V (K[n − 1], L)2 ≥ V (K[n − 2], L, L)V (K).

.

(3.17)

Equality holds if .dim(K) ≤ n − 2 or if K and L are homothetic, but these are not the only cases. [The characterization of the case of equality involves the .(n − 2)-tangential bodies of L.] 1

Proof We assume that .V (K) > 0 (why?). The function .f (t) := V (K + tL) n is concave on .[0, 1] by the BMI. Since n ( ) E n i t V (K[n − i], L[i]), .t → | h(t) := V (K + tL) = i

t ∈ [0, 1],

i=0

is of class .C 2 , f is of class .C 2 as well (note that .V (K) > 0). Hence, .f '' ≤ 0 on 1 .[0, 1]. Since .f (t) = h(t) n , we get [ ] 1 1 2 '' −2 1 − n ' n h (t) + h(t)h (t) . f (t) = h(t) n n

1 1 h(t) n −1 h' (t), .f (t) = n

'

''

Therefore .f '' (0) ≤ 0 yields h(0)h'' (0) ≤

.

n−1 ' 2 h (0) . n

(3.18)

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Furthermore, n ( ) E n i−1 it V (K[n − i], L[i]), .h (t) = i '

i=1

h'' (t) =

n ( ) E n i(i − 1)t i−2 V (K[n − i], L[i]). i i=2

Plugging .h'' (0) = n(n − 1)V (K[n − 2], L, L), .h' (0) = nV (K[n − 1], L) and .h(0) = V (K) into (3.18), it follows that V (K)n(n − 1)V (K[n − 2], L, L) ≤

.

n−1 2 n V (K[n − 1], L)2 , n

and hence V (K)V (K[n − 2], L, L) ≤ V (K[n − 1], L)2 ,

.

u n

which is the required inequality.

Replacing K or L in (3.17) by the unit ball, we obtain more special inequalities, for example (in .R3 ) π w(K)2 ≥ S(K)

.

or

S(K)2 ≥ 6π w(K)V (K).

Next we describe two basic applications. Claim If .K ∈ Kn is symmetric with respect to the origin o, that is .K = −K, then among all parallel hyperplane sections of K the central sections have maximal section volume. To verify the claim, let .H (u, t) = {x ∈ Rn : = t} for .u ∈ Sn−1 , .t ∈ R. If .H (u, t) ∩ K /= ∅, then .H (u, −t) ∩ K /= ∅ and .

1 1 (H (u, t) ∩ K) + (H (u, −t) ∩ K) ⊂ H (u, 0) ∩ K. 2 2

Hence (writing again v for the volume in a hyperplane) v(H (u, 0) ∩ K)

.

1 n−1

( ≥v

) 1 n−1 1 1 (H (u, t) ∩ K) + (H (u, −t) ∩ K) 2 2

1 1 1 1 v (H (u, t) ∩ K) n−1 + v (H (u, −t) ∩ K) n−1 2 2 1 1 1 1 = v (H (u, t) ∩ K) n−1 + v (H (u, t) ∩ (−K)) n−1 2 2

≥

1

= v(H (u, t) ∩ K) n−1 , which yields .v(H (u, 0) ∩ K) ≥ v(H (u, t) ∩ K).

3 Geometric and Functional Inequalities

113

The second application concerns a uniqueness property of the top order area measures. Theorem 3.32 (Uniqueness of Top Order Area Measures) Let .K, L ∈ Kn with .dim K = dim L = n. Then .Sn−1 (K, ·) = Sn−1 (L, ·) if and only if K and L are translates. Proof If .K, L are translates of each other, the equality of the area measures is clear. Assume now .Sn−1 (K, ·) = Sn−1 (L, ·). Then f 1 hL (u) dSn−1 (K, u) n Sn−1 f 1 hL (u) dSn−1 (L, u) = V (L). = n Sn−1

V (K[n − 1], L) =

.

In the same way, we obtain .V (L, . . . , L, K) = V (K). The MI therefore yields that V (L)n ≥ V (K)n−1 V (L)

and

.

V (K)n ≥ V (L)n−1 V (K),

which implies that .V (K) = V (L). But then we have equality in both inequalities, and hence K and L are homothetic. Since K and L have the same volume, they must be translates of each other. u n The Brunn–Minkowski inequality has been discussed also for general compact sets. A similar extension can be obtained for the Minkowski inequality (see [42, Sections 5.2.1-5]). Definition 3.33 (Outer Relative Surface Area for Compact Sets) For .X, Y ∈ Cn the lower (outer) relative surface area of X with respect to Y is defined by S+ (X, Y ) := lim inf

.

ε↓0

V (X + εY ) − V (X) . ε

The upper (outer) relative surface area of X with respect to Y is defined by S + (X, Y ) := lim sup

.

ε↓0

V (X + εY ) − V (X) . ε

If .S + (X, Y ) = S+ (X, Y ) =: S(X, Y ), then we say that the outer relative surface area of X with respect to Y exists. In Chapter essentially the same concept is considered, where the .lim inf and the lim sup are denoted as lower and upper anisotropic outer Minkowski content of X with respect to Y .

.

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Theorem 3.34 If .X, Y ∈ Cn , then S + (X, Y ) ≥ S+ (X, Y ) ≥ nV (X)

.

n−1 n

1

V (Y ) n .

Let .X, Y ∈ Cn with .V (X), V (Y ) > 0. Suppose that .S(X, Y ) exists. Then .S(X, Y ) = 1 n−1 nV (X) n V (Y ) n holds if and only if .X, Y are homothetic convex bodies. n−1 1 If only .S+ (X, Y ) = nV (X) n V (Y ) n is available, then still the volume kernels of .X, Y are homothetic convex bodies. Here the volume kernel of a set .X ∈ Cn is the set of all .x ∈ X such that .X ∩ U has positive volume for each neighborhood U of x. The volume kernel of X is again a compact set which has the same volume as X, which justifies its name.

3.3.2 The Aleksandrov–Fenchel Inequality A generalization of Minkowski’s second inequality states that V (M[n − 2], K, L)2 ≥ V (M[n − 2], K, K)V (M[n − 2], L, L),

.

(3.19)

for convex bodies .K, L, M ∈ Kn . The case considered in Theorem 3.31 is recovered by choosing .M = K. The resolution of the long-standing open problem of characterizing the equality case for the general quadratic Minkowski inequality (3.19) has been achieved by Shenfeld and van Handel [60]. The even more general Aleksandrov–Fenchel inequality (AFI) is obtained if the sequence of the .n − 2 bodies .M, . . . , M in (3.19) is replaced by arbitrary convex bodies .M1 , . . . , Mn−2 , so that V (M1 , . . . , Mn−2 , K, L)2 ≥ V (M1 , . . . , Mn−2 , K, K)V (M1 , . . . , Mn−2 , L, L) (3.20)

.

is obtained. The Aleksandrov–Fenchel inequality is connected to various fields in mathematics and has many and surprising applications. A recent characterization of the equality cases of (3.20), if all bodies involved are polytopes, is provided in [61]. See also Sections 7.3, 7.6 and 7.7 of Schneider’s monograph [59] for further information on the Aleksandrov–Fenchel inequality, its equality cases and applications. A complete understanding of the equality cases is not yet available and is the subject of current research. In the following, for integers .1 ≤ i ≤ j ≤ n and convex bodies .Ki , . . . , Kj ∈ Kn , we write .Ki··j = (Ki , . . . , Kj ) (with or without brackets) for a given finite sequence of .j − i + 1 convex bodies. The sequence is empty (and omitted) if .i > j . We write .Knn for the set of convex bodies with non-empty interiors.

3 Geometric and Functional Inequalities

115

Theorem 3.35 (Aleksandrov–Fenchel Inequality) Let .K1 , . . . , Kn ∈ Kn . Then V (K1 , K2 , K3..n )2 ≥ V (K1 [2], K3..n ) V (K2 [2], K3..n ).

(AFI)

.

By the symmetry properties of mixed volumes this is equivalent to (3.20). For n = 2 the AFI boils down to the MI. For this reason, we focus on dimension .n ≥ 3 in the following. For .m ∈ {2, . . . , n} and convex bodies .K1 , K2 , Km+1 , . . . , Kn ∈ Kn , we consider the function defined by

.

1

fm (t) := V ((K1 + tK2 ) [m], Km+1..n ) m ,

.

t ≥ 0.

We will see now that the validity of (AFI) is closely related to the fact that .fm is a concave function on .[0, ∞). The following argument is adjusted from Cordero-Erausquin, Klartag, Merigo, Santambrogio [33] (see also [43, Section 3.5] for further details). We start by determining the second derivative of the function .fm . Lemma 3.36 Let .m ∈ {2, . . . , n}, .K1 , K2 , Km+1 , . . . , Kn ∈ Knn and .t ≥ 0. Then fm'' (t) = −(m − 1)fm (t)1−2m ( ) × V (K1 , K2 , K3..n )2 − V (K1 [2], K3..n )V (K2 [2], K3..n ) ,

.

where .K t := K1 + tK2 and .K3..n := (K t [m − 2], Km+1..n ) for .t ≥ 0. Proof We define .hm (t) := fm (t)m for .t ≥ 0. Then .fm and .hm are of class .C 2 and '' .fm (t)

( ) 1 1−m ' 1−2m '' 2 hm (t)hm (t) + hm (t) , = fm (t) m m

Using the Minkowski linearity of mixed volumes, we obtain hm (t) = V (K t [m], Km+1..n ),

.

h'm (t) = mV (K t [m − 1], K2 , Km+1..n ), h''m (t) = m(m − 1)V (K t [m − 2], K2 [2], Km+1..n ).

t ≥ 0.

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Hence we get 1−m ' hm (t)2 m ( = −m(m − 1) V (K t , K2 , K t [m − 2], Km+1..n )2

h''m (t)hm (t) +

.

) −V (K t [2], K t [m − 2], Km+1..n )V (K2 [2], K t [m − 2], Km+1..n ) ) ( = −m(m − 1) V (K1 , K2 , K3..n )2 − V (K1 [2], K3..n )V (K2 [2], K3..n ) , from which the assertion follows.

u n

Note that in writing .K3..n for .(K t [m − 2], Km+1..n ) the dependence on m and t is not made explicit. Furthermore, it should be observed that for .m = 2 the sequence .K3..n := (Km+1..n ) is independent of t. The following proposition is an immediate consequence of Lemma 3.36. Proposition 3.37 (a) If (AFI) holds for all convex bodies, then .fm is a concave function for all convex bodies. (b) For fixed convex bodies .K1 , . . . , Kn ∈ Knn , (AFI) holds if and only if .f2 is concave. In particular, (AFI) holds for all convex bodies if and only if .f2 is always concave. The next lemma shows that a converse of Proposition 3.37 (a) holds not only for m = 2, but also for .m = 3. This observation will be a key fact in the proof of the Aleksandrov–Fenchel inequality which proceeds by induction on the dimension n of the space.

.

Lemma 3.38 If .f3 is a concave function for all convex bodies, then (AFI) holds for all convex bodies. For .n = 3 we already know by the BMI that .f3 is concave. Hence the AFI holds in full generality in three-dimensional space. The proof of Lemma 3.38 uses an algebraic extension of mixed volumes (see Lemmas 3.5 and 3.6 in [43]). Since it is sufficient to prove the AFI for a dense class of bodies, it is sufficient to prove the result for polytopes. In order to simplify formulas for mixed volumes of polytopes, we restrict the consideration to tuples of polytopes having the same combinatorial type (strongly isomorphic polytopes, polytopes having the same atype). The polytopes within such a combinatorial class are determined by their support vectors. This finally allows us to calculate derivatives with respect to these support vectors. We start with some preparations.

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We write .Pnn for the set of n-dimensional polytopes in .Rn . Polytopes in .Pnn are called n-polytopes. For vectors .u1 , . . . , uN ∈ Sn−1 and .h = (h1 , . . . , hN )T ∈ RN , we now consider polyhedral sets of the form P[h] :=

N n

.

H − (ui , hi ).

i=1

Clearly, if .h ∈ RN + , then .o ∈ P[h] and .P[h] is a polytope if and only if the vectors n−1 .u1 , . . . , uN ∈ S are not contained in any hemisphere. Further, .o ∈ int(P[h] ) if and only if .h1 , . . . , hN > 0. The vector h is called the vector of support numbers of .P[h] if .u1 , . . . , uN are the exterior unit facet normals of .P[h] , that is, if .dim(P[h] (ui )) = n − 1 for .i = 1, . . . , N . In this case, the support numbers are uniquely determined by .P[h] , since .h(P[h] , ui ) = hi . Strongly Isomorphic Polytopes As explained above, in order to prove the Aleksandrov–Fenchel inequality, it is sufficient to establish the inequality for a dense class of convex bodies such as convex polytopes. One of the classical proofs (due to Aleksandrov) proceeds in this way and in fact employs the subclass of simple polytopes such that the n polytopes for which the mixed volume is evaluated have the same a-type in the sense of the subsequent definitions. As we will see below, for polytopes having the same a-type simplified representations for the mixed volumes are available. Definition 3.39 (a) A polytope .P ∈ Pnn is simple if each vertex of P is contained in precisely n facets of P . (b) Two polytopes .P1 , P2 ∈ Pnn are strongly isomorphic (or analogous) if the condition .dim(P1 (u)) = dim(P2 (u)) is satisfied for all .u ∈ Sn−1 . Clearly, strong isomorphism of n-polytopes is an equivalence relation, the equivalence class of a polytope .P ∈ Pnn is called the a-type of P . For .Q ∈ Pnn we write .Q ∈ a(P ) if Q and P belong to the same class. The following lemma (see [43, Lemma 3.7] for explicit references to [59]) collects several geometric facts about strongly isomorphic polytopes that will be used in the following. Lemma 3.40 (a) If .P1 , P2 ∈ Pnn are strongly isomorphic, then the support sets .P1 (u) and .P2 (u) are also strongly isomorphic, for each .u ∈ Sn−1 . (b) If .P1 , . . . , Pm ∈ Pnn are strongly isomorphic, then all polytopes .α1 P1 + · · · + αm Pm with .α1 , . . . , αm ≥ 0 and .α1 + · · · + αm > 0 are strongly isomorphic.

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(c) If .P = P[h] ∈ Pnn is simple and has exterior facet normals .u1 , . . . , uN ∈ Sn−1 , then there is some .β > 0 such that any two of the polytopes .P[h+α] with .α = (α1 , . . . , αN )T and .|αi | ≤ β are strongly isomorphic. (d) For any .(K1 , . . . , Kn ) ∈ (Kn )n there is a sequence .(P1(m) , . . . , Pn(m) ) ∈ (Pnn )n , (m) .m ∈ N, such that .P → Kj as .m → ∞ (in the Hausdorff metric), for j j = 1, . . . , n, and .P1(m) , . . . , Pn(m) are simple and strongly isomorphic for each .m ∈ N. .

Mixed Volumes of Strongly Isomorphic Polytopes Let .P ∈ Pnn be a simple polytope with facet normals .u1 , . . . , uN ∈ Sn−1 . Then { } C(P ) := h ∈ (0, ∞)N : P[h] ∈ a(P )

.

is an open convex cone. In fact, if .h, h' ∈ C(P ), then .P[h] + P[h' ] ∈ a(P ) by Lemma 3.40 (b), and thus (P[h] + P[h' ] )(ui ) = P[h] (ui ) + P[h' ] (ui ) = P[h+h' ] (ui )

.

implies that .P[h] + P[h' ] = P[h+h' ] . The fact that .C(P ) is open follows from Lemma 3.40 (c). Lemma 3.41 Let .P ∈ Pnn be a simple polytope with exterior facet normals .u1 , . . . , uN ∈ Sn−1 . For .i = 1, . . . , n, let .Pi = P[h(i) ] ∈ a(P ) with (i)

hj = h(P[h(i) ] , uj ) for .j = 1, . . . , N . Then there are real numbers .aj1 ···jn , for .j1 , . . . , jn ∈ {1, . . . , N }, depending only on .a(P ) (and independent of the support numbers of the polytopes) and symmetric in the lower indices, such that .

V (P1 , . . . , Pn ) =

N E

.

j1 ,...,jn =1

(1)

(n)

aj1 ···jn hj1 · · · hjn .

The map .C(P )N e (h(1) , . . . , h(n) ) |→ V (P[h(1) ] , . . . , P[h(n) ] ) is of class .C ∞ . For a (twice continuously) differentiable function .F on an open subset of .RN we write .TF(h) = (F1 (h), . . . , FN (h))T ∈ RN for the gradient of .F at h and N N,N for the Hessian matrix of .F at h, with 2 2 .T F(h) := ∂ F(h) = (Fij (h)) i,j =1 ∈ R N respect to the standard basis of .R . Proof (of the AFI) We proceed by induction. The theorem has already been proved in the cases where .n ∈ {2, 3}. Hence let .n ≥ 3 (or even .n ≥ 4) and assume the theorem holds in smaller dimensions.

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Let .P ∈ Pnn be a simple polytope and fix .Kn−3 := K4..n := (P4 , . . . , Pn ) with n-polytopes .Pr ∈ a(P ) for .r = 4, . . . , n. Then the function .F : C(P ) → (0, ∞) defined by F(h) := V (P[h] [3], Kn−3 )

.

has the following properties. (i) .F is .C ∞ and positively homogeneous of degree 3 on the open convex cone .C(P ). (ii) For .h ∈ C(P ), we have Fi (h) =

.

) 3 (n−1) ( V P[h] (ui )[2], Kn−3 (ui ) > 0, n

where .Kn−3 (ui ) = (P4 (ui ), . . . , Pn (ui )). The derivative follows from an explicit formula for the mixed volume of strongly isomorphic polytopes and the symmetry of mixed volumes. (iii) By the induction hypothesis, the AFI holds for the mixed volume .V (n−1) in ⊥ .u . Hence, since for .λ ∈ [0, 1] we have i ) ( .P[(1−λ)h+λh' ] (ui ) = (1 − λ)P[h] + λP[h' ] (ui ) = (1−λ)P[h] (ui )+λP[h' ] (ui ), 1

the map .C(P ) e h |→ Fi (h) 2 is concave. (iv) Let .J := {(i, j ) ∈ {1, . . . , N }2 : dim(P (ui ) ∩ P (uj )) = n − 2}. For .(i, j ) ∈ J and .h ∈ C(P ), it follows from an explicit formula for the mixed volume of strongly isomorphic polytopes that Fij (h) =

.

1 3 2 V (n−2) / n n − 1 sin (ui , uj ) ( ) P[h] (ui ) ∩ P[h] (uj ), Kn−3 (ui , uj ) > 0,

where .Kn−3 (ui , uj ) = (P4 (ui ) ∩ P4 (uj ), . . . , Pn (ui ) ∩ Pn (uj )) and (ui , uj ) ∈ (0, π ) denotes the angle enclosed by .ui and .uj , that is, .cos / (ui , uj ) = (as in the proof of Theorem 3.12 (a)). Note that all these intersections are .(n − 2)-dimensional since .(i, j ) ∈ J . Further, we have .Fii (h) < 0 and .Fij (h) = 0 if .(i, j ) ∈ {(r, s) ∈ {1, . . . , N }2 \ J : r /= s}. Since any two facets are connected by a sequence of facets such that the intersection of successive facets has dimension .n − 2, it follows that the matrix 2 .T F(h) is irreducible. ./

The following Lemma 3.42 shows that these properties imply that .C(P ) e h |→ 1 F(h) 3 is concave. Now from the approximation Lemma 3.40 (d) and the continuity of mixed volumes it follows that 1

L |→ V (L[3], K3 , . . . , Kn ) 3 ,

.

L ∈ Kn ,

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is a concave map, and therefore .f3 is concave for all convex bodies. Hence, AFI holds in n-dimensional Euclidean space, which completes the induction step. Lemma 3.42 Let .C ⊂ (0, ∞)N be an open, convex cone. Let .F : C → R be a function of class .C 3 such that (a) .F is positively 3-homogeneous, 1 (b) .C e h |→ Fi (h) 2 is concave and .Fi (h) > 0 for .h ∈ C, 2 (c) .T F(h) is irreducible and .Fij (h) ≥ 0 for .i /= j and .h ∈ C. 1

Then .C e h |→ F(h) 3 is a concave function. In the proof of the analytic Lemma 3.42, the following fact from linear algebra, which is based on a special case of the Perron–Frobenius theorem, is used. For details, we refer to [43, Section 3.5]. Lemma 3.43 Let .M = (mij ) ∈ RN,N be symmetric, irreducible and such that n−1 .mij ≥ 0 for .i /= j . Suppose that .v1 ∈ S is a positive eigenvector of M with corresponding eigenvalue .λ1 > 0. If M does not have any eigenvalues in .(0, λ1 ), then M ≤ λ1 v1 ⊗ v1 .

.

This completes the proof of the Aleksandrov–Fenchel inequality.

u n

3.3.3 Generalized Brunn–Minkowski and Minkowski Inequalities We already observed that the following theorem, which is a generalized version of the Brunn–Minkowski inequality, is a consequence of the Aleksandrov–Fenchel inequality. Theorem 3.44 (GBMI) For .m ∈ {2, . . . , n} and .Km+1 , . . . , Kn ∈ Kn , let .K = (Km+1 , . . . , Kn ). Then the map 1

Kn e L |→ V (L[m], K) m

.

is concave. From the inequalities derived up to this point, a variety of strong geometric inequalities can be deduced. For example, we obtain a general version of Minkowski’s inequality.

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Theorem 3.45 (GMI) For .m ∈ {2, . . . , n} and .K1 , K2 , Km+1 , . . . , Kn ∈ Kn , let .K = (Km+1 , . . . , Kn ). Then V (K1 [m − 1], K2 , K)m ≥ V (K1 [m], K)m−1 V (K2 [m], K).

.

Proof Let .K = (Km+1 , . . . , Kn ). The function .fm defined by 1

fm (t) := V ((K1 + tK2 ) [m], K) m ,

t ≥ 0,

.

is concave, and hence fm' (t) |t=0+ =

.

1 1 V (K1 [m], K) m −1 · m · V (K1 [m − 1], K2 , K) ≥ fm (1) − fm (0). m

This and the GBMI for mixed volumes imply that 1

V (K1 [m], K) m −1 V (K1 [m − 1], K2 , K)

.

1

1

≥ V ((K1 + K2 ) [m], K) m − V (K1 [m], K) m 1

1

1

≥ V (K1 [m], K) m + V (K2 [m], K) m − V (K1 [m], K) m 1

= V (K2 [m], K) m , and hence V (K1 [m − 1], K2 , K) ≥ V (K1 [m], K)

.

m−1 m

1

V (K2 [m], K) m , u n

which is the required inequality.

The GBMI and the GMI Are Essentially Equivalent In fact, using the GMI and the Minkowski linearity of mixed volumes, we get V ((K1 + K2 ) [m], K) = V ((K1 + K2 ) [m − 1], K1 + K2 , K)

.

= V ((K1 + K2 ) [m − 1], K1 , K) + V ((K1 + K2 ) [m − 1], K2 , K) ≥ V ((K1 + K2 ) [m], K)

m−1 m

+ V ((K1 + K2 ) [m], K) = V ((K1 + K2 ) [m], K) which yields the GBMI.

1

V (K1 [m], K) m

m−1 m

m−1 m

(

1

V (K2 [m], K) m

) 1 1 V (K1 [m], K) m + V (K2 [m], K) m ,

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3.4 Lp Brunn–Minkowski Theory In the beginning of this section we briefly describe the basics of the .Lp Brunn– Minkowski theory. Thereafter we present the .Lp Brunn–Minkowski inequality, in the case .p ≥ 1 and in the more delicate case .0 ≤ p < 1, and we focus in particular on the so-called log-Brunn–Minkowski inequality, corresponding to the case .p = 0.

3.4.1 The Lp Addition The .Lp Brunn–Minkowski theory is based on the .Lp addition of convex bodies, introduced by Firey in [35], that we are now going to recall. Throughout this part we will be working with convex bodies containing the origin. Recall that Kno = {K ∈ Kn : o ∈ K}.

.

Moreover, .Kn(o) denotes the family of convex bodies containing the origin in their interior. Note that .K ∈ Kno if and only if .hK ≥ 0, and .K ∈ K(o) if and only if .hK (x) > 0 for every .x /= o. We start from the definition of p-means of non-negative numbers. Definition 3.46 Let .t ∈ [0, 1]. . For .p > 0 and .a, b ≥ 0 we set Mp (a, b; t) := ((1 − t)a p + tbp )1/p .

.

. For .p = 0 and .a, b > 0 we set M0 (a, b; t) := a 1−t bt .

.

. For .p < 0 and .a, b > 0 we set Mp (a, b; t) := ((1 − t)a p + tbp )1/p .

.

. For .p ≤ 0 and .a, b ≥ 0 such that .ab = 0, we set Mp (a, b; t) := 0.

.

. For .a, b ≥ 0 we set M∞ (a, b; t) := max{a, b};

.

M−∞ (a, b; t) = min{a, b}.

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123

Remark 3.47 Let .p, q ∈ [−∞, ∞] with .p ≤ q. For every .t ∈ [0, 1] and for every a, b ≥ 0:

.

Mp (a, b; t) ≤ Mq (a, b; t).

.

Proposition 3.48 Let .p ≥ 1. For every .K, L ∈ Kno , and for every .α, β ≥ 0, the function .h : Sn−1 → R+ defined by )1/p ( h(x) = αhK (x)p + βhL (x)p

.

is 1-homogeneous and convex. The proof of the previous result follows from basic properties of p-means; the condition .p ≥ 1 is crucial for convexity. Definition 3.49 Let .p ≥ 1. For .K, L ∈ Kno , and for .α, β ≥ 0, we define the .Lp linear combination of K and L, denoted by α · K +p β · L,

.

through the relation ( p p )1/p hα·K+p β·L := αhK + βhL .

.

Note that .hα·K+p β·L ≥ 0, and therefore .α · K +p β · L ∈ Kno . Hence we have a new algebraic structure on .Kn , for every .p ≥ 1. The standard case, i.e. that of Minkowski addition, corresponds to .p = 1. A comment about notation: the product “.·” depends on p like the sum “.+p ”. For simplicity, we avoid to indicate it explicitly by a further index p. Exercise 3.50 What is the .L2 addition of two orthogonal segments centered at the origin in the plane? The notion of .Lp addition is at the basis of the so-called .Lp Brunn–Minkowski theory (or Brunn–Minkowski–Firey theory) of convex bodies. This theory began with the work of Firey (see [35]), and it was then developed by Lutwak (starting from [52]). The .Lp Brunn–Minkowski theory constitutes now an important part of convex geometry; a recent survey can be found, for instance, in [59, Chapter 9]. Remark 3.51 Let .p, q ≥ 1, be such that .p ≤ q, and let .α, β ≥ 0, .K, L ∈ Ko . Then, by Remark 3.47, we have hα·K+p β·L ≤ hα·K+q β·L ,

.

whence α · K +p β · L ⊂ α · K +q β · L

.

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(clearly, in the previous relations, the symbol “.·” is the product corresponding to the Lp addition on the left-hand side, and the product corresponding to the .Lq addition on the right-hand side).

.

3.4.2 The First Variation of the Volume with Respect to the Lp Addition and the Lp Surface Area Measure In analogy with the formula which expresses the first variation of the volume with respect to the Minkowski addition, (see formula (1.2) in Chap. 1), we present the corresponding formula for the .Lp addition. Theorem 3.52 Let .K, L ∈ Kno , and let .p ≥ 1. Then .

lim

ε→0+

f V (K +p ε · L) − V (K) 1 = hL (u)p hK (u)1−p dSn−1 (K, u). ε p Sn−1 (3.21)

Note that the relation p

p

p

hα·K+p β·L = αhK + βhL

.

tells us that the .Lp addition behaves linearly with respect to the pth power of support functions. Hence formula (3.21) shows that the measure 1−p

hK (·)Sn−1 (K, ·)

.

is the first variation of the volume with respect to the .Lp addition. This measure is called the .Lp surface area measure of K. A corresponding Minkowski-type problem can be posed; for more details we refer the reader to [59, Chapter 9].

3.4.3 The Lp Brunn–Minkowski Inequality for p ≥ 1 Let .p ≥ 1. If .K ∈ Kno and .λ ≥ 0, then the support function of .λ · K is hλ·K = λ1/p hK .

.

Hence the multiplication by a non-negative real number .λ in the .Lp sense, coincides with the standard dilation of a factor .λ1/p . In particular this changes the degree of homogeneity of the volume: for .K ∈ Kno and .λ ≥ 0, Vn (λ · K) = λn/p Vn (K).

.

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125

Let us now see the .Lp version of the Brunn–Minkowski inequality. Theorem 3.53 Let .p ≥ 1. Let .K0 , K1 ∈ Kno , and let .t ∈ [0, 1]. Then V ((1 − t) · K0 +p t · K1 )p/n ≥ (1 − t)V (K0 )p/n + tV (K1 )p/n .

.

Proof Assume that .V (K0 ) = V (K1 ) = 1. By Remark 3.51, the monotonicity of the volume and the classical Brunn–Minkowski inequality, we have: V ((1 − t) · K0 +p t · K1 ) ≥ V ((1 − t)K0 + tK1 ) [ ]n ≥ (1 − t)V (K0 )1/n + tV (K1 )1/n

.

]n/p [ = (1 − t)V (K0 )p/n + tV (K1 )p/n . Hence the inequality is proved under the assumption that .K0 and .K1 have volume 1. The general case follows from a standard argument based on homogeneity (see also (3.13) and the related remark). u n

3.4.4 The Case 0 ≤ p < 1 and the log-Brunn–Minkowski Inequality In this part we extend the definition of .Lp addition (or better, of .Lp -convex linear combination) to the case .0 ≤ p < 1. The construction requires the notion of a Wulff shape, also known as the Aleksandrov body.

Wulff Shape and Aleksandrov Body With a continuous and non-negative function .f : Sn−1 → R we associate the set K[f ] := {x ∈ Rn : ≤ f (u) for every u ∈ Sn−1 }.

.

Clearly, .K[f ] is closed and convex, being the intersection of closed half-spaces. Moreover, as f is continuous on .Sn−1 , it is bounded; this implies that .K[f ] is bounded. Note also that .o ∈ K[f ], since .f ≥ 0; if we assume that f is strictly positive, then the origin is an interior point of K. We conclude that .∅ /= K[f ] ∈ Kn . The convex body .K[f ] is called the Wulff shape or the Aleksandrov body of f . It is easy to see that hK[f ] ≤ f

.

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and we have equality (for every point of .Sn−1 ) if and only if the 1-homogeneous extension of f is a convex functions (i.e. f is a support function). .K[f ] can also be characterized as follows: its support function is the largest 1-homogeneous convex function which is bounded from above by f . The Convex Linear Combination of Order p with p ≥ 0 Let .p ≥ 0, .K0 , K1 ∈ Kno , and .t ∈ [0, 1]. Set, for .u ∈ Sn−1 , ft (u) := Mp (hK0 (u), hK1 (u); t),

.

that is, .ft is the p-mean of the support functions of .K0 and .K1 (with weights .1 − t and t). If .p < 1, in general this is not a support function. The p convex linear combination of .K0 and .K1 is defined as (1 − t) · K0 +p t · K1 := K[ft ].

.

In more explicit terms: (1−t)·K0 +p t ·K1 = {x ∈ Rn : ≤ Mp (hK0 (u), hK1 (u); t) for every u ∈ Sn−1 }.

.

Remark 3.54 If .p ≥ 1, .ft is a support function, so that this definition is consistent with the one of .Lp addition provided before. In the case .p = 0, we get t n−1 (1 − t) · K0 +0 t · K1 = {x ∈ Rn : ≤ h1−t }. K0 (u)hK1 (u) for every u ∈ S

.

This is the so-called log Minkowski convex linear combination of .K0 and .K1 . Exercise 3.55 Let .K0 and .K1 be two orthogonal segments in the plane, centered at the origin. What is .

1 1 · K0 +0 · K1 ? 2 2

The log-Brunn–Minkowski Inequality In [18], Böröczky, Lutwak, Yang and Zhang proved the following version of the Brunn–Minkowski inequality for the 0 addition, in the two-dimensional case. Theorem 3.56 Let .K0 , K1 ∈ K2o be centrally symmetric with respect to the origin. If .t ∈ [0, 1], then V ((1 − t) · K0 +0 t · K1 ) ≥ V (K0 )1−t V (K1 )t .

.

(3.22)

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127

For .t ∈ (0, 1) equality holds if and only if .K0 and .K1 are dilates of each other, or they are parallelograms with parallel sides. Remark 3.57 Similarly to the case .p ≥ 1, by the monotonicity of p-means with respect to p (see Remark 3.10), and by the definition of .Lp convex linear combination for .p ≤ 1, we have the inclusion (1 − t) · K0 +0 t · K1 ⊂ (1 − t) · K0 +p t · K1

.

for every .K0 , K1 ∈ Kno , .t ∈ [0, 1], and .p ≥ 0. Hence, by a standard homogeneity argument (see the proof of Theorem 3.53, (3.13) and the related remark), inequality (3.22) implies, under the same assumptions on .K0 , K1 and t, and for .n = 2, that V ((1 − t) · K0 +p t · K1 )p/n ≥ (1 − t)V (K0 )p/n + tV (K1 )p/n ,

.

for every .p > 0. Remark 3.58 The assumption of central symmetry is necessary (see [54]), even if some extensions of this result to the non-symmetric case have been established (see [64]). Theorem 3.56 led to the following conjecture. Conjecture 3.59 Let .p ∈ (0, 1). If .K0 , K1 ∈ Kn are centrally symmetric with respect to the origin and .t ∈ (0, 1), then V ((1 − t) · K0 +p t · K1 )p/n ≥ (1 − t)V (K0 )p/n + tV (K1 )p/n .

.

In particular, in the limiting case .p = 0, V ((1 − t) · K0 +0 t · K1 ) ≥ V (K0 )1−t V (K1 )t .

.

This conjecture has been intensively studied in recent years, and much progress has been made. An updated survey on the state of the art can be found in [46].

3.4.5 The log-Brunn–Minkowski Inequality for Unconditional Bodies In this part we present a result due to Saroglou [58], which proves the validity of the log-Brunn–Minkowski conjecture for unconditional convex bodies. A convex body .K ∈ Kn is called unconditional if it is symmetric with respect to all coordinate hyperplanes of .Rn : x = (x1 , . . . , xn ) ∈ K

.

⇒

(±x1 , . . . , ±xn ) ∈ K,

for every possible choice of the signs “.+” and “.−”.

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Theorem 3.60 If .K0 , K1 ∈ Kno are unconditional and .t ∈ [0, 1], then V ((1 − t) · K0 +0 t · K1 ) ≥ V (K0 )1−t V (K1 )t .

.

Proof For general points .x, y ∈ Rn we denote their coordinates by .(x1 , . . . , xn ), .(y1 , . . . , yn ). We set K01−t K1t := {(±|x1 |1−t |y1 |t , . . . , ±|xn |1−t |yn |t ) : x ∈ K0 , y ∈ K1 }.

.

Again, all choices of the signs are considered; in particular, .K01−t K1t is symmetric with respect to all coordinate hyperplanes. We will prove the following two facts: (i) (1 − t) · K0 +0 t · K1 ⊃ K01−t K1t ;

.

(ii) V (K01−t K1t ) ≥ V (K0 )1−t V (K1 )t .

.

To prove the first inclusion, we start from t n−1 (1 − t) · K0 +0 t · K1 = {x ∈ Rn : ≤ h1−t }. K0 (u)hK1 (u) for every u ∈ S

.

Then, we need to prove that .z = (z1 , . . . , zn ) ∈ K01−t K1t implies, for .u ∈ Sn−1 , t ≤ h1−t K0 (u)hK1 (u).

.

As .K01−t K1t , .K0 and .K1 are symmetric with respect to all coordinate hyperplanes, it is sufficient to consider the case in which all coordinates of z and u are non-negative. There exist .x = (x1 , . . . , xn ) ∈ K0 and .y = (y1 , . . . , yn ) ∈ K1 (which we may also assume to have non-negative coordinates) such that, for .i ∈ {1, . . . , n}, zi = xi1−t yit .

.

3 Geometric and Functional Inequalities

129

Then =

n E

.

t xi1−t yit u1−t i ui

i=1

=

n E (xi ui )1−t (yi ui )t i=1

≤

( n E

)1−t ( xi ui

i=1

n E

)t yi ui

i=1

t = ()1−t ()t ≤ h1−t K0 (u)hK1 (u),

where we used Hölder’s inequality and the definition of support function. Hence we have proved claim (i). In order to prove claim (ii), let Rn+ := {x = (x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0},

.

and let f := 1K0 ∩Rn ,

.

g := 1K1 ∩Rn ,

+

+

h := 1(K 1−t K t )∩Rn 0

+

1

(we recall that .1A is the standard indicator function of a set A). We also set f¯(x) ¯ := f¯(x¯1 , . . . , x¯n ) = f (ex¯1 , . . . , ex¯n )ex¯1 +···+x¯n ,

.

¯ z). By the definition of and we give similar definitions for .g¯ = g( ¯ y) ¯ and .h¯ = h(¯ 1−t t .K 0 K1 , it follows that ¯ z) ≥ f¯1−t (x) ¯ g¯ t (y) ¯ h(¯

.

for .x, ¯ y¯ ∈ Rn and .z¯ = (1 − t)x¯ + t y. ¯ Then, by the Prékopa–Leindler inequality (see Theorem 3.27), f .

¯ z) d¯z ≥ h(¯ Rn

(f R

f¯(x) ¯ dx¯ n

)1−t (f Rn

)t g( ¯ y) ¯ dy¯

.

On the other hand, by a simple change of variable we see that f .

f ¯ z) d¯z = h(z) dz = V ((K01−t K1t ) ∩ Rn+ ), h(¯ n n R R+

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and, similarly, f .

f ¯(x) f (x) dx = V (K0 ∩ Rn+ ), f ¯ d x ¯ = Rn Rn+ f f g( ¯ y) ¯ d y ¯ = g(y) dy = V (K1 ∩ Rn+ ). Rn Rn+ u n

This concludes the proof of (ii) and of the present theorem.

3.5 Inequalities of Brunn-Minkowski Type for Variational Functionals In this section we present some functionals, coming from the world of calculus of variations and elliptic PDE’s, which, when restricted to convex bodies, surprisingly share some significant properties with the ordinary volume, including inequalities of Brunn–Minkowski type, representation formulas, and variational formulas. Our first and main example will be the electrostatic capacity; the second part of the section is devoted to other well-known functionals.

3.5.1 The Electrostatic Capacity of a Convex Body The standard definition (see, for instance, [47]) of the electrostatic capacity of a convex body .K ∈ Kn , .n ≥ 3, is {f .

Cap(K) := inf

} |Tu(x)| dx : u ∈ 2

Rn

Cc1 (Rn ),

u ≥ 1 on K

(in the 2-dimensional case this notion is naturally replaced by logarithmic capacity, see, for instance, [30]). Here .Cc1 (Rn ) denotes the set of functions in .C 1 (Rn ) with compact support. For simplicity, let us assume that K has non-empty interior. Then .Cap(K) can also be written as f .

Cap(K) =

Rn \K

|Tu(x)|2 dx

where u is the solution of the following exterior boundary value problem ⎧ in Rn \ K, ⎨ Au = 0 . u=1 on ∂K, ⎩ lim|x|→∞ u(x) = 0.

(3.23)

3 Geometric and Functional Inequalities

131

The function u is uniquely determined (by problem (3.23) and by the maximum principle), and it is called the equilibrium potential of K. Another consequence of the maximum principle is that .0 ≤ u(x) ≤ 1 for every .x ∈ Rn \ K. As a functional from .Kn to .R, the capacity has the following properties: . it is continuous with respect to the Hausdorff metric; . it is rigid motion invariant; . it is homogeneous of degree .n − 2: for .K ∈ Kn and .λ ≥ 0, .

Cap(λK) = λn−2 Cap(K).

3.5.2 The Brunn–Minkowski Inequality for the Capacity The electrostatic capacity satisfies an inequality of Brunn–Minkowski type, as shown by the following result. Theorem 3.61 Let .K0 , K1 ∈ Kn , .n ≥ 3, and let .t ∈ [0, 1]. Then .

Cap((1 − t)K0 + tK1 )1/(n−2) ≥ (1 − t) Cap(K0 )1/(n−2) + t Cap(K1 )1/(n−2) . (3.24)

If .K0 , K1 ∈ Knn and equality holds in (3.24), then .K0 and .K1 are homothetic. The previous theorem is due to Borell (see [15]), who proved (3.24), and to Caffarelli, Jerison and Lieb (see [27]), who characterized equality conditions. The similarity between (3.24) and the Brunn–Minkowski inequality for the volume is clear. Both assert the concavity in .Kn of a certain functional, raised to the reciprocal of its homogeneity order, with respect to the Minkowski addition. Outline of a Proof of Theorem 3.61 Here we sketch a proof of the Brunn–Minkowski inequality for the capacity. The complete argument can be found in [32]. Step 1. Let K be a convex body with non-empty interior in .Rn , .n ≥ 3, and let u be the equilibrium potential of K, i.e. the solution to problem (3.23). Extend u to be identically 1 on K. A simple consequence of the maximum principle is that 0 s} = (1 − t){u0 > s} + t{u1 > s}.

.

In other words, the super-level sets of .u˜ t are the Minkowski convex linear combinations of the corresponding super-level sets of .u0 and .u1 . In particular we have u˜ t = 1 on ∂K,

and

.

lim u(x) ˜ = 0.

|x|→∞

(3.27)

Step 4. The most delicate part of the proof is to compare .u˜ t with the equilibrium potential .ut of .Kt . This is done showing that .u˜ t is a viscosity sub-solution of the Laplace equation in .Rn \ Kt : Au˜ t ≥ 0

.

in Rn \ Kt .

(3.28)

In order to prove (3.28) one needs to evaluate and estimate the Hessian matrix of u˜ at a point z, in terms of the Hessian matrices of .u0 at x and of .u1 at y, where .x, y, z are such that .z = (1 − t)x + ty and .

u(z) ˜ = u0 (x) = u1 (y).

.

The details can be found in [32]. Together with the boundary conditions in (3.23), (3.27) and the maximum principle, we then deduce that u˜ t ≤ ut

.

in .Rn .

(3.29)

3 Geometric and Functional Inequalities

Step 5.

133

By (3.29), (3.26) and (3.25), we deduce: .

Cap(Kt ) ≥ min{Cap(K0 ), Cap(K1 )}.

A standard homogeneity argument leads then to the Brunn–Minkowski inequality for capacity.

3.5.3 A Variational Formula for the Capacity and the Corresponding Minkowski Problem We start this part with a formula for capacity which parallels the representation formula (1.1) for the volume. Let .K ∈ Kn , and assume that it has non-empty interior. Let u be the equilibrium potential of K. Then .Tu can be appropriately extended to almost every point of .∂K, with respect to the .(n − 1)-dimensional Hausdorff measure restricted to .∂K, and the extension is such that Tu ∈ L2 (∂K).

.

We refer the reader to the article [47] for a detailed presentation of these facts. Hence Cap we can define a measure .Sn−1 (K, ·) on .Sn−1 , as follows: for every Borel subset .ω of .Sn−1 f Cap .S |Tu(x)|2 dHn−1 (x). n−1 (K, ω) := −1 νK (ω)

−1 Recall that .νK is the Gauss map of K, and .νK (ω) is the set of points of .∂K where the outer unit normal is defined, and it belongs to .ω. The following property can be proved (if K has non-empty interior): Cap

Sn−1 (K, ·) is not concentrated on any great subsphere of Sn−1 .

.

Moreover, translation invariance of capacity implies that f Cap . y dSn−1 (K, y) = 0. n−1 S

(3.30)

(3.31)

The following two formulas can be found in the paper [47]. Proposition 3.62 Let .K ∈ Kn , .n ≥ 3, have non-empty interior, and let u be the equilibrium potential of K. Then f 1 Cap hK (y) dSn−1 (K, y). (3.32) . Cap(K) = n − 2 Sn−1

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Moreover, for every .L ∈ Kn with non-empty interior, .

lim

ε→0+

f Cap(K + εL) − Cap(L) Cap = n−1 hL (y) dSn−1 (K, y). ε S

(3.33)

Comparing (3.32) and (3.33) with the corresponding formulas that we have Cap seen for the volume, i.e. (1.1) and (1.2), it becomes clear that .Sn−1 (K, ·) is the counterpart, for capacity, of the area measure. For this reason it is called the capacitary area measure of K. Remark 3.63 As in the case of the volume, (3.32) can be obtained from (3.33), using the homogeneity of capacity. The proof of (3.33), as presented in [47], is first made in the smooth case, where a (demanding) argument based on integration by parts formulas is exploited. The general case is then achieved by approximation. Based on the formulas provided in Proposition 3.62, Jerison (see [47]) considered the Minkowski problem for electrostatic capacity, which requires to find a convex body with prescribed capacitary area measure. A complete solution to this problem, in terms of existence, uniqueness and regularity, was provided in [47]. We cover Jerison’s existence and uniqueness result in the following statement. Theorem 3.64 (Jerison) Let .n > 3 and let .μ be a non-negative Borel measure on Sn−1 . Then .μ is the capacitary area measure of a convex body K if and only if

.

(a) .μ is not concentrated on any great sub-sphere of .Sn−1 and (b) .μ is centered in the sense that f . y dμ(y) = 0. Sn−1 Moreover, the convex body K is unique up to translations. In dimension .n = 3 the corresponding statement is slightly different. It is worth noting that, just as for the ordinary Minkowski problem, the uniqueness part follows from the characterization of equality conditions in the Brunn–Minkowski inequality (3.24).

3.5.4 The Torsion and the First Dirichlet Eigenvalue of the Laplacian In this section we present two more functionals defined on .Kn , namely the torsion and the first Dirichlet eigenvalue of the Laplacian, which, like the capacity, have several features in common with the volume. In particular, (a) they satisfy an inequality of Brunn–Minkowski type;

3 Geometric and Functional Inequalities

135

(b) they admit a representation formula and a formula for the first variation, similar to (3.32) and (3.33); (c) a Minkowski-type problem has been posed and solved for these functionals. As for the capacity, these functionals are classical examples in the calculus of variations.

The Torsion Let .K ∈ Kn , and assume that K has non-empty interior. The variational definition of the torsion of K , denoted by .τ (K), is the following: {f } f 2 1 1,2 K |Tu(x)| dx := inf (f |u(x)| dx > 0 . . )2 : u ∈ W0 (int(K)), τ (K) K |u(x)| dx K Here, following the standard notation, we denote by .W 1,2 (int(K)) the Sobolev space of functions with (weak) gradient in .L2 (int(K)), and by .W01,2 (int(K)) the closure of .Cc∞ (int(K)) with respect to the norm of .W 1,2 (int(K)). The previous minimum problem admits in fact a minimizer u, which is the unique solution of the following boundary value problem: { .

Au = −2 in int(K), u=0 on ∂K.

(3.34)

Then .τ (K) can be expressed as f τ (K) =

|Tu|2 dx.

.

K

The torsion is rigid motion invariant, continuous with respect to the Hausdorff metric, and homogeneous of order .n + 2: for .K ∈ Kn and .λ > 0, τ (λK) = λn+2 τ (K).

.

The Brunn–Minkowski inequality for .τ was established by Borell (see [16]), and equality conditions were characterized in [29]. Theorem 3.65 Let .K0 , K1 ∈ Kn and let .t ∈ [0, 1]. Then τ ((1 − t)K0 + tK1 )1/(n+2) ≥ (1 − t)τ (K0 )1/(n+2) + tτ (K1 )1/(n+2) .

.

(3.35)

If .K0 , K1 ∈ Knn and equality holds in (3.35), then .K0 and .K1 are homothetic.

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As in the case of the capacity, the gradient of u admits an .L2 (∂K) extension to .∂K, and we have the following formulas (see [31, Theorem 3.1]): τ (K) =

.

f 1 τ hK (y) dSn−1 (K, y); n + 2 Sn−1

moreover, for every .L ∈ Kn with non-empty interior, .

lim

ε→0+

f τ (K + εL) − τ (K) τ = n−1 hL (y) dSn−1 (K, y). ε S

τ (K, ·) is defined in a similar way as the In the previous formulas, the measure .Sn−1 capacitary measure:

f :=

τ .Sn−1 (K, ω)

−1 νK (ω)

|Tu(x)|2 dHn−1 (x),

for every Borel subset .ω of .Sn−1 . Translation invariance and the condition that .int(K) /= ∅ imply that τ Sn−1 (K, ·) is not concentrated on any great subsphere of Sn−1 ,

.

f .

S

n−1

τ y dSn−1 (K, y) = 0.

τ (K, ·), which represents the first variation of .τ at K, induces a The measure .Sn−1 Minkowski problem which has been solved in [31].

Theorem 3.66 Let .μ be a non-negative Borel measure on .Sn−1 . If (a) .μ is not concentrated on any great sub-sphere of .Sn−1 and (b) .μ is centered, i.e. f . y dμ(y) = 0, Sn−1 then there exists a convex body K with non-empty interior such that τ Sn−1 (K, ·) = μ(·).

.

Moreover, K is unique up to translations.

3 Geometric and Functional Inequalities

137

3.5.5 The First Dirichlet Eigenvalue of the Laplacian Let .K ∈ Kn , and assume that K has non-empty interior. Consider the following minimum problem for the so-called Rayleigh quotient: {f λ(K) := inf

.

|Tu(x)|2 dx K f 2 K u (x) dx

}

f :u∈W

1,2

2

u (x) dx > 0 .

(int(K)), K

There exists a minimizer u for this problem, which can be chosen to be the solution of ⎧ ⎨ Au = −λ(K)u in int(K), . (3.36) u>0 in K, ⎩ u=0 on ∂K. The (positive) number .λ(K) is called the first Dirichlet eigenvalue of the Laplacian. It can be also characterized as the best constant .λ such that the following Poincaré inequality, f u2 dx ≤

.

K

1 λ

f |Tu|2 dx K

is valid for every .u ∈ C 1 (int(K)), with compact support contained in .int(K). As in the previous examples, it is a continuous and translation invariant functional on .Kn ; it is homogeneous of order .−2. The Brunn–Minkowski inequality for .λ has been proved by Brascamp and Lieb in [23] (see also [16] for a different proof), while equality conditions have been characterized in [29]. Theorem 3.67 Let .K0 , K1 ∈ Kn and let .t ∈ [0, 1]. Then λ((1 − t)K0 + tK1 )−1/2 ≥ (1 − t)λ(K0 )−1/2 + tλ(K1 )−1/2 .

.

(3.37)

If .K0 , K1 ∈ Knn and equality holds in (3.37), then .K0 and .K1 are homothetic. Following the lines that we have seen for the capacity and for the torsion, a Minkowski-type problem can be posed for .λ. This problem was firstly considered by Jerison in [48], who proved the existence of a solution. Uniqueness was then established in [29], based on the equality conditions in (3.37).

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3.5.6 Remarks . We have seen that the validity of the classical Brunn–Minkowski inequality goes beyond convex sets. By this point of view, the picture for the functionals that we have seen in this part is not complete. The Brunn–Minkowski inequality for the torsion and for the first Dirichlet eigenvalue of the Laplacian can be extended to non-convex sets (with sufficiently smooth boundary). A corresponding extension for the capacity is an interesting open problem. . The solutions to the boundary value problems (3.23), (3.34) and (3.36) inherit specific qualitative properties from the convexity of the domain K (we have already seen it in the case of the capacity). The capacitary equilibrium is quasiconcave; the solution of (3.34) is power concave, the solution of (3.36) is log-concave (see [29] for references about these results). It is interesting to notice that the argument of many of the proofs of geometric properties of the solutions, can be adapted to prove the Brunn–Minkowski inequality for the relevant functionals. . The results that we have seen in this section have been object of numerous extensions. References to a first group of results, where Cap, τ or λ are replaced by other variational functionals (like p-capacity, p-torsion, eigenvalues of other elliptic operators), can be found in [29]. More recently, extensions where the usual Minkowski addition is replaced by the p addition have been studied. The literature in this area is rapidly growing; possible references are [50, 62, 65].

3.6 Brascamp–Lieb–Barthe Inequalities and Reverse Isoperimetry In this section, we discuss a dual pair of analytic inequalities which had a major impact on convex geometry. Conversely, research in convexity has motivated to a large extent the further development of these and related analytic inequalities.

3.6.1 Brascamp–Lieb and Barthe Inequalities The following theorem combines two analytic-geometric inequalities which have proved to be extremely useful and have inspired a lot of research. See Barthe [5] for the following result, Brascamp, Lieb, Luttinger [22, 24, 51] for contributions which inspired much of this work and Barthe, Wolff [7] for a recent account of the state of the art. Many remarkable special cases will be discussed after the next theorem has been stated. The first inequality will be addressed as the Brascamp–Lieb inequality (BLI), the reverse inequality is the Barthe inequality (BI). The latter involves an upper integral since the integrand on the left side need not be measurable in general. An outline of the proof of Theorem 3.68 is given in Chapter of this volume.

3 Geometric and Functional Inequalities

139

Theorem 3.68 (General Brascamp–Lieb and Barthe Inequality) For .i = E 1, . . . , k, let .ci > 0, .ni ∈ N with . ki=1 ci ni = n. Let .Bi : Rn → Rni be linear and surjective maps with Euclidean adjoint .Bi∗ : Rni → Rn . Let .fi : Rni → R be integrable. Then )ci k (f 1 || ci f (B x) dx ≤ f (x ) dx √ i i i i i D i=1 Rni Rn i=1

f .

k ||

and f .

∗

Rn

sup

≥

{ k ||

fi (zi ) : x = ci

i=1

√ D

} ci Bi∗ zi , zi

∈R

ni

dx

i=1

k (f || i=1

k E

)ci

Rni

fi (xi ) dxi

,

where (E ) ⎧ ⎫ k ∗ ⎨ det ⎬ i=1 ci Bi Ai Bi .D := inf : Ai ∈ Rpd (ni , ni ) ||k ci ⎩ ⎭ i=1 (det Ai ) and .Rpd (ni , ni ) are the positive definite (real) .ni × ni -matrices.

Remarks n . If ki=1 Ker(Bi ) /= {o}, then D = 0 and the stated inequalities hold trivially. E . If ni = n, Bi = In , ci = 1/pi and ki=1 1/pi = 1 and if fi is replaced by 1/c fi i then BLI turns into Hölder’s inequality. In this situation we have D = 1. To see this, one can use an inequality for determinants: Let Ai ∈ Rpd (n, n), E i = 1, . . . , k, and ci > 0 with ki=1 ci = 1. Then

.

det

( k E i=1

) ci Ai

≥

k ||

(det Ai )ci

i=1

with equality if and only if A1 = · · · = Ak . . If k = 2, n1 = n2 = n, B1 = B2 = In , c1 = 1 − λ, c2 = λ with λ ∈ [0, 1], we obtain the PLI from the BLI. Similarly, a multivariate version of the PLI is obtained. . Young’s convolution inequality is another special case of the BLI.

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. Meanwhile continuous versions [6, 8, 25] and extensions to other spaces [9, 21] have been dealt with by various authors, and a variety of arguments has been developed in order to study the structure and to obtain generalizations of Brascamp–Lieb and Barthe type inequalities and to resolve the problem of obtaining a classification of the cases where equality occurs in these inequalities (see [8, 11, 12, 28, 34]). . From a geometric viewpoint, the following special rank one case is useful: Choose ni = 1, let Bi : Rn → R be given by Bi (x) := for some ui ∈ Rn \ {o}, where span{u1 , . . . , uk } = Rn (otherwise D = 0). Let ci > 0 with E k ∗ ∗ n i=1 ci = n and Bi : R → R , Bi (t) = tui , which is the Euclidean adjoint map of Bi . The preceding inequalities now take the form f .

)ci k (f 1 || ci f (t) dt f () dx ≤ √ i i i D i=1 R Rn i=1 k ||

and f .

{

∗

Rn

sup

k ||

fi (zi ) : x = ci

i=1

k E

} ci zi ui , zi ∈ R

k √ || dx ≥ D

i=1

(f

i=1

R

)ci fi (t) dt

,

where (E ) ⎫ ⎧ k ⎨ det ⎬ c a u ⊗ u i i i i i=1 .D = inf : a > 0 . ||k i ci ⎭ ⎩ i=1 ai Here we read ui ⊗ ui = ui as a linear map or as an n × n-matrix ui · uT i . If the vectors u1 , . . . , uk form a decomposition of the identity (by which we mean that a relation of the form (3.38) holds), then we obtain that D = 1. This is the special case described in the next theorem, which has been particularly useful in geometric contexts. Theorem 3.69 (Geometric Version) Let fi : R → [0, ∞) be measurable, ci > 0, let ui ∈ Sn−1 for i = 1, . . . , k be distinct unit vectors, and suppose that k E .

ci ui ⊗ ui = id.

(3.38)

i=1

Then f .

k ||

Rn i=1

fi () dx ≤ ci

k (f || i=1

)ci R

fi (t) dt

3 Geometric and Functional Inequalities

141

and f .

{

∗

Rn

sup

k ||

fi (zi ) : x = ci

i=1

k E

} ci zi ui , zi ∈ R

dx ≥

i=1

k (f ||

)ci R

i=1

fi (t) dt

.

Remarks . If k = n, then c1 = . . . = cn = 1 and u1 , . . . , un is an orthonormal basis. Hence the inequalities turn into equalities. . If equality holds in either inequality and none of the functions fi is identically zero or a scaled version of a Gaussian, then there is an origin symmetric regular cross polytope in Rn such that u1 , . . . , uk are among its vertices (see, e.g., Barthe [5, 6] and Lutwak, Yang, Zhang [53]). . Conversely, equality holds in these inequalities if each fi is a scaled version of the same centered Gaussian, or if k = n and u1 , . . . , un form an orthonormal basis. . A thorough discussion of the rank one BLI can be found in Carlen, CorderoErausquin [28]. The higher rank case, due to Lieb [51], is reproved and further explored by Barthe [5], including a discussion of the equality case, and is again carefully analyzed by Bennett, Carbery, Christ, Tao [11]. In particular, see Barthe, Cordero-Erausquin, Ledoux, Maurey [9] for a review of the relevant literature and an approach via Markov semigroups in a quite general framework. . Barthe [5] provided concise proofs for the geometric form of the BLI (GeoBLI) based on mass transportation (for a survey with selected applications, see Ball [4]). . The geometric form of the inequalities can be obtained by showing D = 1 as a consequence of the assumed representation of the identity. First, taking traces we get n = Tr(id) = Tr

( k E

.

) ci ui ⊗ ui

=

i=1

k E

ci Tr(ui ⊗ ui ) =

i=1

k E i=1

and span{u1 , . . . , uk } = Rn . We show that D = 1. Choosing a1 = · · · = ak = a > 0, we get

.

det

( k E i=1

) ci ai ui ⊗ ui

( = det a ·

k E i=1

) ci ui ⊗ ui

= an

ci ,

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A. Colesanti and D. Hug

and k || .

aici = a

Ek

= an,

i=1 ci

i=1

E √ hence D ≤ 1. Next we show that D ≥ 1. Set wi := ci ui . Then ki=1 wi ⊗wi = id and thus ( k ) ) ( E .1 = det(id) = det wi ⊗ wi = det W W T , W := (w1 . . . wk ) ∈ Rn,k . i=1

We set WI := det(wi : i ∈ I )2 for I ⊂ {1, . . . , k} with |I | = n, where k ≥ n, and get E

1 = det(W W T ) =

.

(3.39)

WI .

|I |=n

In the same way,

.

det

( k E

) ai wi ⊗ wi

=

E

aI WI ≥

|I |=n

i=1

||

(aI )WI ,

|I |=n

aI :=

||

ai ,

(3.40)

i∈I

where (3.39) and the inequality of arithmetic and geometric means were used. The exponent of ai in the product on the right side of (3.40) is given by E .

i∈I,|I |=n

WI =

E |I |=n

WI −

⎛ ⎞ k E WI = 1 − det ⎝ wj ⊗ wj − wi ⊗ wi ⎠

E i ∈I,|I / |=n

j =1

) = 1 − det (In − wi ⊗ wi ) = 1 − 1 − |wi |2 = |wi |2 = ci . (

This shows that also D ≥ 1. Next we provide a direct argument for Theorem 3.69. We start with a lemma. Assertion (i) of Lemma 3.70 has just been established in showing that D ≥ 1. Lemma 3.70 Let ci > 0, let ui ∈ Sn−1 for i = 1, . . . , k be distinct unit vectors, and suppose that k E .

i=1

ci ui ⊗ ui = id.

3 Geometric and Functional Inequalities

143

(i) For any t1 , . . . , tk > 0, we have ( .

det

k E

) ti ci ui ⊗ ui

≥

i=1

(ii) If z =

Ek

k ||

tici .

i=1

for θ1 , . . . , θk ∈ R, then

i=1 ci θi ui

|z|2 ≤

k E

.

ci θi2 .

i=1

Assertion (ii) can be seen from (E √ )2 √ ci θi · ci E E ≤ ci θi2 · ci 2 E = ci θi2 · |z|2 .

|z|4 =

.

For the proof of Theorem 3.69 we first assume that each fi is a positive 2 continuous probability density. Let g(t) = e−π t for t ∈ R be the Gaussian density. For i = 1, . . . , k, we consider the transportation map Ti : R → R satisfying f

f

t

.

−∞

fi (s) ds =

Ti (t) −∞

g(s) ds.

It is easy to see that Ti is bijective, differentiable and fi (t) = g(Ti (t)) · Ti' (t),

.

t ∈ R.

(3.41)

To these transportation maps, we associate the smooth transformation o : Rn → Rn given by o(x) =

k E

.

ci Ti () ui ,

x ∈ Rn ,

i=1

which satisfies do(x) =

k E

.

ci Ti' () ui ⊗ ui .

i=1

In this case, do(x) is positive definite and o : Rn → Rn is injective.

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A. Colesanti and D. Hug

Therefore, using first (3.41), then Lemma 3.70 (i) with ti = Ti' (), the definition of o and Lemma 3.70 (ii), and finally the transformation formula, the following argument leads to the BLI f .

k ||

R

n

fi ()ci dx

i=1

( k ||

f =

Rn

≤ ≤ ≤

Rn f R f

n

R

n

g(Ti ())

i=1

( k ||

f

)( ci

e

) Ti' ()ci

dx

i=1

) −π ci Ti ()2

k ||

( k E

det

i=1

) ci Ti' () ui

⊗ ui

dx

i=1

e−π |o(x)| det (do(x)) dx 2

e−π |y| dy = 1. 2

The BLI for arbitrary non-negative integrable functions fi follows by scaling and approximation. For the reverse BLI (BI), we consider the inverse Si of Ti , and hence f

f

t

.

−∞

g(s) ds =

Si (t)

−∞

fi (s) ds,

whence g(t) = fi (Si (t)) · Si' (t),

.

t ∈ R.

In addition, dw(x) =

k E

.

ci Si' () ui ⊗ ui

i=1

holds for the smooth transformation w : Rn → Rn given by w(x) =

k E

.

ci Si () ui ,

x ∈ Rn .

i=1

In particular, dw(x) is positive definite and w : Rn → Rn is injective.

(3.42)

3 Geometric and Functional Inequalities

145

Therefore, the transformation formula, Lemma 3.70 (i), and (3.42) imply that f .

k ||

∗

R

n

sup

E x= ki=1 ci θi ui i=1

f ≥

∗

Rn

≥

Rn

Rn

R

k ||

sup

E w(y)= ki=1 ci θi ui i=1

fi (Si ())

⎞ fi (θi )ci ⎠ det (dw(y)) dy

ci

i=1

n

) det )(

fi (Si ())

i=1

( k ||

f =

⎝

( k ||

f ≥

⎛

( k ||

f

fi (θi )ci dx

ci

g()

) ci Si' () ui

i=1 k ||

⊗ ui

dy

)

Si' ()ci

dy

i=1

) ci

( k E

f dy =

R

i=1

e−π |y| dy = 1. 2

n

The BI for arbitrary non-negative integrable functions fi follows by scaling and approximation. Example (Application of the BI, Lower Bound E for the Volume of a Zonotope) Let c1 , . . . , ck > 0, let u1 , . . . , uk ∈ Sn−1 with ki=1 ci ui ⊗ ui = id. Furthermore, let β1 , . . . , βk > 0 be given. We claim that V

( k E

.

) βi [−ui , ui ] ≥ 2n

i=1

k ( )ci || βi i=1

ci

.

For the proof, we consider for i ∈ {1, . . . , k} the indicator function fi (t) := 1[−βi /ci ,βi /ci ] (t), t ∈ R, which is 1 if t ∈ [−βi /ci , βi /ci ] and zero otherwise. If x ∈ Rn is such that { k } k || E ci .1 = sup fi (zi ) : x = ci zi ui , zi ∈ R { = sup

i=1 k || i=1

i=1

1[−βi ,βi ] (zi ci ) : x =

k E i=1

} ci zi ui , zi ∈ R ,

146

A. Colesanti and D. Hug

then x ∈ V

Ek

( k E

.

i=1 βi [−ui , ui ].

)

Hence we deduce from BI that

f

βi [−ui , ui ] ≥

Rn

i=1

≥

{

∗

sup

k ||

fi (zi ) : x = ci

i=1

k (f ||

} ci zi ui , zi ∈ R

dx

i=1

)ci R

i=1

k E

fi (t) dt

=

) m ( || 2βi ci i=1

ci

=2

n

k ( )ci || βi i=1

ci

.

Equality holds if and only if k = n, the vectors u1 , . . . , un form an orthonormal basis and c1 = · · · = cn = 1. Example (Application of the BLI, Upper Bound for the E Volume of a Gauge Body) Let c1 , . . . , ck > 0, let u1 , . . . , uk ∈ Sn−1 with ki=1 ci ui ⊗ ui = id. Furthermore, let α = (α1 , . . . , αk ) with αi > 0 for i = 1, . . . , k and 1 ≤ p < ∞ be given. Then { ||x||α,p :=

k E

.

} p1 αi ||

p

,

x ∈ Rn ,

i=1

defines a norm with unit ball (gauge body) K, that is, K := {x ∈ Rn : ||x||α,p ≤ 1},

.

in particular || · ||α,p = || · ||K . The body K depends on various parameters. In the special case p = 1, k = n, we have c1 = · · · = cn = 1 and u1 , . . . , un is an orthonormal basis of Rn . Then } { −1 −1 .K = conv ±α 1 u1 , . . . , ±αn un is a cross polytope with volume V (K) =

.

2n (α1 · · · αn )−1 . n!

For general parameters, we now provide the sharp upper bound )n ( ) ci k ( 2n r 1 + p1 || ci p ) ( .V (K) ≤ , αi r 1 + pn i=1 for which equality is achieved in the above special case of a cross polytope.

3 Geometric and Functional Inequalities

147

To verify this, we first derive a general representation for the volume of a convex body K which contains the origin in its interior. We claim that V (K) =

.

(

f

1

r 1+

n p

)

R

n

( p) exp −||x||K dx

(3.43)

for p > 0 and K ∈ Kn with o ∈ int(K). For a convex body K containing the origin o in its interior, the radial function ρ(K, ·) : Rn \ {o} → (0, ∞) is defined by ρ(K, x) := max{s ≥ 0 : sx ∈ K} for x ∈ Rn \ {o}. Hence we have ρ(K, x) = 1/||x||K for x ∈ Rn \ {o}. Using polar coordinates, the radial function ρ(K, ·) of K and the (n − 1)-dimensional Hausdorff measure Hn−1 restricted to Sn−1 , we get f .

exp Rn

{

p} −||x||K

f dx = =

S f

=

∞

f

0 ∞

n−1

S n−1

f =

f

S f S

n−1

0

{ p} exp −r p ||u||K r n−1 dr dHn−1 (u)

n−1 ||u||−n (u) K dH

f

∞ 0

ρ(K, u)n dHn−1 (u) n−1 f

= nV (K) p1 =

{ p} exp −||ru||K r n−1 dr dHn−1 (u)

n pr

( ) n p

∞

n

e−t t p

−1

e−s s n−1 ds

f

p

∞

e−t t

n−1 p

0

dt

0

( ) V (K) = V (K)r 1 + pn ,

which yields the asserted representation for the volume of K. Now, for i ∈ {1, . . . , k}, we consider } { αi p .fi (t) := exp − |t| , ci

t ∈ R.

1 − pt

p−1 p

dt

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A. Colesanti and D. Hug

Using (3.43) and BLI, we get V (K) =

.

=

≤

=

{

f

1

}

k E

) ( exp − αi || Rn r 1 + pn i=1

p

dx

f || k 1 ) ( fi ()ci dx n n R r 1+ p i=1 (

1

r 1+

n p

)

m (f ||

R

i=1 k ||

1

) ( r 1 + pn i=1

)ci fi (t) dt

[ ( )1 ( )]ci 1 ci p , r 1+ 2 αi p

E which yields the claim since ki=1 ci = n. Again equality holds if and only if K is a cross polytope.

3.6.2 Reverse Isoperimetric Problem In this section, we consider n-dimensional compact convex sets in .Rn . The class of these bodies was denoted by .Knn . It has already been emphasized that the isoperimetric ratio .S(K)n /V (K)n−1 is minimized among all .K ∈ Knn by Euclidean balls, but the functional .S n /V n−1 on .Knn is clearly unbounded from above. This can be seen, for .n = 2, by considering a sequence of rectangles with increasingly larger aspect ratios. To prevent such a degeneration, we consider for each convex body .K ∈ Knn a transformation of K by an invertible linear map for which the isoperimetric ratio is as small as possible. More formally, we define } { S(oK)n : o ∈ GL(n) . . ir(K) := inf V (oK)n−1 Here the minimization is over all bijective linear transformations (translations are irrelevant, since volume and surface area are translation invariant). The new functional .ir(·) has the following properties: . .ir is affine invariant, upper semi-continuous and hence attains its maximum. . Petty [55] (see also Giannopoulos, Papadimitrakis [37]) has shown the following facts: The infimum is attained, in fact, there is a unique .K0 ∈ {oK : o ∈ GL(n)}, the .GL(n) class of K, such that .

ir(K) =

S(K0 )n . V (K0 )n−1

3 Geometric and Functional Inequalities

149

The minimizer is characterized by the isotropy condition f .

S

n−1

u ⊗ u dSn−1 (K0 , u) =

S(K0 ) id. n

The determination of the maximum for the new functional .ir(·) and of its extremizers turned out to be an inspiring problem. In the Euclidean plane, the solution was achieved by elementary geometric arguments (see Gustin [41] and Behrend [10]). In the following, we write .T2 for a regular triangle and .W2 for a square in .R2 . Since the functionals under consideration are affine invariant, we can assume that .T2 and .W2 are circumscribed to the unit ball .B 2 . Theorem 3.71 (Gustin, Behrend (Symmetric Case)) (a) If .K ∈ K22 , then .

ir(K) ≤ ir(T2 ).

Equality holds if and only if K is a triangle. (b) If .K ∈ K22 and .K = −K, then .

ir(K) ≤ ir(W2 ).

Equality holds if and only if K is a parallelogram. A careful analysis of the arguments of Gustin [41] and Behrend [10] shows that the previous result can be strengthened by providing a quantitative improvement (see Böröczky, Hug [17] and Böröczky, Fodor, Hug [19] for the symmetric case). To state these classical (and then also more recent) results, we first need suitable notions of distance which allow us to quantify to which extent two convex bodies deviate from each other. Such a measure of deviation should not change if one of the two bodies is replaced by its image under an invertible affine transformation (in the case of origin symmetric bodies we consider images under invertible linear maps), so as to reflect the invariance of the functionals under consideration. For general n .K, L ∈ Kn , we define the (geometric) Banach–Mazur distance of K and L by δBM (K, L):= log min{λ≥1 : K − x⊂o(L − y) ⊂ λ(K − x), o∈ GL(n), x, y∈Rn }.

.

If .K, L ∈ Knn are both symmetric with respect to the origin, that is, if .K = −K and .L = −L, this definition simplifies, and we arrive at δBM (K, L) = log min{λ ≥ 1 : K ⊂ oL ⊂ λK, o ∈ GL(n)}.

.

Another way to quantify the distance between convex bodies is obtained by modifying the symmetric difference metric which measures the volume deviation of two sets. For convex bodies .K, L ∈ Knn , we denote by .KAL := (K \ L) ∪ (L \ K)

150

A. Colesanti and D. Hug

the symmetric difference of K and L. The usual symmetric difference metric of K and L is defined as the volume of .KAL. To obtain an affine invariant measure of deviation, as described above, we set .αK := V (K)−1/n , .αL := V (L)−1/n and then define } { δvol (K, L) := min Hn (o(αK K)A(x + αL L)) : o ∈ SL(n), x ∈ Rn .

.

If K and L are origin symmetric, the translations in the definition of .δvol (K, L) are omitted. These definitions induce metrics on the affine equivalence classes of convex bodies, where two convex bodies .K, L ∈ Knn are said to be affinely equivalent, if .K = αL for an invertible affine transformation .α of .Rn (in the case of origin symmetric convex bodies, we consider linear equivalence classes). We refer to [17, Lemma 8.2] and the literature cited there for relations between .δvol and .δBM . Theorem 3.72 (Böröczky, Hug and Böröczky, Fodor, Hug (Symmetric Case)) (a) Let .K ∈ K22 , and let .ε ∈ [0, 1/72). If .δBM (K, T2 ) ≥ 72 ε, then .

ir(K) ≤ (1 − ε) ir(T2 ).

(b) Let .K ∈ K22 be origin symmetric, and let .ε ∈ [0, 1). If .δvol (K, W2 ) ≥ ε or .δBM (K, W2 ) ≥ ε, then .

( ε ) ir(W2 ). ir(K) ≤ 1 − 54

It turned out that the corresponding problems in dimensions .n ≥ 3 were much more challenging and required new ideas and tools from geometric functional analysis. In fact they remained completely open until Keith Ball [1, 2] discovered the connection between the geometric Brascamp–Lieb inequality, volume ratios and the reverse isoperimetric problem. K. Ball managed to establish the higherdimensional inequalities for .ir(·) and resolved various other (related) geometric extremal problems. However, the characterization of the equality cases had to wait until F. Barthe resolved the equality cases in the geometric form of BLI and BI [5], as stated in Theorem 3.69. Since certain associated ellipsoids play a crucial role in the resolution of the reverse isoperimetric problem (and in fact in many other contexts as well), we next define two of them. For a given convex body .K ∈ Knn , the maximal (with respect to volume) inscribed ellipsoid is usually called the John ellipsoid .EJ (K) of K, the minimal (with respect to volume) circumscribed ellipsoid is the Loewner ellipsoid .EL (K) of K.

3 Geometric and Functional Inequalities

151

Theorem 3.73 (a) For each .K ∈ Knn there is a unique ellipsoid .EJ (K) which is contained in K and has maximal volume among all ellipsoids contained in K. (b) For each .K ∈ Knn there is a unique ellipsoid .EL (K) which contains K and has minimal volume among all ellipsoids containing K. (c) Both ellipsoids are affinely associated with K, that is, if .α is an invertible affine transformation of .Rn and .K ∈ Knn , then .EJ (αK) = αEJ (K) and .EL (αK) = αEL (K). Next we state John’s classical result [44] (see also [3, Proposition 1]) which provides necessary and sufficient conditions which ensure that the unit ball is the John ellipsoid (the Loewner ellipsoid) of .K ∈ Knn . These conditions involve the common contact points of the unit sphere and the boundary of K. Theorem 3.74 (F. John, K. Ball) (a) Let .K ∈ Knn and .EJ (K) = B n . Then there are .s ∈ N with .n + 1 ≤ s ≤ 1 n−1 ∩ ∂K such that 2 n(n + 3), .c1 , . . . , cs > 0 and .u1 , . . . , us ∈ S s E .

ci ui = o

and

i=1

s E

ci ui ⊗ ui = id.

(3.44)

i=1

The converse is also true. If ⊂ K and there are .s, ci , ui as above, then EJ (K) = B n . (b) If .K = −K, then the centeredness condition is omitted and .n ≤ s ≤ 12 n(n + 1). (c) An analogous result holds for the Loewner ellipsoid. Moreover, .EJ (K) = B n if and only if .EL (K ◦ ) = B n , where .K ◦ is the polar body of K. n .B

.

In honor of Fritz John, the decomposition of the identity in (3.44) has been called John decomposition (of the identity). Various proofs for Theorem 3.74 and characterization results of John-type for general pairs of convex bodies have been suggested (see, e.g., Gordon, Litvak, Meyer, Pajor [38] and Gruber, Schuster [40]).

3.6.3 Volume Ratios and Reverse Isoperimetry For a given convex body .K ∈ Knn , we consider the inner and the outer volume ratio ( .

vr(K) :=

V (K) V (EJ (K))

)1

(

n

,

VR(K) :=

V (EL (K)) V (K)

Clearly, the functionals .vr and .VR are . bounded from below by 1 (sharp) and bounded from above, . affine invariant, . continuous.

)1

n

.

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A. Colesanti and D. Hug

The following theorem solves the problem of maximizing the volume ratio vr. The result is due to Ball [1, 2], the characterization of the equality cases is due to Barthe [5]. We write .[K] for the class of all convex bodies which are affinely equivalent to K (that is, which are obtained from K by an invertible affine transformation). We write .Tn for a regular simplex circumscribed to the unit ball .B n and .Qn for a cube circumscribed to .B n . Hence the class .[Tn ] contains all n-dimensional simplices and the class .[Qn ] contains all n-dimensional parallelepipeds.

.

Theorem 3.75 (Volume Ratios, Ball, Barthe) (a) If .K ∈ Knn , then .vr(K) ≤ vr(Tn ) with equality if and only if .[K] = [Tn ]. (b) If .K ∈ Knn and .K = −K, then .vr(K) ≤ vr(Qn ) with equality if and only if .[K] = [Qn ]. Proof We start with the easier part (b). We show: If .K = −K, .EJ (K) = B n and .Qn is a cube centred at o with edge length 2, then V (K) ≤ V (Qn ) = 2n .

.

In fact, by the characterization theorem for .EJ (K) we get .k ≥ n, .c1 , . . . , ck > 0 and vectors .u1 , . . . , uk ∈ Sn−1 ∩ ∂K with k E .

ci ui ⊗ ui = id

and

ui ∈ / {±uj : j /= i}

for i = 1, . . . , k.

i=1

By elementary geometry .K ⊂ K⊂

k n

.

nk

i=1 H

− (u , 1), i

H − (ui , 1) ∩

i=1

k n

and by symmetry even

H − (−ui , 1) =: L.

i=1

Consider .fi := 1[−1,1] : R → [0, ∞) for .i = 1, . . . , k. Then for .y ∈ Rn we get 1L (y) =

k ||

.

i=1

1[−1,1] () =

k || i=1

fi ()ci .

(3.45)

3 Geometric and Functional Inequalities

153

Hence, first using (3.45) and then Theorem (3.69), we deduce that f V (K) ≤ V (L) =

.

≤

k ||

R

n

i=1

)ci

k (f || i=1

=

fi ()ci dx

k ||

R

fi (t) dt

2ci = 2n .

(3.46)

i=1

If .V (K) = 2n , then .(3.46) must hold with equality. Since .fi /≡ 0 and .fi are not Gaussian, the equality condition in the geometric BLI (due to Barthe) implies that n .k = n and .u1 , . . . , un are an orthonormal basis of .R , hence .L = Qn . On the other hand, .K ⊂ Qn and .V (K) = V (Qn ). This finally shows that .K = Qn . Now we turn to part (a). Let .K ∈ Knn and .EJ (K) = B n . We show that .V (K) ≤ V (Tn ), where .Tn is a regular simplex circumscribed to .B n . By Theorem 3.73 (a), there are .k ≥ n + 1, .c1 , . . . , ck > 0 and .u1 , . . . , uk ∈ Sn−1 ∩ ∂K with k E .

ci ui = o

and

i=1

k E

ci ui ⊗ ui = id.

i=1

n In particular, .K ⊂ ki=1 H − (ui , 1) =: L. We show that .V (L) ≤ V (Tn ). For this, we define for .i = 1, . . . , k / ) ( n n+1 1 ∈ Rn+1 , di := .vi := −ui , √ ci . n+1 n n E It can be checked easily that .|vi |2 = 1, . ki=1 di = n + 1 and k E .

i=1

di vi ⊗ vi = idRn+1 .

For .i = 1, . . . , k, we consider the functions fi : R → [0, ∞),

.

{

t |→

e−t , t ≥ 0, 0, t < 0,

and define F (x) :=

k ||

.

i=1

fi ()di ,

x ∈ Rn+1 .

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A. Colesanti and D. Hug

Theorem (3.69) yields f .

)di k (f || f F (x) dx ≤ (t) dt = 1. i Rn+1 R i=1

(3.47)

We next determine the value of the left side of (3.47) by means of Fubini’s theorem. For this we write x = (y, r) ∈ Rn × R ∼ = Rn+1

.

such that = √

.

/

r n+1

−

n . n+1

For each .y ∈ Rn there is a .j ∈ {1, . . . , k} with . ≥ 0. In fact, if not then . < 0 for .i = 1, . . . , k, hence 0>

k E

.

/ k E

ci =

i=1

\ ci ui , y = 0,

a contradiction.

i=1

Consider .x = (y, r) ∈ Rn+1 . If .r < 0, then . < 0 for some .j ∈ {1, . . . , k}, and hence .F (x) = 0. If .r ≥ 0, then .F (x) /= 0 if and only if . ≥ 0 for .j = 1, . . . , k, hence if and only if r ≤ √ n

for j = 1, . . . , k

.

Observe that for .x = (y, r) with .r ≥ 0 and .y ∈ { F (x) = exp −

k E

.

or √r L n

r y ∈ √ L. n we have

}

} { √ di = exp −r n + 1 .

i=1

Hence by Fubini’s theorem we get f 1≥

.

f Rn+1

F (x) dx =

∞f

0

f =

∞f

√r L n

0

f = 0

√r L n

∞

F (y, r) dy dr } { √ exp −r n + 1 dy dr

√ rn −r n+1 V (L)e dr nn/2

3 Geometric and Functional Inequalities

155

= V (L)

)n ( )n f ∞ ( dt 1 2 t e−t √ √ n n + 1 n +1 0

= V (L) · =

n! n 2

n (n + 1)

n+1 2

V (L) V (K) ≥ . n V (T ) V (T n )

If .V (K) = V (Tn ), then also .V (L) = V (Tn ), and equality must hold in (3.47). Hence .v1 , . . . , vk are among the vertices of an origin symmetric regular cross polytope in .Rn+1 . This yields .k = n + 1 and .v1 , . . . , vn+1 is an orthonormal basis in n+1 . From . = δ it follows that . + 1 = 0 for .i /= j , hence .L = T . .R i j ij i j n n Since also .K ⊂ L and .V (K) = V (L), we conclude that .K = L = Tn . u n Similar results hold for outer volume ratios. Then the analysis is based on the BI. Having resolved the extremal problem for the volume ratio, it is now surprisingly easy to resolve also the reverse isoperimetric problem. Theorem 3.76 (Reverse Isoperimetric Inequality, Ball, Barthe) (a) If .K ∈ Knn , then .ir(K) ≤ ir(Tn ) with equality if and only if .[K] = [Tn ]. (b) If .K ∈ Knn and .K = −K, then .ir(K) ≤ ir(Qn ) with equality if and only if .[K] = [Qn ]. Proof (a) Let .K ∈ Knn and choose an affine transformation .α of .Rn such that .αEJ (K) = B n = EJ (αK) ⊂ αK. Then S(αK) = lim

V (αK + εB n ) − V (αK) ε

≤ lim

V (αK + εαK) − V (αK) ε

.

ε↓0

ε↓0

= V (αK) lim ε↓0

(1 + ε)n − 1 = ε

= V (αK)n = V (αK) ≤ V (αK)

n−1 n

n−1 n

1

nV (αK) n

1

nV (Tn ) n ,

where we used the preceding theorem on volume ratios for the last inequality. Hence .

ir(K) ≤

S(αK)n S(Tn )n n ≤ n V (T ) = = ir(Tn ). n V (αK)n−1 V (Tn )n−1

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A. Colesanti and D. Hug

If .ir(K) = ir(Tn ), then in particular .V (αK) = V (Tn ), and hence .vr(K) = vr(Tn ). It follows that .[K] = [Tn ]. The argument for (b) is similar.

u n

For the reverse isoperimetric problem, quantitative improvements have been studied by Böröczky, Hug [17] and Böröczky, Fodor, Hug [19] (see also the related investigation [20]). From these works, we mention one particular result. Theorem 3.77 (Böröczky, Hug) Let .K ∈ Knn with .EJ (K) = B n , and let .ε ∈ (0, 1). If .δvol (K, T n ) ≥ ε, then .

n n S(K)n 4 S(T ) ≤ (1 − γ ε ) , V (K)n−1 V (T n )n−1

where one may choose .γ = n−250n . A similar result holds for the Banach–Mazur distance. In Böröczky, Fodor, Hug [19] the symmetric case is treated. Results for .Lp zonoids and their polars (of even isotropic measures) have been obtained by Böröczky, Fodor, Hug [19, 20].

References 1. K. Ball, Volumes of sections of cubes and related problems, in Geometric Aspects of Functional Analysis. Lectures Notes in Mathematics, vol. 1376 (Springer, Berlin, 1989), pp. 251–260 2. K. Ball, Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. 44, 351–359 (1991) 3. K. Ball, Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41, 241–250 (1992) 4. K. Ball, Convex geometry and functional analysis, in Handbook of the Geometry of Banach Spaces, vol. I (North-Holland, Amsterdam, 2001), pp. 161–194 5. F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134, 335–361 (1998) 6. F. Barthe, A continuous version of the Brascamp–Lieb inequalities, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1850 (Springer, Berlin, 2004), pp. 53–63 7. F. Barthe, P. Wolff, Positive Gaussian kernels also have Gaussian minimizers. Mem. Am. Math. Soc. 276(1359), v+90 pp. (2022) 8. F. Barthe, D. Cordero-Erausquin, Inverse Brascamp–Lieb inequalities along the heat equation, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1850 (Springer, Berlin, 2004), pp. 65–71 9. F. Barthe, D. Cordero-Erausquin, M. Ledoux, B. Maurey, Correlation and Brascamp–Lieb inequalities for Markov semigroups. Int. Math. Res. Not. 10, 2177–2216 (2011) 10. F. Behrend, Über einige Affininvarianten konvexer Bereiche (German). Math. Ann. 113, 713– 747 (1937) 11. J. Bennett, A. Carbery, M. Christ, T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals. Geom. Funct. Anal. 17, 1343–1415 (2017) 12. J. Bennett, N. Bez, S. Buschenhenke, M.G. Cowling, T.C. Flock, On the nonlinear Brascamp– Lieb inequality. Duke Math. J. 169, 3291–3338 (2020)

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13. F. Besau, S. Hoehner, An intrinsic volume metric for the class of convex bodies in Rn . Commun. Contemp. Math. (in press) 14. V.I. Bogachev, Measure Theory, vol. II (Springer, Berlin, 2007) 15. C. Borell, Capacitary inequalities of the Brunn–Minkowski type. Math. Ann. 263, 179–184 (1983) 16. C. Borell, Greenian potentials and concavity. Math. Ann. 272, 155–160 (1985) 17. K.J. Böröczky, D. Hug, Isotropic measures and stronger forms of the reverse isoperimetric inequality. Trans. Am. Math. Soc. 369, 6987–7019 (2017) 18. K.J. Böröczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn–Minkowski inequality. Adv. Math. 231, 1974–1997 (2012) 19. K.J. Böröczky, F. Fodor, D. Hug, Strengthened volume inequalities for Lp zonoids of even isotropic measures. Trans. Am. Math. Soc. 371, 505–548 (2019) 20. K.J. Böröczky, F. Fodor, D. Hug, Strengthened inequalities for the mean width and the l-norm. J. Lond. Math. Soc. (2) 104, 233–268 (2021) 21. R. Bramati, Brascamp–Lieb inequalities on compact homogeneous spaces. Anal. Geom. Metr. Spaces 7, 130–157 (2019) 22. H.J. Brascamp, E.H. Lieb, Best constants in Young’s inequality, its converse and its generalization to more than three functions. Adv. Math. 20, 151–173 (1976) 23. H.J. Brascamp, E.H. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976) 24. H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974) 25. S. Brazitikos, A. Giannopoulos, Continuous version of the approximate geometric Brascamp– Lieb inequalities. J. Geom. Anal. 32, Paper No. 174, 23 pp. (2022) 26. E.M. Bronshtein, L.D. Ivanov, The approximation of convex sets by polytopes. Siberian Math. J. 16, 852–853 (1975) 27. L. Caffarelli, D. Jerison, E. Lieb, On the case of equality in the Brunn–Minkowski inequality for capacity. Adv. Math. 117, 193–207 (1996) 28. E.A. Carlen, D. Cordero-Erausquin, Subadditivity of the entropy and its relation to Brascamp– Lieb type inequalities. Geom. Funct. Anal. 19, 373–405 (2009) 29. A. Colesanti, Brunn–Minkowski inequalities for variational functionals and related problems. Adv. Math. 194, 105–140 (2005) 30. A. Colesanti, P. Cuoghi, The Brunn–Minkowski inequality for the n-dimensional logarithmic capacity of convex bodies. Potential Anal. 85, 45–66 (2006) 31. A. Colesanti, M. Fimiani, The Minkowski problem for torsional rigidity. Indiana Univ. Math. J. 59, 1013–1039 (2010) 32. A. Colesanti, P. Salani, The Brunn–Minkowski inequality for p-capacity of convex bodies. Math. Ann. 327, 459–479 (2003) 33. D. Cordero-Erausquin, B. Klartag, Q. Merigot, F. Santambrogio, One more proof of the Alexandrov–Fenchel inequality. C. R. Math. Acad. Sci. Paris 357, 676–680 (2019) 34. T.A. Courtade, J. Liu, Euclidean forward-reverse Brascamp–Lieb inequalities: finiteness, structure, and extremals. J. Geom. Anal. 31, 3300–3350 (2021) 35. W.J. Firey, p-Means of convex bodies. Math. Scand. 10, 17–24 (1962) 36. R.J. Gardner, The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39, 355–405 (2002) 37. A. Giannopoulos, M. Papadimitrakis, Isotropic surface area measures. Mathematika 46, 1–3 (1999) 38. Y. Gordon, A.E. Litvak, M. Meyer, A. Pajor, John’s decomposition in the general case and applications. J. Differ. Geom. 68, 99–119 (2004) 39. P.M. Gruber, Convex and Discrete Geometry. Grundlehren der mathematischen Wissenschaften, vol. 336 (Springer, Berlin, 2007) 40. P.M. Gruber, F.E. Schuster, An arithmetic proof of John’s ellipsoid theorem. Arch. Math. (Basel) 85, 82–88 (2005)

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41. W. Gustin, An isoperimetric minimax. Pacific J. Math. 3, 403–405 (1953) 42. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (German) (Springer, Berlin–Göttingen–Heidelberg, 1957) 43. D. Hug, W. Weil, Lectures on Convex Geometry. Graduate Texts in Mathematics, vol. 286 (Springer, Cham, 2020) 44. F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays, ed. by K.O. Friedrichs, O.E. Neugebauer, J.J. Stoker (Interscience Publishers, New York, 1984), pp. 187–204 45. O. Kallenberg, Foundations of Modern Probability, 3rd edn. Probability Theory and Stochastic Modelling, vol. 99 (Springer, Cham, 2021) 46. A.V. Kolesnikov, E. Milman, Local Lp -Brunn–Minkowski inequalities for p < 1. Mem. Am. Math. Soc. 277(1360), v+78 pp. (2022) 47. D. Jerison, A Minkowski problem for electrostatic capacity. Acta Math. 176, 1–47 (1995) 48. D. Jerison, The direct method in the calculus of variations for convex bodies. Adv. Math. 122, 262–279 (1996) 49. J. Lewis, Capacitary functions in convex rings. Arch. Rational Mech. Anal. 66, 201–224 (1977) 50. N. Li, B. Zhu, The Orlicz–Minkowski problem for torsional rigidity. J. Differ. Equ. 269, 8549– 8572 (2020) 51. E.H. Lieb, Gaussian kernels have only Gaussian maximizers. Invent. Math. 102, 179–208 (1990) 52. E. Lutwak, The Brunn–Minkowski–Firey theory, I: Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993) 53. E. Lutwak, D. Yang, G. Zhang, Volume inequalities for subspaces of Lp . J. Differ. Geom. 68, 159–184 (2004) 54. P. Nayar, T. Tkocz, A note on a Brunn–Minkowski inequality for the Gaussian measure. Proc. Am. Math. Soc. 141, 4027–4030 (2013) 55. C.M. Petty, Surface area of a convex body under affine transformations. Proc. Am. Math. Soc. 12, 824–828 (1961) 56. J. Prochno, C. Schütt, E.M. Werner, Best and random approximation of a convex body by a polytope. J. Complexity 71, Paper No. 101652, 19 pp. (2022) 57. S. Reisner, C. Schütt, E. Werner, Dropping a vertex or a facet from a convex polytope. Forum Math. 13, 359–378 (2001) 58. C. Saroglou, Remarks on the conjectured log-Brunn–Minkowski inequality. Geom. Dedicata 177, 353–365 (2015) 59. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 151, second expanded edn. (Cambridge University Press, Cambridge, 2014) 60. Y. Shenfeld, R. van Handel, The extremals of Minkowski’s quadratic inequality. Duke Math. J. 171, 957–1027 (2022) 61. Y. Shenfeld, R. van Handel, The extremals of the Alexandrov–Fenchel inequality for convex polytopes. Acta. Math. (in press) 62. G. Sun, L. Xu, P. Zhang, The uniqueness of the Lp Minkowski problem for q-torsional rigidity. Acta Math. Sci. Ser. B (Engl. Ed.) 41, 1405–1416 (2021) 63. E.M. Werner, Floating bodies and approximation of convex bodies by polytopes. Probab. Surv. 19, 113–128 (2022) 64. D. Xi, G. Leng, Dar’s conjecture and the log-Brunn–Minkowski inequality. J. Differ. Geom. 103, 145–189 (2016) 65. D. Zou, G. Xiong The Lp Minkowski problem for the electrostatic p-capacity. J. Differ. Geom. 116, 555–596 (2020)

Chapter 4

Dualities, Measure Concentration and Transportation Shiri Artstein-Avidan

Abstract In these three lectures we shall review and discuss in detail the fascinating connection between duality transforms (mainly Legendre and polarity, but also those associated with other cost functions), measure transportation, cost sub-gradient mappings and concentration inequalities. The topics of the three letures are:

1. Introduction 2. Transportation of measure 3. Concentration of measure

4.1 Lecture I: A Short Introduction In the introductory lecture we review some background topics that put the main subjects of the course in context, and these are measure transportation—the topic of the second lecture, and measure concentration—the topic of the third one. The two main objects of study are convex bodies and convex functions. One could offer a whole semester working only on the basic facts about these objects, as there is a vast literature, in particular the textbooks of Schneider [48], Gruber [27], Gardner [26], Hug and Weil [30] and my books with Giannopoulos and Milman [6] and [8]. The topic of duality, present in the title of our lecture series, will be present throughout the talks as a connecting theme, to which we point every now and again. We start with a few facts, and even fewer proofs, which will put the main topic of the course in context, and are useful and beautiful on their own right. Let us mention that the intersection of this lecture with the other courses of this program is quite large, but one should see this, hopefully, not as a bug but rather as a feature, helping readers absorb the material better.

S. Artstein-Avidan (O) Tel Aviv University, Tel Aviv, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6_4

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Fig. 4.1 Non-convex sums versus homothety

4.1.1 The Brunn–Minkowski Inequality The Minkowski sum of two sets is defined by .A + B = {x + y : x ∈ A, y ∈ B}. Usually, we assume that A and B are convex bodies. One good reason to do this is that for general sets it need not be the case that .A + A = 2A = {2x : x ∈ A}, as one can see in Fig. 4.1. However, the Brunn–Minkowski inequality, which we formulate next, holds for any measurable pair of sets, and since the proof is simple also for such a general case, this is the way we formulate it here. Theorem 4.1 (Brunn–Minkowski Inequality) Let .A, B, C ⊆ Rn be non-empty and Lebesgue measurable and assume that .C ⊇ A + B. Then Vol(C)1/n ≥ Vol(A)1/n + Vol(B)1/n .

.

(usually one assumes that .A + B measurable as well, and writes .Vol(A + B)1/n ≥ Vol(A)1/n + Vol(B)1/n ). Many proofs exist for this theorem, the simplest one, which was given by Hadwiger and Ohmann [28] uses induction, and approximation by finite unions of disjoint boxes. Proof We start with the case in which A and B are coordinate ||boxes, and as the .A = [0, ai ] and .B = inequality is invariant to translation we may assume || || [0, bi ] in which case .A + B = [0, ai + bi ]. The inequality then amounts to ( n || .

)1/n (ai + bi )

( n )1/n ( n )1/n || || ≥ ai + bi ,

i=1

i=1

(4.1)

i=1

which in turn follows from the arithmetic-geometric means inequality, ( n || .

i=1

ai ai + bi

(

)1/n +

n || i=1

bi ai + bi

)1/n ≤

) n ( ai 1E bi = 1. + n ai + bi ai + bi i=1

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Fig. 4.2 The box union’s splitting

To prove the inequality in the case where A and B are unions of disjoint coordinate boxes we use an induction argument on the total number of boxes in A and B together. If there are m boxes altogether, we just solved for .m = 2, so .m ≥ 3 and we assume to know the theorem when A and B are unions of disjoint coordinate boxes with altogether at most .(m − 1) boxes. So, at least one of the two, say A, has at least two boxes in it. Pick a coordinate hyperplane .H = {xi = c} which separates two of these boxes (it exists as the boxes are disjoint). Let .H + = {x : xi ≥ c} and − = {x : x ≤ c}, and let .A+ = A ∩ H + , .A− = A ∩ H − . We next make the same .H i splitting for B, using a hyperplane .E = {x : xi = d} parallel to H , with d chosen in such a way that .

Vol(B ∩ E − ) Vol(A ∩ H − ) = . Vol(A) Vol(B)

This can clearly be done by continuity of volume, see Fig. 4.2. Calling this fraction .λ ∈ (0, 1) we thus have .Vol(A ∩ H − ) = λVol(A), + − + .Vol(A ∩ H ) = (1 − λ)Vol(A), .Vol(B ∩ E ) = λVol(B) and .Vol(B ∩ E ) = + + − − (1 − λ)Vol(B). Denoting then .B = B ∩ E and .B = B ∩ E , the crucial observation is that .A+ + B + and .A− + B − are disjoint, and are both Minkowski sums of a pair of sets which satisfy the induction hypothesis. This gives Vol(A + B) ≥ Vol(A+ + B + ) + Vol(A− + B − )

.

≥ (Vol(A+ )1/n + Vol(B + )1/n )n + (Vol(A− )1/n + Vol(B − )1/n )n = (1 − λ)Vol((A)1/n + Vol(B)1/n )n + λVol((A)1/n + Vol(B)1/n )n , completing the proof. For the general case we approximate A and B by disjoint u n unions of coordinate boxes and use the continuity of volume. Let us stop for a moment to appreciate what we have seen. The main ingenious part of the proof is in the induction step where, after choosing the hyperplane H ,

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so as to allow induction, we choose a parallel E in such a way that the measure (of B) to its left and to its right correspond to the measures (of A) to the left and to the right of H . This is a very primitive “forefather” of transportation of measure, as we shall see shortly. Before moving on to the next topic, let us mention that by homogeneity the Brunn–Minkowski inequality can also be written as Vol((1 − λ)A + λB)1/n ≥ (1 − λ)Vol(A)1/n + λVol(B)1/n

.

for any .λ ∈ (0, 1) which by the geometric arithmetic means inequality implies Vol((1 − λ)A + λB) ≥ Vol(A)(1−λ) Vol(B)λ .

.

It is an easy exercise to show that knowing the latter for all .λ ∈ (0, 1) implies the original formulation of the Brunn–Minkowski inequality for the same A and B (simply choose the right .λ and use homogeneity).

4.1.2 Polarity and Blaschke–Santaló Inequality Another very famous and basic inequality for convex bodies is concerned with the operation of duality for convex bodies. Given a convex body which includes the origin, it defines a gauge function ||x||K = inf{r > 0 :

.

x ∈ K}, r

which is positively 1-homogeneous and convex. If .0 ∈ int(K) then this function is finite, if K is bounded then the function is bounded away from 0, and if in addition .K = −K then this function is a norm (with unit ball K, of course). A convex body also defines a support function hK (u) = sup ,

.

x∈K

which is a positively 1-homogeneous convex function. Its corresponding sub-levelset .{u : hK (u) ≤ 1} is called the polar body: K ◦ = {y ∈ Rn : ≤ 1 ∀x ∈ K}.

.

It is easy to check that the only self-dual convex body is the Euclidean ball .B2n = E 2 {x : xi = 1} (we show this below, after Theorem 4.11). The Blaschke–Santaló inequality (see e.g. [6, Section 1.5.4]) states that among convex bodies, the volume product, namely the product of the volume of a body and its polar, which is easily seen to be an affine invariant, is maximal for ellipsoids.

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Theorem 4.2 (Blaschke-Santaló) For a centrally symmetric convex body .K ⊂ Rn with non-empty interior s(K) := Vol(K)Vol(K ◦ ) ≤ Vol(B2n )2 ,

.

with equality if and only if K is a centered ellipsoid. Many proofs for this inequality are known. In fact, it is true also for bodies which are not centrally symmetric, so long as their center of mass is at the origin. The proof which will be explained in the course of Bianchi and Gronchi (Theorem 5.21 in this volume) uses Steiner symmetrization, which is a process in which, at the limit, a ball is obtained, and with respect to which the volume product is increasing. We mention the curious fact that the exact minimizers for the reverse inequalities are not known to this day. The “Mahler conjecture” (see [48]) states that for centrally symmetric bodies, cubes are examples of minimizers (not unique, though), and for general convex bodies, centered simplices. The centrally symmetric case in dimension 3 was recently solved [31], see also [25]. The general question, both in the symmetric and in the general case, is open. We shall get back to this inequality and its functional form later in this course.

4.1.3 Concentration of Measure Concentration of measure is a very strong tool, which since its discovery has been in use in a multitude of areas in mathematics, and proving concentration results serves as a good motivation for many developments in Asymptotic Geometric Analysis, see e.g. [6, Chapter 3] and the Notes and Remarks section of the same Chapter with many references. Concentration is the topic of the third talk, to which this section serves as a short introduction. The general notion is as follows: Consider a metric probability space .(X, d, μ), with the measurable sets being the Borel .σ -algebra. Concentration of measure is the phenomenon (which may or may not occur) that sets .A ⊆ X of measure .1/2 (say), when extended slightly, already have very large measure. By an extension we mean to consider At = {x : d(x, A) < t},

.

and “large measure” means that the measure of this extension is close to 1. Note that we always have that .μ(At ) → 1 as .t → ∞, so the issue is usually regarding “how fast” this convergence occurs in terms of t. To illustrate the importance and usefulness of this phenomenon, which was developed by V. Milman and has been a topic of wide study since then (with applications throughout vast areas of mathematics) let us address two issues:

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1. When this happens, Lipschitz functions are “virtually constant” on the space. 2. In spaces like the sphere, Gauss space, and the discrete cube, this is indeed the case.

Why Is Concentration Useful Given a 1-Lipschitz function .f : X → R on a metric probability space .(X, d, μ), we can consider a median M, which is a (not necessarily unique) value for which .μ(x : f (x) ≤ M) ≥ 1/2 and .μ(x : f (x) ≥ M) ≥ 1/2. Denote .A1 = {x : f (x) ≤ M} and .A2 = {x : f (x) ≥ M}, see Fig. 4.3. Note that for y in the intersection of their t-extensions, .|f (y) − M| < t, since there is some x with .f (x) ≤ M and .f (y) − f (x) ≤ d(x, y) < t, so .f (y) < M + t and there is some x with .f (x) ≥ M and .f (x) − f (y) ≤ d(x, y) < t so .f (y) > M − t. Under the assumption of “high concentration” this intersection has a big measure since μ((A1 )t ∩ (A2 )t ) ≥ 1 − μ(X \ (A1 )t ) − μ(X \ (A2 )t )

.

so on this whole big part, .|f − M| < t as claimed. This innocent observation was actually ground-breaking, and is a tool in proving many theorems, first and foremost in the proof of Dvoretzky’s theorem on nearEuclidean section of convex bodies by Milman [40].

Fig. 4.3 Neighborhood of Median for a function on the sphere

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Examples of Concentration A whole course can be devoted to methods for obtaining concentration, and in Lecture III we shall give various methods connected with measure transportation. For this introduction we remark on the “cleanest” (but also least interesting, in some sense) way to prove concentration, which works only for very specific cases: proof via “extremal sets”. Considering all sets of measure .1/2, we can ask what is the smallest possible value of the volume of their t-extension. (There is usually no meaning to asking for the largest extension, as one can take a tiny dense set and get that the t-extension is everything.) Usually it is quite hard figuring out what are these extremal sets, but in three important examples it is actually possible, and this is one way to get “concentration bounds”. Moreover, one can then compare, sometimes, more general probability metric spaces to one of these examples. The Sphere S n−1 ⊂ Rn The sphere, embedded in .Rn , has a natural metric (geodesic; which is equivalent to the one induced by distance in .Rn ). The normalized Lebesgue measure on it (called Haar measure and denoted .σ ) is invariant under rotations, and can be realized by volume of the associated hull-with-0: for .A ⊆ S n−1 σ (A) =

.

1 Vol(C(A)) Vol(B2n )

where C(A) = {sx : s ∈ [0, 1], x ∈ A}.

The following theorem [35, 47] is well known and can be proved in various ways (see e.g. [23]): Theorem 4.3 (Lévy, Schmidt) Let .A ⊆ S n−1 and .t > 0. Then σ (At ) ≥ σ (Bt )

.

where .B = {x ∈ S n−1 : d(x, x0 ) ≤ d} for some .x0 ∈ S n−1 and d is chosen so that .σ (B) = σ (A). In other words, among subsets of the sphere with fixed measure, spherical caps have the smallest t-extension volume. Theorem 4.3 gives a straightforward way to deduce concentration, provided we compute .σ (x : x1 > t) and show that it is very small (here we denoted .x = (x1 , . . . , xn ) so that we have chosen a specific cap, which by rotation invariance of the measure is enough). This allows one to give very precise bounds, σ (At ) ≥ 1 −

.

/ 2 π/8e−t n/2 ,

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but the calculation is tedious, and it is more fun to give a geometric bound as follows (this argument was presented by Ball in [11, Chapter 3]) σ ({x ∈ S n−1 : x1 > t}) =

.

and so long as .t
t}) Vol(B2n )

√ √ 1 − t 2 (that is, .t < 1/ 2), we know

{sx : s ∈ [0, 1], x ∈ S n−1 , x1 > t} ⊆ {y ∈ Rn : d(y, (t, 0))
t}) ≤

.

1 2 Vol((1 − t 2 )1/2 B2n ) = (1 − t 2 )n/2 ≤ e−t n/2 . n Vol(B2 )

We conclude, using Theorem 4.3, that for any .A ⊆ S n−1 with .σ (A) ≥ 1/2, we have −t 2 n/2 . .σ (At ) ≥ 1 − e

Gauss Space The term Gauss space here is used to mean .Rn with the usual metric and the measure 2 1 .dγ (x) = e−|x| /2 dx. This is a “model space” with super nice properties, and (2π )n/2 will be a main object of study for us in this short course. The measure .γ (when we want to emphasize the dimension we write .γn ) is rotation invariant and also a product measure (in some sense it is quite surprising that a rotation invariant product-measure exists). Its marginals are the lower dimensional .γk . Here too the exact solutions to the isoperimetric inequality are known, and are half-spaces. This Theorem goes back to Borell, Sudakov–Tsirelson [16, 49] Theorem 4.4 (Borell, Sudakov-Tsirelson) Let .A ⊂ Rn and .t > 0. Then γ (At ) ≥ γ (Ht )

.

where .H = {x ∈ Rn : x1 ≤ d} for d such that .σ (H ) = σ (A). In particular .Ht = {x : x1 ≤ d + t}. One may prove the theorem using symmetrizations. Once we have Theorem 4.4, computing the measure of .Ht for a measure .1/2 half-space, say, is simply computing one dimensional Gaussians, .γ1 ((−∞, t]) (which in particular does not depend on the dimension). As a corollary we thus get that for .A ⊂ Rn with .γ (A) = 1/2 we have f ∞ 1 1 2 2 .γ (At ) ≥ 1 − √ e−s /2 ds ≥ 1 − e−t /2 . 2 2π t

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We will see many ways to get Gaussian-type concentration in this course, avoiding the need for proving the exact isoperimetric-type result above (although its proof is very beautiful and elegant, and the reader is advised to read the original proofs). In Lecture III we will give several proofs of concentration of this same form.

The Discrete Cube For combinatorics fans, there are many discrete versions of the theorems considered in this course. The set .{−1, 1}n , consisting of vertices of the cube .Q = [−1, 1]n , endowed with the uniform probability measure (counting, normalized) and the Hamming metric .d(x, y) = n1 #{i : xi /= yi } is such an object (called “the discrete cube”). It satisfies concentration in the form 1 2 μ(At ) ≥ 1 − e−2t n . 2

.

Here, again, the exact isoperimetric family of extremal sets is known, and consists of metric caps, by a theorem of Harper [29]. For more on these discrete cases see [6, Section 3.1.5] and the references in the corresponding Notes and Remarks section.

4.1.4 Functional Forms An extremely powerful method which is discussed in depth in the course of Colesanti and Hug (in this volume), is the method of functionalization of geometric inequalities. It turns out that geometric notions, and corresponding inequalities, have analytic counterparts (sometimes more than one) which are of similar form, and sometimes of much stronger force. We mention two of these, which play a special role in this course. Before this, we need to explain which functions we plan to work with, and what are the operations corresponding to polarity and to Minkowski addition. The key player here will be the well-known Legendre transform, which we shall get to know more intimately in the next lecture, but will already encounter in the next section.

Convex Functions, Inf-Convolution and the Legendre Transform The class of functions we shall consider is the class of proper lower semi-continuous convex functions from .Rn to .(−∞, ∞], which we denote .Cvx(Rn ). A main reason for considering this class comes from the fact that replacing convex bodies with log-concave measures leads to a useful generalization of the theory. To this end we recall briefly what these measures are.

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Definition 4.5 A Borel measure .μ on .Rn is called log-concave if for any .0 < λ < 1 and any .A, B ⊆ Rn such that .(A, B, λA + (1 − λ)B) are measurable, ( ) μ λA + (1 − λ)B ≥ μ(A)λ μ(B)1−λ .

.

A measurable function .f : Rn → [0, ∞) is called log-concave if .f = exp(−ϕ) where .ϕ : Rn → (−∞, ∞] is a convex function, that is, ϕ((1 − λ)x + λy) ≤ (1 − λ)ϕ(x) + λϕ(y).

.

The connection between log-concavity of measures and functions was established by C. Borell in [17]. Assume the support of a measure .μ does not belong to any affine hyperplane. Then .μ is log-concave if and only if .μ is absolutely continuous and its density f is a log-concave function. This is the case .p → 0 and .s → 0 of a more general theorem about pconcave functions, which correspond to .s(p)-concave measures (notions we have not defined, but the reader may readily find in the literature (see e.g. [6, Section 1.4] and [8, Theorem 9.1.2]). The following theorem is of Borell [17]. Theorem 4.6 (Borell) Let .μ be a measure on .Rn and assume the affine hull of the support of .μ has full dimension n. Then for .s ≤ 1/n, the measure .μ is s-concave if and only if .dμ = f (x)dx where .f ≥ 0 is locally integrable and p-concave with 1 .p = s/(1 − sn) ∈ (− , ∞]. n The most important example of a log-concave measure is the usual Lebesgue measure on .Rn , assigning to a set K its measure .Vol(K). It is log-concave (and in fact .1/n-concave) by the Brunn–Minkowski inequality (Theorem 4.1.1). Its density is a trivial log-concave function: the constant function. Its restriction to a convex body is log-concave as well, as one may easily prove, as well as its marginals to sub-spaces of lower dimensions. Moreover, the closure of the family of measures on .Rn attained by taking marginals of uniform measures on convex sets in .Rm with .m > n is the set of all log-concave measures. More precisely one can prove (see e.g. [8, Chapter 9]) Theorem 4.7 Let .f = exp(−ϕ) be a log-concave function, namely .ϕ ∈ Cvx(Rn ) f is convex and lower semi continuous. Assume .0 < f < ∞. f Then there exists a sequence of convex bodies .Ks ⊂ Rn+s with .Voln+s (Ks ) ≤ f such that, letting .fs = πRn 1Ks denote the n-dimensional marginal of .1Ks , we have .fs → f locally uniformly. Another important example for a log-concave measure is the Gaussian measure 2 on .Rn which has the log-concave density .e−|x| /2 /(2π )n/2 and we denoted above by .γ or .γn . As we shall see shortly, its density function is in some sense the functional analogue of the geometric object: “the Euclidean ball .B2n ”. To functionalize the Brunn–Minkowski inequality, we need to define Minkowski addition of functions.

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Definition 4.8 (Inf-Convolution) Given two lower semi continuous convex functions, .ϕ, ψ : Rn → (−∞, ∞], let their inf-convolution be defined by (ϕOψ)(x) = inf (ϕ(y) + ψ(z)) .

.

y+z=x

From the point of view of epi-graphs, which are .

epi(ϕ) = {(x, y) ∈ Rn × R : y ≥ ϕ(x)} ⊂ Rn+1 ,

the inf-convolution is standard Minkowski addition, and we have, as one can prove directly from the definitions (see [8, Chapter 9] once more) that Lemma 4.9 Let .ϕ, ψ : Rn → (−∞, ∞] be lower semi continuous convex functions. Then .ϕOψ is a lower semi continuous convex function which satisfies .

epi(ϕOψ) = epi(ϕ) + epi(ψ).

It is useful to introduce averages, not just sums, so we define the .λ-homothety of a function by .λ · ϕ(x) = λϕ(x/λ) and together with the Minkowski addition we get thus the Minkowski average of two convex, or log-concave, functions: Definition 4.10 Given two lower semi continuous convex functions, .ϕ, ψ : Rn → (−∞, ∞], and some .λ ∈ (0, 1) let their inf-.λ-average be defined by (ϕOλ ψ)(x) =

.

inf

(1−λ)y+λz=x

((1 − λ)ϕ(y) + λψ(z)) .

Given two upper semi continuous log-concave functions, .f1 , f2 : Rn → [0, ∞), and some .λ ∈ (0, 1) let their sup-.λ-average be defined by (f1 *λ f2 )(x) =

.

( sup (1−λ)y+λz=x

) f1(1−λ) (y)f2λ (z) .

To functionalize the Blaschke–Santaló inequality (Theorem 4.2) we need to define the dual of a function. At this point we recall the definition of the classical Legendre transform .L defined for functions .φ : Rn → (−∞, ∞] by (Lφ)(x) = sup ( − φ(y)) .

.

(4.2)

y

Let us list some useful properties of this transform. 1. The transform always produces a convex function, .Lφ ∈ Cvx(Rn ) for all .φ : Rn → (−∞, ∞]. 2. The transform .L : Cvx(Rn ) → Cvx(Rn ) is a bijection, and in fact 3. The transform .L : Cvx(Rn ) → Cvx(Rn ) is an involution, .LLφ = φ for any n .φ ∈ Cvx(R ).

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4. The transform reverses order, .φ ≤ ψ implies .Lφ ≥ Lψ. ˆ denotes “regularized ˆ 5. Therefore, .L(max(φ, ψ)) = min(Lφ, Lψ) where .min minimum”, that is, the largest lower semi continuous convex function below all functions participating in the minimum. 6. The transform pulls back standard addition to the inf-convolution, namely .Lφ + Lψ = L(φOψ) for any .φ, ψ ∈ Cvx(Rn ). The proofs of these properties are either easy or classical, and we provide some of them in this lecture. Of particular interest is the last property which connects Legendre transform with Minkowski addition, and we prove it as Lemma 4.13. To illustrate the importance of Legendre transform in the field, we mention without proof the following characterization theorem which was proved in [2]. Theorem 4.11 (Artstein–Milman) Assume a transform .T : Cvx(Rn ) → Cvx(Rn ) satisfies 1. .TTφ = φ 2. .φ ≤ ψ implies .Tφ ≥ Tψ Then, .T is essentially the classical Legendre transform, namely there exist a constant .C0 ∈ R, a vector .v0 ∈ Rn , and a symmetric transformation .B ∈ GLn such that (Tφ)(x) = (Lφ)(Bx + v0 ) + + C0 .

.

For the usual polarity on convex bodies, the only self-polar set is the Euclidean ball .B2n . Indeed, clearly for any norm one has (letting .|| · ||∗ denote the dual norm, that is, the norm corresponding to the polar body of the unit ball) |x|2 = ≤ ||x||||x||∗

.

so that if .||x|| = ||x||∗ then .||x|| > |x|, and as this is true for all x we get .B2n ⊇ K for the unit ball K of .||x||, in which case .K ◦ ⊆ B2n , so that .K = K ◦ implies .K = B2n . Similarly, for the Legendre transform, the only self-dual function is .|x|2 /2. Indeed, by the definition of .L we have for any .x, y ∈ Rn that ϕ(x) + Lϕ(y) > ,

.

so that in particular ϕ(x) + Lϕ(x) > |x|2 .

.

Assume .ϕ = Lϕ, then we get .ϕ(x) > |x|2 /2, but since the right-hand side is self-dual, and .L reverses order, we get that .Lϕ(x) ≤ |x|2 /2 as well, and we have equality.

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For this reason, one often uses .|x|2 /2 as the analogue for the Euclidean ball in the world of convex functions, and .exp(−|x|2 /2) (normalized, or not) as its analogue in the world of log-concave functions. Since we will be using the Legendre transform frequently, we discuss in more depth some of its properties. Fact For a convex function .ϕ : Rn → (−∞, ∞] we have: .Lϕ(0) = − inf ϕ, and ∞ = h and .L(ϕ + c) = Lϕ − c. In addition for a closed convex K we have .L1 K K ∞ .LhK = 1 . K Here we denote .1∞ K the function attaining 0 on the set K and .+∞ on its complement. Indeed, for any y there is some .x ∈ ∂K which has y as a “direction of normal” at x, that is, .hK (x) = supz∈K is attained at .z = αy for some .α ≥ 0. If .y /∈ K then .α < 1 and so . − hK (x) > 0 and by 1-homogeneity the supremum satisfies .

sup − hK (x) = ∞.

x∈Rn

If on the other hand .y ∈ K then . ≤ hK (x) for any x and by picking .x = 0 the supremum is 0. Lemma 4.12 Let .ϕ : Rn → (−∞, ∞] and not constant .+∞. Then .LLϕ is the largest lower semi continuous convex function which is below .ϕ (also called the “convex enelope” of .ϕ). In particular, for proper lower semi continuous convex .ϕ we have that .LLϕ = ϕ. Proof We start by discussing what we have to prove. First of all, .LLϕ is below .ϕ since LLϕ(x) = sup − Lϕ(y) = sup infn − + ϕ(z) ≤ ϕ(x).

.

y∈Rn z∈R

y∈Rn

Secondly, it is in the right class, obviously (we stated this in the previous fact). If we show that on the class of proper lower semi continuous convex functions .L is an involution, then for any function we know that .LLϕ is in this class. Any .ψ ≤ ϕ which is in this class satisfies .Lψ ≥ Lϕ so .LLψ ≤ LLϕ and as .LLψ = ψ we know .LLϕ is bigger. So, all we need to prove is that on the class .Cvx(Rn ) of proper lower semi continuous convex functions (with values in .(−∞, ∞]) it holds that .LLϕ = ϕ. To this end we use that .ϕ is the supremum of affine functions below it (this is “separation” which one always needs for claims of this sort, it is a claim about the epi-graph of .ϕ, which is a convex set in .Rn+1 , together with the fact that we can “forget” about hyperplanes which are vertical). So, we use that ϕ = sup{( − c) : − c ≤ ϕ}.

.

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The latter condition, after moving terms from one side to another, can be written as c ≥ Lϕ(y). Concluding,

.

ϕ = sup{( − c) : c ≥ Lϕ(y)} ≤ sup( − Lϕ(y)) = LLϕ.

.

y

u n

This completes the proof of the Lemma.

Finally, we elaborate on the last fact we listed above (item 6) namely that the Legendre transform induces the “summation for convex functions” given by the inf-convolution, or summation of epi-graphs, as a pullback of standard addition of functions. This is in some sense a “generalization” of the fact that we have already observed, that .hK + hT = hK+T . Lemma 4.13 Let .ϕ, ψ : Rn → (−∞, ∞] be lower semi continuous, convex and proper. Then .L(Lϕ + Lψ) = ϕOψ, namely L(Lϕ + Lψ)(z) = inf{ϕ(x) + ψ(y) : x + y = z}.

.

Proof Indeed, by the previous theorem we can take .L of both sides and show equality. Compute: L(ϕOψ)(y) = sup ( − (ϕOψ)(x))

.

x

= sup( − inf (ϕ(z) + ψ(w))) z+w=x

x

= sup ( + − ϕ(z) − ψ(w)) = Lϕ(y) + Lψ(y). z,w

u n

The Prékopa–Leindler Inequality Based on the functional analogues introduced in the previous section for volume and for Minkowski addition, we can now present the inequality of Prékopa and Leindler which is considered the “functional analogue” for the Brunn–Minkowski inequality. It is the following statement [34, 42] (see also [6, Chapter 1]). Theorem 4.14 (Prékopa–Leindler) Let .f, g, h : Rn → R+ be measurable functions, and let .λ ∈ (0, 1). We assume that f and g are integrable, and that for every .x, y ∈ Rn h((1 − λ)x + λy) ≥ f (x)1−λ g(y)λ .

.

(4.3)

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Then, )1−λ (f

(f

f .

Rn

h≥

Rn

f

)λ

Rn

g

.

Up to integrability issues of the function .exp(−ϕOλ ψ), which we can resolve by using outer integral, we may restate the Prékopa–Leindler inequality as Theorem 4.15 Let .ϕ, ψ : Rn → (−∞, ∞] be lower semi continuous convex functions, and let .λ ∈ (0, 1). Then, )(1−λ) (f

(f

f .

Rn

exp(−ϕOλ ψ) >

Rn

exp(−ϕ)

Rn

)λ exp(−ψ) .

f If we let .Vol(ϕ) = exp(−ϕ) we get the following formulation of a Brunn– Minkowski type inequality for functions: Vol(ϕOλ ψ) ≥ Vol(ϕ)(1−λ) Vol(ψ)λ .

.

Let us make a few remarks. First, it is indeed a generalization for the Brunn– Minkowski inequality. Indeed, Proof of the Brunn–Minkowski Inequality, Using Prékopa–Leindler Let K and T be non-empty compact subsets of .Rn , and .λ ∈ (0, 1). We define .f = 1K , .g = 1T , and .h = 1(1−λ)K+λT . It is easily checked that the assumptions of Theorem 4.14 are satisfied, therefore )1−λ (f

(f

f Voln ((1 − λ)K + λT ) =

.

Rn

h≥

Rn

f

Rn

)λ = Voln (K)1−λ Voln (T )λ .

g

u n

This completes the proof.

Another remark is that the Prékopa–Leindler inequality is in a sense opposite to Hölder’s inequality. Indeed, the function h in the assumption of the theorem satisfies in particular that h(z) ≥ f (z)1−λ g(z)λ ,

.

whereas from Hölder it follows that

Rn

)1−λ (f

(f

f .

f (z)1−λ g(z)λ dz ≤

Rn

f (x)dx

Rn

)λ g(x)dx

,

which is an inequality in the opposite direction (but for a function smaller than h).

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A third remark is that for .f = g = h, the condition in the theorem is precisely that f is log-concave. Of course, for .f = g = h the conclusion of the inequality is trivial, but this becomes less trivial if one considers, for example, .f, g and h to be a given log-concave function restricted to K, T and .(1 − λ)K + λT respectively. In fact, this is precisely the statement that if a density of a measure is log-concave then so is the measure, which we mentioned after Definition 4.5. Finally, for the geometric interpretation, these “volumes of functions” are merely the measures of the epi-graphs with respect to the density on .{(x, z) : x ∈ Rn , z ∈ R+ } given by .e−z dzdx. Indeed, e−ϕ(x) =

f

∞

e−z dz

.

f Rn

so that

ϕ(x)

e−ϕ(x) dx =

f

Rn

f

∞

e−z dzdx =

f

e−z dzdx.

epi(ϕ)

ϕ(x)

Once we know that the measure on .Rn+1 given by .e−z dzdx is log-concave then Theorem 4.14 follows directly, and we do know this if we accept the fact mentioned above that a log-concave density makes for a log-concave measure. However, since to prove this fact one usually uses the Prékopa–Leindler inequality, this argument is somewhat circular (not completely: it means if we know this for a specific log-linear density, we deduce it for all log-concave densities). The full proof of Theorem 4.14 appears in the course of Colesanti and Hug (it is given as Theorem 3.27 in this volume). However, as the proof involves a part directly connected with one of the main topics in this course, in the following section we discuss some (main) elements of this proof in dimension one.

Functional Blaschke–Santaló Inequality The classical Blaschke–Santaló inequality (Theorem 4.2) states that the volume product .s(K) is maximized for Euclidean balls and their linear images. Recall once more that Euclidean balls (of radius 1) are the only self-dual bodies in .Rn . For logconcave functions, with duality given by the Legendre duality, one thus anticipates an inequality of similar form. Define for an even function .ϕ the quantity f s(ϕ) =

.

e−ϕ

f

e−Lϕ .

Note that when considering an upper bound we may as well restrict to convex functions, since .LLϕ ≤ ϕ and thus .s(ϕ) ≤ s(Lϕ). The only self-dual function 2 n .ϕ(x) = |x| /2, gives .s(ϕ) = (2π ) . It turns out that also in the functional setting, the product .s(ϕ) is maximized when the function .ϕ is self-dual. The not-necessarily even case is the following theorem from [5]. The even version was given first by Ball [10]. See also [24] and [32, 33].

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Theorem 4.16 Let .ϕ : Rn → (−∞, +∞] be a proper function, and assume that f . x exp(−ϕ(x))dx = 0 then f .

f exp(−ϕ)

exp(−Lϕ) ≤ (2π )n .

We will not prove this theorem, which is more in the vein of theorems from the course of Colesanti and Hug. See [8, Chapter 9] for a detailed discussion, proofs and many more references.

4.1.5 A Key Element from the Proof of Prékopa–Leindler Inequality In this last part of the first lecture we detail some specific elements from the proof of Theorem 4.14. The full proof appears in the course “Geometric and Analytic Inequalities” and we will here only address the case of dimension .n = 1, with the additional assumption that f and g are continuous and strictly positive. This will lead us smoothly to the second lecture. f f Without loss of generality one may assume . f = g = 1 by dividing both by constants (and h by the corresponding average constant). We may thus think of f and of g as densities of probability measures. We can take the uniform measure m on .[0, 1] as a model space. A monotone increasing transport f map from .([0, f 1], dm) to .(R, f (x)dx) is a mapping .x : (0, 1) → R such that . A f (x)dx = x −1 (A) dm, f x(t) which is equivalent to . −∞ f = t. See Fig. 4.4. In other words, we define .x, y : (0, 1) → R by the equations f

x(t)

.

−∞

f f =t

f R

f = t and

y(t)

−∞

f g=t

Fig. 4.4 Transportation of the measure m on .[0, 1] to .f (x)dx on .R

R

g = t.

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In view of our assumptions, x and y are differentiable, and for every .t ∈ (0, 1) we have f f ' .x (t)f (x(t)) = f = 1 and y ' (t)g(y(t)) = g = 1. R

R

We next define .z : (0, 1) → R by z(t) = (1 − λ)x(t) + λy(t).

.

Since x and y are strictly increasing, z too is strictly increasing, and the arithmeticgeometric means inequality implies that z' (t) = (1 − λ)x ' (t) + λy ' (t) ≥ (x ' (t))1−λ (y ' (t))λ .

.

Hence, we can estimate the integral of h making the change of variables .s = z(t), as follows: f

f .

R

h=

1

h(z(t))z' (t)dt

0

f ≥

1

h((1 − λ)x(t) + λy(t))(x ' (t))1−λ (y ' (t))λ dt

0

f ≥

(

1

f 0

1−λ

1 (x(t))g (y(t)) f (x(t)) λ

)1−λ (

1 g(y(t))

)λ dt = 1.

This completes the proof of the Prékopa–Leindler inequality in dimension one, under the mild additional assumptions on f and g. Recapping: the fact that x was a transport map implies (in the normalized case) that .x ' (t)f (x(t)) = 1, and similarly .y ' (t)g(y(t)) = 1. Thus z takes .([0, 1], dm) to some density .ξ on .R which should solve .z' (t)ξ(z(t)) = 1 namely .ξ(z) = 1/z' (t (z)). Since .z(t) = (1 − λ)x(t) + λy(t) we know )−1 ( z' (t) = (1 − λ)x ' (t) + λy ' (t) ≥ x ' (t)(1−λ) y ' (t)λ = f (x(t))(1−λ) g(y(t))λ

.

and so ξ(z) = 1/z' (t (z)) ≤ f (x(t))(1−λ) g(y(t))λ ≤ h(z)

.

where the last inequality is the original assumption on h joined with the fact that z(t) = (1 − λ)x(t) + λy(t). Note that the choice of .([0, 1], dm) is quite arbitrary, and any other reference measure (with strictly positive continuous density function) would work just as well.

.

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Monotonicity of the transport map was crucial here, since we used the arithmeticgeometric means inequality which can only be applied to positive numbers. To prove Theorem 4.14 in higher dimensions one usually uses induction, but in fact one can provide a direct proof with a similar transportation argument, after developing these tools in .Rn . For us this serves as motivation to understand what general measure transportation is, and more importantly, what monotone transportation should mean in higher dimensions. This is the topic of the second lecture.

4.2 Lecture II: Transportation of Measure Transportation of measure has to do with optimal ways of transferring a pile of sand of some shape into a hole in the ground with some (other) shape (See Fig. 4.5). The meaning of the word “optimal” here is of crucial significance. For a beautiful introduction to this topic, its origins and many applications, we refer the reader to [51]. These questions turn out to have very interesting relations with convexity theory and with measure concentration. Before diving into these exciting connections, let us discuss what measure transportation is.

4.2.1 The Set Up Pairings of Measures We are given sets X and Y , and on them probability measures .μ ∈ P (X) and ν ∈ P (Y ). Here we usually assume that X and Y are subsets of .Rn with the Borel .σ -algebra, but much of the theory works in the setting of general measure spaces. A transport plan is a measure .π ∈ P (X × Y ) with marginals .μ and .ν respectively, that .

Fig. 4.5 Transportation of a sand pile to a hole

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S. Artstein-Avidan

Fig. 4.6 Transport plan (left) and transport map (right)

is, such that .π(X × B) = ν(B), π(A × Y ) = μ(A) for measurable .A ⊆ X, B ⊆ Y . We denote the family of all transport plans between .μ and .ν by .||(μ, ν). A transport plan is sometimes induced by a transport map, that is, by a measurable function .T : X → Y satisfying .μ(T −1 (B)) = ν(B) for all measurable .B ⊆ Y . In such a case one sometimes writes that .ν = T #μ. The transport plan corresponding to T is then the measure .πT ∈ ||(μ, ν) which is concentrated on the graph of T , .r(T ) = {(x, T x) : x ∈ X} ⊂ X × Y , and is the pull-back under the canonical projection of .ν (or .μ) to this set, see Fig. 4.6. We mention (and this is an easy exercise) that the condition .π ∈ ||(μ, ν) is equivalent to the condition that for any .ϕ ∈ L1 (μ), .ψ ∈ L1 (ν) it holds that f

f

f

ϕ(x)dπ(x, y) =

.

X×Y

X

f ψ(y)dπ(x, y) =

ϕ(x)dμ and X×Y

ψ(y)dν. Y

For a transport map T this means f

f

f

ϕ(y)dν(y) =

.

Y

ϕ(y)dπT (x, y) = f

X×Y

=

ϕ(T x)dπT (x, y) X×Y

ϕ(T x)dμ(x). X

Remark 4.17 Whether it is enough to take only continuous bounded .ϕ, ψ, and further restrict to functions tending to 0 at .∞, can depend on how general a setting one wants to work with. We here assume X and Y to be Polish (complete separable metric spaces), .ν and .μ to be Borel probability measures (so, the .σ -algebras generated by open sets) and further that .X, Y are locally compact (each point has a compact neighborhood). For these spaces the Riesz representation theorem states that .M(X) = C0 (X)∗ . Remark 4.18 We would like to emphasize a detail which may give some extra motivation. In the case where .μ and .ν are absolutely continuous on .Rn , say

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dμ(x) = f (x)dx, dν(y) = g(y)dy, a transport mapping T satisfying .ν = T #μ, if differentiable, must satisfy as above that

.

f

f ϕ(y)g(y)dy =

.

ϕ(T x)f (x)dx,

which by the change of variables formula is equal to f .

f ϕ(y)g(y)dy =

ϕ(T z)g(T z)| det(DT )(z)|dz

(.DT (z) denoting the differential of T at the point .z ∈ Rn ). This holding for any test function .ϕ means that f (x) = g(T (x))| det(DT (x))|,

.

which is a highly non-linear differential equation in T . (So, when we soon optimize a quantity over all transport maps, this will be a very non-linear constraint.) A priori it is not clear why should there exist such a T , but not only will we show it exists in certain quite general situations, we can make sure that it has a very special form (for example, the most useful one is .T = ∇ϕ for a convex .ϕ) and satisfies other nice properties.

Cost Functions In optimal transport problems one is given a measurable cost function .c : X × Y → (−∞, ∞], (sometimes assumed non-infinite, sometimes assumed non-negative), and to every transport plan .π , one associates its total cost f c(x, y)dπ.

.

X×Y

An optimal plan is one that minimizes this total cost, and a priori it is not clear under which conditions such an optimal plan exists, or is unique (up to measure zero). When the plan is induced by a mapping .T : X → Y , the total cost can be written as f . c(x, T x)dμ, X

and another important question is under what assumptions is an optimal plan induced by a map. More formally,

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S. Artstein-Avidan

Definition 4.19 The optimal transportation cost between .μ and .ν is the value f C(μ, ν) = inf{

c(x, y)dπ : π ∈ ||(μ, ν)}.

.

X×Y

A plan .π ∈ ||(μ, ν) for which .C(μ, ν) = an optimal transport plan.

f X×Y

c(x, y)dπ , when exists, is called

Not much needs to be assumed in order to show that an optimal plan exists, since ||(μ, ν) is convex, and since the functional being infimized is linear in .π . Therefore it is enough to make some assumptions involving continuity, and use compactness arguments, as we shall discuss below.

.

Remark 4.20 In a probabilistic language, one is given probability measures .μ, ν and trying to find a joint distribution, namely a pair of random variables .(U, V ) with U distributed as .μ and V as .ν (not necessarily independent!) which minimizes .E(c(U, V )). This is sometimes called a “coupling” of .μ and .ν. Remark 4.21 Restricting to transport plans induced by maps, the optimal transport f problem goes back to Monge [41] and can be stated as infimizing . X c(x, T x)dμ over all measurable maps .T : X → Y with .ν = T #μ. Here the mere existence of a map is not clear and also the type of argument involving compactness does not work as there is high non-linearity in the constraint (mentioned in Remark 4.18 above). The relaxation to plans is due to Kantorovich, making this into a linear problem, and the associated transport problem is now called “Monge-Kantorovich”. For background and much information on the history (and current state) of optimal measure transportation see Villani’s book [51].

Doubly Stochastic Measures and Matrices When the sets X and Y are finite and of the same size (say, .{xi }ni=1 , {yi }ni=1 ), and the measures are normalized counting measures, a transport map is simply a pairing (or matching, in combinatorial language) of the two sets of points. A transport plan can then be represented by a matrix .π = (πi,j ) with all elements non-negative (with this being a permutation matrix in the case of a plan induced by a map). The fact that the plan transports one measure to the other is captured in the E fact that the sum of every row, and every column, of the matrix, is 1, namely . i πi,j = 1 = E π . Such non-negative matrices are called bi-stochastic. Clearly the class of i,j j 2

bi-stochastic matrices is convex and bounded (as a subset of .Rn , as .πi,j ∈ [0, 1]). In particular it is the convex hull of its extreme points. Clearly permutation matrices are bi-stochastic and are extreme points of this set. The fact that they are the only extreme points is a nice exercise, see e.g. [15]. This polytope is called the Birkhoff polytope. E The transportation problem in this case amounts to infimizing . πi,j c(xi , yj ) on the Birkhoff polytope. Since such a linear function is minimized on an extreme

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point, it is minimized on one of the .n! permutation matrices. In other words, in this case we see that the best plan is induced by a map (it might not be unique, if no additional condition on the cost is assumed). This map is called an optimal matching of the two sets X and Y . Remark 4.22 An interesting projection of the Birkhoff polytope is the permutation polytope. Given a point .x = (x1 , . . . , xn ) consider all its permutations .σ (x) = (xσ (1) , . . . , xσ (n) ) and take their convex hull, .P (x). It is easy to check that setting .Tx A = Ax for an .n×n matrix A, we get that .Tx Pn = P (x) where .Pn is the Birkhoff E E polytope. The permutation polytope lies in the hyperplane .{v : vi = xi }. In fact, so long as there is some .xi /= xj , the dimension of .P (x) is .(n − 1). In the general setting, however, it is not so easy to understand the structure of the extreme points of .||(μ, ν). Clearly sometimes “atoms need to be split”, if we think of two discrete measures with non-matching weights. So one cannot expect extreme points to consist only of maps in general. We will not dive deep into these questions in the general setting, but mention that there is some structure of the support set which must be maintained. As an example, let us quote and prove one theorem of J. Lindenstrauss [37] which was discovered independently and at the same time by Douglas [22]. We call a probability measure on the unit square, .π ∈ P ([0, 1]2 ), “doubly stochastic” if its marginal onto each of the two coordinates is uniform. In our notation we can write it .π ∈ ||(m, m) where m denotes the uniform measure on .[0, 1]. Theorem 4.23 (J. Lindenstrauss/R.G. Douglas) A measure .π ∈ ||(m, m) is an extreme point of .||(m, m) if and only if the subspace of .L1 (π ) consisting of functions of the form .f (x) + g(y) where .f, g ∈ L1 (m) is norm-dense in .L1 (π ). Remark 4.24 Note that in the case where functions of the form .f (x) are dense in L1 (π ), the measure .π must be concentrated on a graph;

.

Proof Non-extremeness of .π simply means we may write it as the average of two doubly stochastic measures, call them .π − η and .π + η, so in other words nonextremeness means that there exists some measure .η /= 0 with .|η(S)| ≤ π(S) for all .S ⊆ [0, 1]2 and .η([0, 1] × A) = η(A × [0, 1]) = 0 for all Borel .A ⊆ [0, 1]. Taking the density of .η with respect to .π to be .F (x, y) (using the Radon–Nikodym f theorem) this simply means .|F (x, y)| ≤ 1 and, at the same time, that . f (x)dη = f g(y)dη = 0 for all .f, g ∈ L1 (m). Rephrasing, f .π is not extreme if and only if there is a non-zero element .F ∈ L∞ (π ) with . (f (x) + g(y))F (x, y)dπ = 0 for all .f, g ∈ L1 (π ). Since .L1 (π )∗ = L∞ (π ), the proof is complete. u n Note that the above theorem already means that doubly stochastic measures on .[0, 1]2 which are extreme points of the set of all doubly stochastic measures, must be singular with respect to Lebesgue measure. Indeed, otherwise one may find a rectangle .(x1 , y1 ), (x1 , y2 ), (x2 , y1 ), (x2 , y2 ) of density points of the measure, which in turn leads to non-extremality (one can prove this as an exercise, see [37] for details).

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We also mention that denseness of functions of the form .f (x) + g(y) cannot be replaced by equality, namely there can be elements of .L1 (π ) which do not “split” in this way, as the following example, again from [37], shows. Example (J. Lindenstrauss) Let .λi be a sequence of positive numbers which E2n−1 sum to 1. Take .x0 = y0 = 0, .x1 = λ1 , .y1 = λ1 + λ2 , .xn = i=1 λi , E2n .yn = i=1 λi . Define the measure .π to be supported on segments joining .(xn , yn ) with .(xn+1 , yn+1 ), and joining .(xn+1 , yn ) with .(xn+2 , yn+1 ). On each segment .[(xn , yn ), (xn+1 , yn+1 )] we take the uniform measure with total mass .λ2n+1 , and on the segment .[(xn+1 , yn ), (xn+2 , yn+1 )] uniform with total mass .λ2n+2 . The total mass is 1, so that .π is a probability measure, and moreover the marginals are m, that is, uniform measures on .[0, 1]. The fact that .π is extreme follows from the theorem we have just shown (this is an easy exercise). However, not every .h ∈ L1 (π ) is a sum of .f (x) ∈ L1 (m) and .g(y) ∈ L1 (m), .μ almost everywhere. Indeed, we can take h to be constant on the corresponding segments, and the condition on the constants making it in .L1 (π ) is easy to write. Such a function cannot be split as a sum. Note that in the discussions above no special attention was given yet to the cost function c. It turns out that when transport problems are considered for special spaces .X, Y and a cost function which is well-behaved, not only can one show that optimal plans are induced by transport maps, these transport maps are of very special and “monotone” form. In particular, the most important and useful case for convexity is when .X = Y = Rn and .c(x, y) = |x − y|2 . This is the subject matter of the next section.

4.2.2 Brenier’s Theorem: Statement and Applications An extremely useful theorem in our field states, essentially, that under mild assumptions on two probability measures on .Rn , when the quadratic cost function is considered .c(x, y) = |x − y|2 , one may find a transport plan between the two measures which is not only optimal, it is concentrated on a set (in the product space) which is the graph of a function (in other words, the transport plan is in fact a transport map). Moreover, the transport map .T : Rn → Rn is given by the gradient of a convex function. This means that the differential of T is positive semi definite at every point, which is a very useful property, as we shall see below.

Statement of the Theorem A fundamental measure transport theorem in convex geometric analysis is the following [19, 20, 38, 39]

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Theorem 4.25 (Brenier–McCann) Let .μ, .ν ∈ P (Rn ) and assume that .μ is absolutely continuous with respect to the Lebesgue measure. Then, there exists a convex function .ϕ : Rn → (−∞, ∞] such that .∇ϕ : Rn → Rn is defined .μ-almost everywhere, and .(∇ϕ)#μ = ν. This theorem is intimately connected with optimal transport since not only is T = ∇ϕ a transport map between the two measures, it is easily seen, and will follow from our analysis below that it is optimal with respect to the quadratic cost. We shall see that it is equivalent to the fact that the support of the transport plan .π is a cyclically monotone set with respect to c, which in turn is equivalent to the fact that it lies on the graph of a sub-gradient of a convex function. All of these notions are explained below. In the next sections we will show some applications of the above theorem. In the sections following them we develop some of the general theory, which will then allow us to give a short proof of Brenier’s theorem (along with many other Breniertype theorems). Before embarking on this journey, however, we remark on a connection with the Legendre transform. Assume some transport map is given by .∇ϕ. Note that for a convex function .ϕ, its gradient .∇ϕ is defined almost everywhere in its domain. In fact, at every interior point of the domain the sub-gradient .∂ϕ(x) is defined, and it is not single-valued only in a set of Lebesgue measure zero. Here

.

∂ϕ(x) = {y : ∀z, ϕ(z) ≥ ϕ(x) + }.

.

Looking at the supremum in the definition of the Legendre transform Lϕ(y) = sup − ϕ(x),

.

x

we see that it is attained at x for a certain y, if and only if .∂ϕ(x) e y. It is not a hard exercise to check that the domain of .Lϕ is precisely the image of .∇ϕ. Using that .LL = Id (see Lemma 4.12), we have that .∇Lϕ(∇ϕ(x)) = x, that is, the gradients of .ϕ and of .Lϕ are inverse maps. So, if .∇ϕ maps .μ to .ν, then .∇Lϕ maps .ν back to .μ. Application: Another Proof of the Brunn–Minkowski Inequality One may use Brenier’s theorem to give yet another proof for the Brunn–Minkowski inequality for two convex bodies. Given two convex bodies .K0 and .K1 in .Rn , define 1 1 .μ = Vol(K0 ) Vol|K0 and .ν = Vol(K1 ) Vol|K1 . Use Theorem 4.25 to find a measure preserving transformation between them, which is given by .T = ∇ϕ for a convex .ϕ. Then the set .(I + ∇ϕ)(K0 ) will be a subset of .K0 + K1 . The fact that T is measure preserving implies that for all .x ∈ K0 we have .det(DT )(x) = det(∇ 2 ϕ)(x) = Vol(K1 ) Vol(K0 ) . Next we compute f .Vol(K0 + K1 ) ≥ Vol((I + T )(K0 )) = det(D(I +T ) (x))dx. K0

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As .(DT )(x) is the differential of the gradient of a convex function, it is a positive 1) definite matrix (with determinant . Vol(K Vol(K0 ) ). Denoting the eigenvalues of .DT (x) by n .(λi (x)) i=1 , the eigenvalues of .DI +T = I +DT at x are thus .(1+λi (x)). We already figured out (in the induction basis for our box-proof of the Brunn–Minkowski inequality), using the Arithmetic-Geometric inequality, see (4.1), that ( n || .

)1/n (1 + λi )

≥1+

i=1

( n ||

)1/n λi

i=1

( =1+

Vol(K1 ) Vol(K0 )

)1/n .

Putting these together we arrive at ( ) )n ( Vol(K1 ) 1/n n 1+( ) .Vol(K0 + K1 ) ≥ dx = Vol(K0 )1/n + Vol(K1 )1/n . Vol(K0 ) K0 f

Note that we could have instead used the more general fact that for positive definite .A, B we have .det(A + B)1/n ≥ det(A)1/n + det(B)1/n , which is one of Aleksandrov’s inequalities for positive definite matrices (see e.g. [6, Appendix B]).

Application: Mixed Volumes Mixed volumes are be discussed in depth the course of Ludwig and Mussnig (in this volume). Let us show briefly how Brenier’s theorem gives a short proof for Minkowski’s theorem on the polynomiality of volume, namely thatE as a function of m ∈ Rm , for fixed convex bodies .(K )m , the quantity .Vol( m λ K ) is a .(λi ) i + i=1 i i i=1 i=1 homogeneous polynomial of degree n. Usually one calls its coefficients the “mixed volumes” and writes Vol(

m E

.

i=1

λ i Ki ) =

m E i1 =1

···

m E in =1

V (Ki1 , . . . , Kin )

n ||

λij ,

j =1

where in the notation we use that the coefficient of a monomial only depends on the Ki ’s where i belongs to the monomial. In addition, we normalize V to be symmetric with respect to its arguments. It is part of Minkowski’s theorem that .V ≥ 0 as a function on n-tuples. We can show all of this using the Brenier map of Theorem 4.25. Choose (somewhat arbitrarily) one measure to be the Gaussian measure .γ on .Rn , and for 1 .i = 1, . . . , m let .νi = Vol(Ki ) Vol|Ki . Brenier’s theorem applied to the pair .γ and .νi gives us for each i a convex function .ϕi such that .∇ϕi transports .γ to .νi . In particular, and strangely this is all we will be using, this means that .∇ϕi (Rn ) = Ki , as sets, and as a consequence .∇(λi ϕi )(Rn ) E = λ i Ki . Em n The key observation here is that .∇( m i=1 λi ϕi )(R ) = i=1 λi Ki . One inclusion is immediate, just as we did in the proof above, where .(I + ∇ϕ)(K0 ) .

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was clearly a subset of .K0 + K1 . However, we claim that here we in fact have an equality, ∇(

m E

.

λi ϕi )(Rn ) =

i=1

m E

∇(λi ϕi )(Rn ).

i=1

The reason is intimately connected with the inf-convolution properties we discussed in Sect. 4.1.4. Indeed, for a convex function .η, the image of .∇η is in fact, up to boundary, the domain of the Legendre transform .Lη. Indeed, if .y ∈ ∂η(x) then .x ∈ ∂Lη(y) and in particular .Lη(y) < ∞. If .Lη(y) < ∞ and y is not on the boundary of theE domain, then there is some x for which .x ∈E∂Lη(y) and so .y ∈ ∂η(x). So, m m n .∇( i=1 λi ϕi )(R ) is simply the domain of .L( i=1 λi ϕi ). But as the Legendre transform convolution, the domain of this sum is E maps summation to Einfimum m simply . m λ dom(Lϕ ) = λ K . i i=1 i i=1 i i Once this is settled, we get that Vol(

m E

.

λi Ki ) = Vol(∇

i=1

m E

λi ϕi (Rn ))

i=1

f =

Rn

det(D∇

f

Em

(x))dx i=1 λi ϕi

=

Rn

det(

m E

λi ∇ 2 ϕi (x))dx

i=1

which is clearly a polynomial, and in fact, as the matrices are positive definite, one can use the theory of mixed discriminants to show that the coefficients of the various monomials are non-negative.

Application: Brascamp–Lieb Inequality Another very impressive application of the Brenier map is the Brascamp–Lieb inequality [18] and its reverse form by Barthe [13] (see also [14] and [12]). As this topic is thoroughly discussed in the lectures of Colesanti and Hug in this volume, let us only discuss the setting briefly, quote the theorem, and point at the specific part of the proof where Brenier’s theorem is used. The setting is as follows.ELet .m > n. Suppose we are given .c1 , . . . , cm > 0 and m .n1 , . . . , nm ∈ N satisfying . j =1 cj nj = n. For each .j = 1, . . . , m, we are given a n surjective linear .Bj : R → Rnj and assume that .∩m j =1 Ker(Bj ) = {0}. Define two + + n n m 1 operators .I, K : L1 (R ) × · · · × L1 (R ) → R by f I (f1 , . . . , fm ) =

m ||

.

Rn j =1

c

fj j (Bj x)dx

f and

K(h1 , . . . , hm ) =

∗ Rm

m(x)dx,

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f∗ where . denotes outer integral and where m(x) = sup

{ || m

.

c hjj (yj )

| yj ∈ R

nj

and

j =1

m E

} cj BjT yj

=x .

j =1

Let .E, F be the largest and smallest constants for which K(h1 , . . . , hm ) > E·

m (f ||

.

R

j =1

nj

hj

) cj

and

I (f1 , . . . , fm ) ≤ F ·

m (f || j =1

R

nj

fj

)cj

nj hold true for all .hj , fj ∈ L+ 1 (R ). For a .k × k symmetric, positive definite matrix A denote by .gA (x) = exp(−) a “centered Gaussian”. Franck Barthe proved the following theorem.

Theorem 4.26 (Barthe) The constants E and F can be computed using centered Gaussian functions. That is, { } K(g1 , . . . , gm ) E = inf || (f )cj | gj is a centered Gaussian, j = 1, . . . , m m n j =1 R j gj

.

and } { I (g1 , . . . , gm ) F = sup || (f )cj | gj is a centered Gaussian, j = 1, . . . , m . m n j =1 R j gj

.

Moreover, letting D denote the largest real number for which

.

det

m (E

m ) || cj BjT Aj Bj > D · (det Aj )cj ,

j =1

(4.4)

j =1

for all .Aj ∈ S + (Rnj ), we have E=

.

√ 1 D and F = √ . D

The applications of this theorem are far reaching, especially in its geometric form (a case where D can be computed and is equal to 1). This appears in the course of Colesanti and Hug. For the proof, we note that letting {

K(g1 , . . . , gm ) .Eg = inf )cj | gj is a centered Gaussian , j = 1, . . . , m ||m ( f n j =1 R j gj

}

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and {

} I (g1 , . . . , gm ) .Fg = sup )cj | gj is a centered Gaussian , j = 1, . . . , m , ||m ( f n j =1 R j gj our objective is to show that E = Eg =

√

.

1 D and F = Fg = √ . D

This follows once we establish √ .

1 D = Eg > E > D · F > D · Fg = D · √ . D

(4.5)

The second inequalities from the left and from the right are trivial. The rightmost and leftmost are sophisticated linear-algebra, and we do not discuss them here (see the original papers of Barthe [12–14], or [8, Section 4.4]) The following proposition is in fact the main step for the proof of Theorem 4.26, and implies the middle inequality .E ≥ D · F in (4.5). Recall that D is defined in (4.4), and we may clearly assume that .D > 0. We will show (using Brenier’s map) the following proposition. nj Proposition 4.27 Assume that the functions .hj , fj ∈ L+ 1 (R ), .1 ≤ j ≤ m, satisfy

f

f .

R

nj

fj =

R

nj

hj = 1.

Then, K(h1 , . . . , hm ) > D · I (f1 , . . . , fm ).

.

Proof We omit some technical details regarding the domain, range and differentiability of the involved mappings, as we mainly want to illustrate the usefulness of Brenier’s map. By Theorem 4.25, for every j there exists a mapping .Tj = ∇ϕj mapping the density .fj (x)dx to .hj (y)dy, where .ϕj is convex. This means (at differentiability points) that .

( ) det DTj (x) · hj (Tj x) = fj (x).

We use these maps to define a map .O : Rn → Rn by O(y) :=

m E

.

j =1

cj BjT (Tj (Bj (y))).

(4.6)

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S. Artstein-Avidan

E T By linearity its differential is given by .DO (y) = m j =1 cj Bj DTj (Bj y)Bj . Using the definition of the constant D in (4.4), the determinant of .DO is bounded from below .

m ( ) || det DO (y) > D · | det DTj (Bj y)|cj > 0,

(4.7)

j =1

and in particular .DO is positive definite (and symmetric of course). It follows that O is injective. Using (4.6) and (4.7) we may write

.

f K(h1 , . . . , hm ) =

.

Rn

sup

m { ||

sup O(Rn )

m { ||

m E

} cj BjT xj dx

j =1 c

hjj (xj ) | x =

j =1

f =

|x=

j =1

f >

c hjj (xj )

| det DO (y)| sup

m E

} cj BjT xj dx

j =1 m { ||

c

hjj (yj ) | O(y) =

j =1

f >

O(Rn )

>D·

| det DO (y)|

m ||

} cj BjT yj dy

j =1 c

hjj (Tj (Bj y)) dy

j =1

f || m

| det DTj (Bj y)|

j =1

=D·

m E

f || m

cj

m ||

c

hjj (Tj (Bj y)) dy

j =1 c

fj j (Bj y) dy

j =1

= D · I (f1 , . . . , fm ). u n

4.2.3 Our Plan for the Proof for Brenier’s Theorem The proof of Brenier’s theorem which we shall show is structured f as follows. Pick, among all measures .π ∈ ||(μ, ν), the one which minimizes . cdπ . The fact that a minimizer exists follows from some compactness argument, and we show this (using Prokhorov’s theorem) in Sect. 4.2.4. Here the specific choice of the quadratic cost is of no special importance, only some regularity of the cost is needed. We then show that a minimizing measure is concentrated on a subset of the product space with special structure, a so-called “cyclically monotone set”. This

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means, essentially, that (up to measure zero) no N -tuple .(xi , yi )N i=1 can be found in EN EN 2 2 the support, such that . i=1 |xi − yi | > i=1 |xi − yσ (i) | for some permutation .σ . It is quite intuitive that this be the case, as otherwise one could make some perturbation of the measure, not changing the marginals, and lowering the total cost. Here, again, changing the condition of “cyclically monotone” to a form of cyclic monotonicity involving a more general cost c, works quite well, under some regularity assumptions on the cost. Finally, the last step is to use Rockafellar’s theorem, which we will presently describe, which states that any cyclically monotone set is in fact the graph of the sub-gradient of a convex function. Together with the first steps, this means that the support of the optimal plan .π lies on the graph of the gradient of some convex .ϕ, and in particular, .π is not only a transport plan, but is a map. Here the generalization to other costs is also possible, but one must change the notion of “sub-gradient” to the less obvious notion of a “c-sub-gradient”. This is called the Rockafellar–Rochet– Rüschendorf theorem, and applies to costs with values in .R. We will discuss this part in depth as it is less well-known. In fact, quite recently the author jointly with Sadovsky and Wyczesany [9] generalized Rockafellar’s theorem to capture the case of costs which are allowed to obtain the value .+∞, and at the same time found a new and transparent proof for the original Rockafellar–Rochet–Rüschendorf theorem. We will describe this development here as well.

4.2.4 Existence of a Minimizer This subsection is devoted to the compactness argument which promises the existence of a minimizing measure in .||(μ, ν). This argument is not directly connected to the theme of this lecture series, and we only include it in the notes for completeness. In fact, the more sophisticated proofs further in these notes will give other methods to show existence of a minimizing plan, but it is good to have in mind the original straightforward argument as well. Note that for the quadratic cost the infimal total cost is bounded from below by 0, and if all plans have infinite cost then any plan is minimal in an empty sense. So we may as well assume that the total cost is finite. When working with a general cost (not bounded from below), this step sometimes requires some additional care. In general terms the way to do this is by compactness and lower semi continuity (this is a general “method” in such cases). Compactness allows to take an infimizing sequence and extract a converging sub-sequence, and lower semi continuity implies that the limit of the sub-sequence converges to where it should. First step (extracting a sub-sequence): For the compactness we make use of a well-known theorem from [43] (See also Theorem 3.17 in this volume). Theorem 4.28 (Prokhorov) Let S be a Polish space and .K ⊆ P (S) a subset of the set of probability measures on S. The subset K is pre-compact in the weak*

190

S. Artstein-Avidan

topology (that is, its closure is sequentially compact) if and only if K is tight, that is, for any .ε > 0 there exists some .Aε ⊆ S with .μ(Aε ) ≥ 1 − ε for any .μ ∈ K. Consider .K = ||(μ, ν), which is non-empty as it includes the product measure μ × ν. Let us explain why K is “tight". Fixing some .δ > 0 we can pick compact .A ⊆ X and .B ⊆ Y with .μ(A), ν(B) ≥ 1 − δ (this is called inner regularity of the measures). Clearly for .π ∈ ||(μ, ν) we have that for .C = A × B .

π(X × Y \ C) ≤ π(X × (Y \ B)) + π((X \ A) × Y ) = ν(Y \ B) + μ(X \ A) ≥ 1 − 2δ.

.

This means that we can find a compact set such that for all .π ∈ ||(μ, ν) its measure is big. The name for this property of .||(μ, ν) is, as explained above, “tight”, and Prokhorov’s theorem implies we can extract a converging sub-sequence for any sequence in .||(μ, ν). Next, .||(μ, ν) is closed in weak* topology. This is trivial: take a sequence .πk ∈ ||(μ, ν) which converges in weak* to .π . This means that f

f .

f (x, y)dπk (x, y) →

f (x, y)dπ(x, y)

for every .f ∈ C0(b) (X × Y ) (continuous, compact support). Using this for functions depending only on x and then for functions depending only of y we see that .π ∈ ||(μ, ν). Next consider a minimizing sequence .πk for the total cost. As .||(μ, ν) is sequentially compact, we may extract a converging sub-sequence, and re-index to have .πk → π in the weak* sense. The second step relies on the lower semi continuity of the cost function c, in this case the quadratic cost (but one may consider a lower semi continuous cost function). It is a well-known fact that weak convergence .μk → μ is equivalent to the inequality f .

lim inf

f f dμk ≥

f dμ

for all non-negative lower semi continuous f . Alternatively, we can just use that as c is non-negative and lower semi continuous we can find a sequence of continuous functions which is non-decreasing and converges point-wise to c, .cj → c. We may now use monotone convergence to get f .

f cdπ = lim j

cj dπ

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f f but by weak* convergence . cj dπ = limk cj dπk and by monotonicity of .cj we get f .

f cdπ = lim lim j

k

f cj dπk ≤ lim k

cdπk

as claimed. Note that in general we would have to move to a .lim inf with respect to k in the last inequality but we know that the limit exists having chosen a minimizing sequence. This completes the proof of the existence of a minimizing measure. Remark 4.29 For the students less familiar with Prokhorov’s theorem, let us illustrate the part most relevant to us, in .Rn . First note that tightness is necessary: For example in .R, if a set of probability measures is not tight, we can choose a sequence of probability measures such that for some .ε0 , .μk ((−k, k)) < 1 − ε0 . This means that if a limit measure (weak*) existed it would have to satisfy (note that an open indicator is lower semi continuous) μ(−x, x) ≤ lim inf μk (−x, x) ≤ lim inf μk (−k, k) ≤ 1 − ε0

.

and this for any x, impossible. So tightness is indeed necessary for weak* compactness. To see how tightness is sufficient (which is the condition we use) let us give a sketch for the proof in .Rn . A probability measure is determined by the function n .F (x) = μ(w : wi ≤ xi ∀i) which is a function on .R , with values in .[0, 1], increasing in each argument, converging to 0 as each argument tends to .−∞ and converging to 1 when all arguments tend to .+∞. It is upper semi continuous (with respect to the standard partial order). It also satisfies (inclusion/exclusion) that μ(

n ||

.

i=1

(ai , bi ]) =

E

(−1)#(θ) F ((ai + θi di )ni=1 ) ≥ 0

θ∈{0,1}n

(here .di = bi − ai ). Moreover, if we are given such an F , we can associate to it a probability measure in the natural way. We mention that since F increases on lines with non-negative coefficients, it is continuous almost everywhere. To show Prokhorov’s theorem, we are given the measures .μk and want to extract a converging sub-sequence. Instead, we will use the associated functions .Fk and extract a subsequence converging to some F , and we will see that the probability measure associated with this F is a limit of the sub-sequence of measures. We consider an enumeration of .Qn ⊂ Rn and using sub-sequences and diagonalization, find a sub-sequence of .Fk which converges point-wise on every .q ∈ Qn . This defines a function F on the rationals, and we extend it to .Rn by defining .F (x) = inf{F (q) : x ≤ q}. One can readily check that F satisfies that it is a distribution function of a measure. The tightness comes into checking the limits of F as .xi → 0 and as .min xi → ∞. It can also be checked that at points of continuity of F , we have usual convergence of .Fk to F . Finally, putting all of these together we get that the measures .μk converge weakly to .μ.

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4.2.5 Cost Duality To see more clearly why Brenier’s theorem has that particular form, namely why a gradient appears as the structural form of an optimal transport, and more interestingly to anticipate which other forms of transport maps can one expect if we optimize with respect to another cost function, we introduce the Kantorovich duality theorem along with the cost-transform for functions. The duality theorem serves as a motivation for the transform and, as we shall see, also for Brenier’s theorem.

Kantorovich Duality Theorem: Statement and Some Discussion Recall the definition of the total cost f .C(μ, ν) = inf{ c(x, y)dπ : π ∈ ||(μ, ν)}. X×Y

(Below we sometimes denote this quantity also by .Wc (μ, ν), see Sect. 4.3.6) It turns out that one may express this infimum in a dual way, as a supremum of integrals of pairs of functions satisfying some constraint. (Note the visual resemblance with the pairs of functions in the Lindenstrauss theorem). Theorem 4.30 Let .X, Y be Polish spaces and .μ ∈ P (X), .ν ∈ P (Y ) probability measures. Assume .c : X × Y → [0, ∞] is lower semi continuous. Then f f .C(μ, ν) = sup{ ϕdμ + ψdν : (ϕ, ψ) ∈ L1 (μ) × L1 (ν), X

Y

ϕ(x) + ψ(y) ≤ c(x, y) (μ, ν)−a.e.}. Moreover, the infimum in the definition of .C(μ, ν) is attained, and the supremum in the above formula does not change if one allows only bounded and continuous pairs .(ϕ, ψ). The fact that the infimum is attained was discussed in the previous section. The fact that the supremum does not change if we restrict to bounded and continuous functions is due to the fact that we can approximate any function in .L1 by bounded and continuous ones. We shall prove the theorem using an infinite dimensional minmax principle, but it is useful to notice that an inequality is immediate: Remark 4.31 If .ϕ, ψ are as above then for .π ∈ ||(μ, ν), f f f f . ϕdμ + ψdν = (ϕ(x) + ψ(y)) dπ ≤ X

Y

X×Y

c(x, y)dπ, X×Y

since the inequality holds .μ, ν a.e. which implies it holds .π a.e. as well. Since this is true for any .π ∈ ||(μ, ν), one may infimize over all these, and we get an inequality in the Kantorovich Duality Theorem.

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The proof of the theorem itself uses an infinite dimensional linear programming duality theorem, in which, strangely, Legendre transform on infinite dimensional spaces comes up. Theorem 4.30 is an easy consequence of the following theorem, where E is a general linear space, which can be (and will be, in our application) infinite dimensional. For the proof of Theorem 4.32 see [51]. We do comment on the geometry behind it after the statement. Here for a function .η : E → (−∞, ∞] where E is a linear space with a canonical pairing with its dual space . : E × E ∗ → R, the Legendre transform .η∗ is defined on the dual space .E ∗ by η∗ (y) = sup ( − η(x)) ,

.

x∈E

generalizing to infinite dimensions the definition we had in (4.2). Theorem 4.32 Let .ϕ, ψ : E → (−∞, ∞] be two convex functions and assume that for some .z ∈ E both are finite and .ϕ is continuous (this is crucial!). Then .

inf (ϕ(x) + ψ(x)) = max∗ (−ϕ ∗ (−y) − ψ ∗ (y))

x∈E

y∈E

( ) (= − min∗ ϕ ∗ (−y) + ψ ∗ (y) . y∈E

Note that part of the statement in the theorem is that on the right hand side there is an actual extremizer. To give some geometric intuition, let us see what the theorem means in finite dimensions, where we have already established some facts regarding the Legendre transform. Note first that L(ϕ + ψ)(0) = sup (−ϕ(x) − ψ(x)) = − inf (ϕ(x) + ψ(x)) .

.

x

x

Therefore, the left hand side in the formula claimed in Theorem 4.32 is simply −L(ϕ + ψ)(0). However, by Lemmas 4.12 and 4.13 (which we discussed in finite dimensions only) we know

.

L(ϕ + ψ)(0) = (LϕOLψ) (0).

.

In other words, we get that .

inf(ϕ + ψ) = − inf (Lϕ(x) + Lψ(y)) z+y=0

which is very similar to the claim in the theorem. The missing ingredients are that we did not show that the infimum is in fact attained as a minimum, and that we have used lemmas that we discussed only in finite dimensions. This paragraph serves only to provide some intuition, for the proof of the theorem, which is essentially a clever use of the Hahn–Banach theorem, see [51]. We next use this theorem to prove the Kantorovich Duality Theorem. Proof of Theorem 4.30 Using Theorem 4.32 We shall define our two convex func(b) tions, on the linear space .E = C0 (X × Y ) of continuous functions .f : X × Y → R

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with compact support, equipped with the supremum norm. Its dual space is the space of f all Radon measures (signed) on .X × Y , with the canonical pairing . = f dπ. The convex functions we choose are o(f ) = 1∞ {f ≥−c}

.

and f w(f ) =

.

f ϕdμ +

ψdν

if f (x, y) = ϕ(x) + ψ(y)

and .+∞ on functions f for which there is no such representation. Notice that such a representation is unique up to adding a constant to .ϕ and subtracting it from .ψ, so that .w is well defined. Convexity of .o is immediate, as is convexity of .w (in fact, linearity on its domain). In order to use Theorem 4.32 we need to find a point in their common domain at which one of the two convex functions is continuous. This point will be the function .f ≡ 1 which is clearly a bounded and continuous function on .X × Y . The function f belongs to the domain of .o since c is nonnegative, and .o is continuous at f since on a .1/2-neighborhood (in the .sup norm) of f , the function .o is constant 0. Finally, f is in the domain of .w since 1 can be written as the combination of two constant functions. Next we compute the Legendre transforms of .o and .w, which are now functions on the dual space, namely on the space of Radon measures. We expect (and this will indeed be the case) that .o∗ be a linear function (as .o was an indicator) and that the dual of .w be an indicator as .w was linear on its domain (we can write .w as . on its domain). Indeed, for a measure .π we get (f

∗

o (π ) = sup

.

f

o∗ (−π ) = sup

f ≥−c

)

f

f dπ − o(f ) = sup f

f (−f )dπ = sup

f ≤c

f ≥−c

f dπ

f dπ.

If .π is not a non-negative measure then there exists some very negative f (in the area were .π “is negative”) which forces the .sup to be f .+∞. On the other hand if .π is non-negative then we can get arbitrarily close to . cdπ by choosing appropriate f (we might not be able to take .c itself if it is not bounded, but we can get as close as we want), so we see that f ∗ .o (−π ) = cdπ

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for .π a non-negative measure, and .+∞ for measures outside the cone of nonnegative measures. As for .w ∗ , let us compute w ∗ (π ) = sup

.

(f

) f dπ − w(f )

f

(f

=

f f (x, y)dπ −

sup

f ϕdμ −

) ψdν .

f (x,y)=ϕ(x)+ψ(y)

If .π ∈ ||(μ, ν) then the expression on the right hand side is 0, and if .π /∈ ||(μ, ν) then for any M there exists a function f for which the difference f .

f (ϕ(x) + ψ(y)) dπ −

f ϕdμ −

ψdν > M.

We conclude that .w ∗ is the indicator of the set of measures .π ∈ ||(μ, ν). Plugging these in we get that .

∗

f

∗

− min∗ {o (−π ) + w (π )} = − min{

cdπ : π ∈ ||(μ, ν)}.

π ∈E

(Note that the fact that the marginals are .μ and .ν is not enough for .π ∈ ||(μ, ν) because we also require .π ≥ 0 which we obtained be the condition of belonging to the domain of .o∗ ). By Theorem 4.32 we get that f .

inf (o(f ) + w(f )) = − min{

cdπ : π ∈ ||(μ, ν)}

which, using that f .

inf (o(f ) + w(f )) =

completes the proof.

inf

ϕ(x)+ψ(y)≥−c

f ϕdμ +

ψdν, u n

The Cost Transform The Kantorovich Duality Theorem, Theorem 4.30, tells us that when searching for the total cost .C(μ, ν) we can finstead look f at pairs .ϕ(x), ψ(y) satisfying .ϕ(x) + ψ(y) ≤ c(x, y) and supremize . ϕdμ+ ψdν. (Here we let .−∞+∞ = −∞.) We can call a pair satisfying this inequality an “admissible pair”. Given some function .ϕ : X → [−∞, +∞], we may associate to it the largest .ψ such that the two constitute an admissible pair. This .ψ we call the “c-transform” of .ϕ.

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Definition 4.33 (The c-Transform) For .ϕ : X → [−∞, +∞], its c-transform is defined to be .ϕ c : Y → R given by ϕ c (y) = inf (c(x, y) − ϕ(x)) .

.

x

Similarly for .ψ : Y → [−∞, +∞], its c-transform is defined to be .ψ c : X → R given by ψ c (x) = inf (c(x, y) − ψ(y)) .

.

y

When .X = Y and .c(x, y) = c(y, x), these amount to the same transform. We abuse notation slightly by using the same notation for both transforms, as we usually assume .X = Y = Rn and assume the cost to be symmetric. (Here, to match the above definition of admissible pairs, one must let .∞ − ∞ = ∞.) The following facts follow directly from the definition. Fact If .ϕ and .ψ are admissible then .ϕ c ≥ ψ. Fact If .ϕ1 ≤ ϕ2 then .ϕ1c ≥ ϕ2c . A third simple fact is that on the image of the transform, namely for .ψ = ϕ c , the transform is an involution. More precisely, Fact For any .ϕ we have that .ϕ ccc = ϕ c . Proof Indeed, the pair .ϕ, ϕ c is admissible and thus by the first Fact above we see that .ϕ ≤ ϕ cc . We may now insert into this inequality the function .ϕ c , getting .ϕ c ≤ ϕ ccc . We may, instead, take the c transform of both sides of this inequality, and by the second Fact above get that .ϕ c ≥ ϕ ccc . u n This last Fact implies that to each cost function there corresponds a natural class of functions on which the cost-induced transform is an order reversing involution. Definition 4.34 (The c-Class) Given a symmetric cost function .c(x, y) : X×X → (−∞, ∞], the corresponding c-class is the image of the mapping .ϕ |→ ϕ c on the class of all functions .f : X → [−∞, ∞]. It readily follows from the above facts and definition that Fact A function .ϕ is in the c-class if and only if .ϕ = ϕ cc . Among functions in the c-class, there is a subfamily which is of special importance. Definition 4.35 (Basic Functions) A function of the form .ϕ(x) = c(x, y0 )+β, for a fixed .y0 ∈ Y and a constant .β ∈ R, is called a basic function for the cost c. When c is not a symmetric cost and X is different than Y , one must distinguish between X-basic functions of the form .ϕ(x) = c(x, y0 ) + β, and Y -basic functions of the form .ψ(y) = c(x0 , y) + β.

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Fact Any basic function associated to a cost c belongs to the c-class. Proof Consider the function .ψy0 ,β : Y → [−∞, ∞] given by .ψy0 ,β (y) = −∞ for y /= y0 and .ψy0 ,β (y0 ) = −β. Then we have1

.

( ) ψ c (x) = inf c(x, y) − ψy0 ,β (y) = c(x, y0 ) + β.

.

y

Therefore, the function .ψ(x) = c(x, y0 ) + β is in the c-class. (One may also check directly that .ϕ = ϕ cc ). u n It is useful to notice that the c-class is always closed under the operation of pointwise infimum. Fact Assume .(ϕα )α∈I is some family of functions which are all in the c-class, where I is some indexing set. Then the point-wise infimum ϕ(x) = inf ϕα (x)

.

α∈I

is also in the c-class. Proof Indeed, by assumption there are function .ψα such that .ϕα = ψαc (in fact, one may take .ψα = ϕαc ). Consider the function .ψ : Y → [−∞, ∞] ψ(y) = sup ψα (y).

.

α

Regardless of whether .ψα is in the c-class or not, we may consider its c-transform and we see that ( ) c .ψ (x) = inf (c(x, y) − ψ(y)) = inf c(x, y) − sup ψα (y) y

y

α

= inf inf (c(x, y) − ψα (y)) = inf ψ (x) = inf ϕα (x) = ϕ(x). c

α

y

α

α

u n The basic functions generate the c-class since, using the definition of the ctransform, we see that Fact Any function .ϕ in the c-class is the infimum of basic functions and vice-versa. Finally, whenever one encounters a class of functions which is closed under point-wise infimums, one may define within this class the operation of .sup(S) ˆ (where S is some subset of functions) which denotes the minimal element in the

that we use the fact that .c(x, y) /= −∞, since for .y /= y0 the function .ψy0 ,β gave the result .+∞ and under our convention, if added to .−∞ this would result in .−∞ and be taken as the infimum, had we allowed .c(x, y) to attain the value .−∞.

1 Note

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class which is greater than all the elements in the set. For example, the class of convex bodies is closed under intersection, and the “convex hull” is the .sup, ˆ a “replacement” for the union operation, which does not preserve convexity. The general definition is as follows: Definition 4.36 (The Operation .sup) ˆ Let .S be a class of functions which is closed under the operation of infimum. Given a family of functions .(ϕα )α∈I for some index set I , one defines sup(ϕ ˆ α : α ∈ I ) = inf{ϕ ∈ S : ϕ ≥ ϕα ∀α}.

.

We conclude with two main examples: The Legendre transform Lϕ(y) = sup ( − ϕ(x))

.

is, up to sign, a cost transform. Indeed, it can be recovered using one of the two 2 ˜ y) = |x−y| following possibilities for cost functions: .c(x, y) = − or .c(x, 2 . To see that .−ϕ c = L(−ϕ) write ϕ c (y) = inf(− − ϕ(x)) = − sup( − (−ϕ(x))) = −L(−ϕ)(y).

.

x

x

One may easily check that the corresponding c-class is the class of upper semi continuous concave functions (which are allowed to attain the value .−∞). Regarding the cost function .c, ˜ we note that ϕ c˜ (y) = inf(|x|2 /2−+|y|2 /2−ϕ(x)) = |y|2 /2+inf(−−(ϕ(x)−|x|2 /2))

.

x

x

so that ϕ c˜ (y) − |y|2 /2 = inf(− − (ϕ(x) − |x|2 /2)) = (ϕ − |x|2 /2)c .

.

x

This means that .c-class ˜ are simply functions in c-class with an added fixed function |x|2 /2, and the induced transform is, up to this “change of appearance”, the same. Another particular cost-induced transform which has recently received much attention is the .A-transform from [1] (see also [4, 7]). It is defined by

.

Aϕ(y) = sup

.

x

( − 1)+ . ϕ(x)

Here one must specify that division by 0 of a positive number gives .+∞ whereas the fraction . 00 gives 0 (this corresponds, after taking a logarithm, to the .±∞ calculus presented above for the cost setting). One may easily verify that the image of .A is the set of so-called geometric convex functions, .Cvx0 (Rn ), namely lower semi continuous convex functions which vanish

4 Dualities, Measure Concentration and Transportation

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at the origin and which are non-negative (and allowed to attain the value .+∞, but not everywhere). As a transform on epi-graphs, .A corresponds to standard polarity and reflection, see [3]. It turns out that .A is induced by the cost function p0 (x, y) = − ln( − 1)+

.

in the following sense .e−v 0 = A(e−v ), as can be verified with a direct computation. In particular, as the .A transform is an order reversing involution on the class of geometric convex functions (see [1]), the .p0 -class consists of .− ln f , where .f ∈ Cvx0 (Rn ). It is of interest that on the class .Cvx0 (Rn ) there are, up to linear variants, precisely two order reversing isomorphisms. These are .A, of course, but also the Legendre transform .L, for which .Cvx0 (Rn ) is an invariant subclass, and which is, as discussed, an order isomorphism on the larger class of all convex functions. This theorem is a close cousin of Theorem 4.11, and appears in [3]. Recapping, we see that in the Kantorovich Duality Theorem, Theorem 4.30, the supremum, which in the formulation runs over all admissible pairs .ϕ ∈ L1 (μ), ψ ∈ L1 (ν), can in fact be restricted to pairs of the form .(ϕ, ϕ c ) for .ϕ in the c-class (or, equivalently, to pairs .(ϕ cc , ϕ c )). One, however, must remain careful, as it may be that these pairs do not belong to .L1 (μ) × L1 (ν), in which case one must consider some truncation as well. Nevertheless, this observation reduces significantly the class on which the supremum in Kantorovich Duality Theorem is to be computed. p

The Cost Sub-Differential In Brenier’s theorem, Theorem 4.25, the gradient of a convex function plays a key role: it is the form of the optimal transport map. For more general cost functions (Brenier’s theorem deals of course with the quadratic cost function) the analogue of the gradient mapping is the so-called c-sub-gradient as we explain below. Definition 4.37 (The c-Sub-Gradient) Given a cost .c : X × Y → (−∞, ∞] and a function .ϕ which is in the c-class, we define ∂ c ϕ = {(x, y) : ϕ(x) + ϕ c (y) = c(x, y) < ∞}.

.

We also define .∂ c ϕ(x) = {y : (x, y) ∈ ∂ c ϕ}. The condition .c(x, y) < ∞ for a pair .(x, y) ∈ ∂ c ϕ may seem unnatural, but it is essential for the theory we present in these lectures, as we shall readily explain.

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One motivation for the definition of the c-sub-gradient comes, as expected, from the Kantorovich Duality Theorem. Indeed, in searching for a pair .(ϕ, ϕ c ) for which the supremum in Kantorovich Duality Theorem is attained, namely f

f

Wc (μ, ν) =

c(x, y)dπ0 (x, y) =

.

f ϕdμ +

ϕ c dν,

X×Y

we note that the inequality .ϕ(x)+ϕ c (y) ≤ c(x, y) always holds, so that for equality to hold we expect the support of the measure .π0 to be concentrated on the equality cases for this inequality, which is precisely the definition of the c-sub-gradient of .ϕ. More precisely Lemma 4.38 Assume .π ∈ ||(μ, ν) and there is some c-class .ϕ ∈ L1 (μ) such that π(∂ c ϕ) = 1. Then .π is an optimal transport plan with respect to c.

.

Proof Indeed, as .π is some plan, an upper bound for .Wc (μ, ν) is given by f

f c(x, y)dπ(x, y) =

.

X×Y

c(x, y)dπ ∂cϕ

f

f =

∂cϕ

(ϕ(x) + ϕ c (y))dπ =

f ϕdμ +

X

ϕ c dν, Y

which by Kantorovich Duality Theorem is also a lower bound. This means that not only is .π optimal, but the supremum in Kantorovich’s theorem is attained, and in the case where .∂ c ϕ is a graph (namely, is uni-valued almost everywhere) .π is in fact a transport map, not just a plan. u n Note that by definition .∂ c ϕ c = {(y, x) : (x, y) ∈ ∂ c ϕ} which is another way of stating that as (set-valued) mappings, the two mappings .x |→ ∂ c ϕ(x) and .y |→ ∂ c ϕ c (y) are inverse to one another. This is precisely what we discussed regarding the Legendre transform in Sect. 4.2.2. We rewrite it here in fuller generality. Lemma 4.39 Let .ϕ : Rn → (−∞, ∞] be a convex lower semi continuous function, and set .ψ(x) = |x|2 /2 − ϕ(x). Let .c(x, y) = |x − y|2 /2, so that .|y|2 /2 − Lϕ(y) = ψ c (y). Then it holds that ∂ c ψ(x) = ∂ϕ(x)

.

where ∂φ(x) = {y : ∀z, φ(z) ≥ φ(x) + }

.

is the usual sub-gradient of a convex function. Proof By definition, ∂ c ψ = {(x, y) : ψ(x) + ψ c (y) = |x − y|2 /2}.

.

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Rearranging we get that ∂ c ψ = {(x, y) : ϕ(x) + Lϕ(y) = }.

.

On the other hand, the sub-gradient can be rewritten as ∂ϕ(x) = {y : − ϕ(x) ≥ sup − ϕ(z) = Lϕ(y)}

.

z

and since the choice .z = x clearly gives equality, we may rewrite ∂ϕ(x) = {y : − ϕ(x) = Lϕ(y)},

.

u n

exactly the same as .∂ c ψ(x).

Cost Cyclic Monotonicity As we have seen above, understanding optimal maps has to do with the possibility of finding a plan supported on the c-sub-gradient of some c-class function. We thus turn to the question of understanding the structure of supports of c-sub-gradients. To this end, we define a property of a subset of .X × Y called c-cyclic monotonicity. Definition 4.40 Given a cost .c : X × Y → (−∞, ∞] , a set .G ⊂ X × Y is called c-cyclically monotone if for any N and any .(xi , yi )N i=1 ⊆ G it holds that for any permutation .σ of .{1, . . . , N } N E .

c(xi , yi ) ≤

i=1

N E

c(xi , yσ (i) ).

i=1

We note the following easy lemma. Lemma 4.41 Let .c : X × Y → (−∞, ∞] and assume .G ⊆ ∂ c ϕ for some c-class .ϕ : X → (−∞, ∞]. The G is c-cyclically monotone. Proof By definition ϕ(xi ) + ϕ c (yi ) = c(xi , yi )

.

and ϕ(xi ) + ϕ c (yσ (i) ) ≤ c(xi , yσ (i) ).

.

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Summing over all i’s we get that the left hand sides have the same sum, and we end up with N E .

c(xi , yi ) ≤

i=1

N E

c(xi , yσ (i) ).

i=1

u n Our main goal in this section is to reverse this observation, and show that in many cases any c-cyclically monotone set has a so-called “potential”, that is, some c c .ϕ such that .G ⊆ ∂ ϕ. When this is possible, and further, when .∂ ϕ is actually a mapping, we will not only get a maximizer in Kantorovich Duality Theorem, but this maximizer will correspond to a transport map (rather than a mere plan). In the case of a quadratic cost, this map will be the gradient of a convex function (which is well-defined almost everywhere) which in addition means its differential is positive definite. It turns out that the above lemma can be reversed in the case of so called “traditional” costs, namely when .c(x, y) ∈ R for all .x ∈ X, y ∈ Y . For costs that are allowed to attain the value .+∞, a slightly stronger condition than ccyclic monotonicity is needed for the existence of a potential, as we will discuss in Sect. 4.2.8. However, before embarking on this important goal of reversing the implication in Lemma 4.41, we show that in the classical case of the quadratic cost on .Rn (or, more generally, any continuous cost), there exists a plan with c-cyclically monotone support. Thus, a reverse Lemma to Lemma 4.41 will in fact produce an optimal transport plan supported on a c-sub-gradient (recall Lemma 4.38). Lemma 4.42 Let .μ and .ν be Borel probability measures on .Rn , and let .c : Rn × Rn → R be a continuous cost function. Then there exists a probability measure .π ∈ ||(μ, ν) with c-cyclically monotone support. E Proof This is clearly the case for finite discrete measures .μ = m1 m i=1 δxi and E m 1 .ν = δ since we simply optimize on a finite set of possibilities and select i=1 yi m the best permutation (see Sect. 4.2.1). Given some general .μ and .ν we construct discrete measures .μk and .νk which converge to them in the weak* topology. For each k we find .πk ∈ ||(μk , νk ) with c-cyclically monotone support. By a standard compactness argument (see Sect. 4.2.4) there exists a weak* sequential limit .π of .πk , which by definition of weak* convergence is in .||(μ, ν) and by continuity of c has c-cyclically monotone support. Indeed, for .(zi , wi ) ∈ support(π ) there are (k) (k) .(z i , wi ) ∈ support(πk ) converging to them, so by continuity of the cost N E .

c(zi , wi ) − c(zi , wσ (i) ) = lim

N E

i=1

This completes the proof of the Lemma.

(k)

(k)

(k)

(k)

c(zi , wi ) − c(zi , wσ (i) ) ≤ 0.

i=1

u n

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203

Note that we have not claimed that the plan .π we constructed is optimal with respect to c, though we do expect this as a limit of optimal plans for very close-by measures.

4.2.6 Rockafellar’s Theorem: Statement and Classical Proof The first main constructive theorem we present states that when the cost considered attains only finite values, given a c-cyclically monotone set G, one may find a “potential” .ϕ which is a c-class function such that .G ⊆ ∂ c ϕ. Theorem 4.43 (Rockafellar–Rochet–Rüschendorf) Let .c : X × X → R be a symmetric cost function, .c(x, y) = c(y, x). Let .G ⊂ X × X be non-empty and c-cyclically monotone. Then there exists some c-class function .ϕ : X → [−∞, ∞] such that .G ⊆ ∂ c ϕ (and in particular, .ϕ(x) /= ±∞ for x such that .(x, y) ∈ G). We provide two proofs for this theorem. Our main goal is to show a new proof from [9] which we find both insightful and simple in nature, and which, more importantly, can be generalized to non-traditional costs in a clear fashion. This is given in the next section. However, for completeness of the exposition, we first present the classical proof, see [44–46]. Proof Fix some element .(x0 , y0 ) ∈ G which we shall call the “pivot”. We will make sure .ϕ(x0 ) = 0. Define (

) m E .ϕ(x) = inf c(x, ym ) − c(x0 , y0 ) + (c(xi , yi−1 ) − c(xi , yi )) , i=1

where the infimum runs over all .m ∈ N and all m-tuples .(xi , yi ) ∈ G, i = 1, . . . , m. The first observation is that this function is indeed in the c-class, namely it is c .ψ for some .ψ. Indeed, this holds because the c-class is closed under infimum, and each of the functions in this infimum is simply of the form .c(x, ym ) − β which (we showed before) is in the c-class as well. The second observation is that .ϕ(x0 ) = 0. Indeed, by picking .m = 0 (or .m = 1 with .(x1 , y1 ) = (x0 , y0 )) it is clear that .ϕ(x0 ) ≤ 0. On the other hand the c-cyclic monotonicity implies that m E .

i=0

c(xi , yi ) ≤ c(x0 , ym ) +

m E

c(xi , yi−1 )

i=1

for any choice of .(xi , yi )m i=1 ⊆ G so that the expression in the infimum, evaluated at .x0 , is at least 0.

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Our goal is thus to show that if .(x, y) ∈ G then .ϕ(x) + ϕ c (y) = c(x, y). We shall do this without computing .ϕ c , but instead showing that for any .z ∈ X we have that c(x, y) − ϕ(x) ≤ c(z, y) − ϕ(z).

.

(4.8)

Before doing this, let us have a short discussion on whether these two claims, (4.8) and .(x, y) ∈ ∂ c ϕ, are indeed the same. The answer is an obvious yes, since this means that the infimum is attained at x. A problem may occur only if we start worrying that perhaps .ϕ(x) = ±∞ (which, the reader will recall, is not allowed for x with .(x, y) ∈ ∂ c ϕ). We clearly have .ϕ(x) ≤ c(x, y0 )−c(x0 , y0 ) which is finite. So one might nevertheless worry that maybe .ϕ(x) = −∞. This definitely could happen if x is some arbitrary point, however here we are assuming .(x, y) ∈ G. In particular, once we show the above inequality, .c(x, y) − ϕ(x) ≤ c(x0 , y) − ϕ(x0 ) = c(x0 , y) so .ϕ(x) cannot be .−∞. To show (4.8) we take some .t > ϕ(x) and use the definition of .ϕ(x) as an infimum to find some .m ∈ N and some .(xi , yi )m i=1 ⊆ G such that t > c(x, ym ) − c(x0 , y0 ) +

.

m E (c(xi , yi−1 ) − c(xi , yi )). i=1

Since .(x, y) ∈ G we may rename them .(xm+1 , ym+1 ) and use the new .(m + 1)-tuple for an upper bound in the definition of .ϕ(z) as an infimum. Namely ϕ(z) ≤ c(z, ym+1 ) − c(x0 , y0 ) +

m+1 E

.

(c(xi , yi−1 ) − c(xi , yi )).

i=1

Plugging in our definitions this becomes .

ϕ(z) ≤ c(z, y) − c(x0 , y0 ) + c(x, ym ) − c(xm , ym ) +

m E

(c(xi , yi−1 ) − c(xi , yi )),

i=1

and using our inequality for t we get .

ϕ(z) − c(z, y) ≤ m E (c(xi , yi−1 ) − c(xi , yi )) < t − c(x, y). −c(x0 , y0 ) + c(x, ym ) − c(x, y) + i=1

As this is true for any .t > ϕ(x), we get that ϕ(z) − c(z, y) ≤ ϕ(x) − c(x, y),

.

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as claimed. This means that .ϕ c (y) + ϕ(x) = c(x, y), namely .(x, y) ∈ ∂ c ϕ, which completes the proof. u n Remark 4.44 The c-cyclic monotonicity of G was only used in the part where we showed that .ϕ(x0 ) = 0.

4.2.7 A New Proof of Rockafellar’s Theorem While the above classical proof is quite short and beautiful, it is not immediate how one may use it to tackle costs which are non-traditional, namely costs for which there exist pairs .(x, y) with .c(x, y) = +∞. To do this, we first reformulate the problem of finding a potential for a given set .G ⊂ X × Y as a question regarding the existence of a solution to a linear system of inequalities. Theorem 4.45 Let .c : X × Y → R ∪ {+∞} be a cost function and let .G ⊆ X × Y . Then there exists a potential for G, namely a c-class function .ϕ : X → [−∞, ∞] such that .G ⊆ ∂ c ϕ, if and only if the following system of inequalities, c(x, y) − c(z, y) ≤ a(x) − a(z),

.

(4.9)

indexed by .(x, y), (z, w) ∈ G, has a solution .a : PX G → R, where .PX G = {x ∈ X : ∃y ∈ Y, (x, y) ∈ G}. Proof Assume that there exists a potential .ϕ : X → [−∞, ∞] such that .G ⊆ ∂ c ϕ. We note that on .PX G the function .ϕ must attain only finite values, because .G ⊆ ∂ c ϕ means in particular that .ϕ(x) + ϕ c (y) = c(x, y) < ∞. For every .z ∈ PX G we have ) ( ϕ(z) = inf c(z, w) − ϕ c (w) ≤ c(z, y) − ϕ c (y).

.

w∈Y

At the same time, since .(x, y) ∈ ∂ c ϕ, ( ) ϕ(x) = inf c(x, w) − ϕ c (w) = c(x, y) − ϕ c (y).

.

w∈Y

Taking the difference of the two equations we get c(x, y) − c(z, y) ≤ ϕ(x) − ϕ(z).

.

Letting .a(x) = ϕ(x) for .x ∈ PX G, we have found a solution for the system of inequalities. For the other direction, assume we have a solution to the system of inequalities. We would like to extend it to some c-class function defined on X. To this end let ϕ(z) =

.

inf {c(z, y) − c(x, y) + a(x)}.

(x,y)∈G

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We will show that the function .ϕ, which is clearly in the c-class, is an extension of the function .a : PX G → R, and that it is a potential for G, namely .G ⊆ ∂ c ϕ. The assumption (4.9) implies that for .z ∈ PX G we have a(z) ≤ c(z, y) − c(x, y) + a(x)

.

and so the infimum in the definition of .ϕ is attained at z itself. In particular, we get that .ϕ is indeed an extension of a. This means that if .(x, y) ∈ G then ϕ c (y) = inf (c(z, y) − ϕ(z))

.

z∈X

= inf

sup

z∈X (x ' ,y ' )∈G

( ) c(z, y) − c(z, y ' ) + c(x ' , y ' ) − ϕ(x ' )

≥ inf (c(z, y) − c(z, y) + c(x, y) − a(x)) z∈X

= c(x, y) − a(x) = c(x, y) − ϕ(x). As the opposite inequality is trivial, we get that .(x, y) ∈ ∂ c ϕ, as required.

u n

Theorem 4.43 will follow from the above joined with the following theorem regarding systems of linear inequalities. Theorem 4.46 Let .{αi,j }i,j ∈I ∈ R, where I is some arbitrary index set, and with αi,i = 0. The system of inequalities

.

αi,j ≤ vi − vj ,

.

i, j ∈ I

(4.10)

has a solution if and only if forE any m and any .i1 , · · · , im ∈ I , and a permutation .σ of .[m] = {1, . . . , m}, we have . m k=1 αik ,iσ (k) ≤ 0. To make the translation, let us consider the set of indices I to be our original set G. EmIf .i = (x, y) and .j = (z, w) we let .αi,j = c(x, y) − c(z, y). The condition . k=1 αik ,iσ (k) ≤ 0 amounts to m E .

c(xi , yi ) − c(xσ (i) , yi ) ≤ 0,

k=1

which is c cyclic monotonicity. The theorem guarantees a solution .(v(x,y) )(x,y)∈G . We would like to define .a(x) = v(x,y) . To do this, we only need to explain why if ' .(x, y) ∈ G and .(x, y ) ∈ G then .v(x,y) = v(x,y ' ) . We do this in the next paragraph. Let us repeat this delicate point: The index set for the inequalities are pairs .((x, y), z) ∈ G × PX G (or, equivalently, pairs .((x, y), (z, w)) ∈ G × G, where we ignore w as it does not appear in the inequalities). The solution vector we are looking for is indexed by .PX G, and we denoted it .(a(x))x∈PX G . In fact, formally, we

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should be using .(a(x, y))(x,y)∈G , which seems to allow multi-valued a. However, note that if .(x, y) and .(x, y ' ) are both in G then 0 = c(x, y) − c(x, y) ≤ v(x,y) − v(x,y ' )

.

and 0 = c(x, y ' ) − c(x, y ' ) ≤ v(x,y ' ) − v(x,y)

.

which means v(x,y) = v(x,y ' ) .

.

In other words, even if we do index the vector by .(x, y) ∈ G instead of .x ∈ PX G, the solution vector depends only on the first coordinate. We thus concentrate from this point onward on proving Theorem 4.46. By summing the inequalities on a permutation .σ of .[m], we see that cyclic monotonicity is a necessary condition for the existence of a solution. The next lemma shows that in the case of finite G this condition is also sufficient. This is a classical fact (which we shall come back to in the infinite case) and we include a proof here for completeness. The main tool is Helly’s theorem (see e.g [48, Theorem 1.1.6]). Lemma 4.47 Let .αi,j ∈ R ∪ {−∞} for .i, j ∈ [m], with .αi,i = 0. The following are equivalent: (a) There exists a vector .v ∈ Rm such that for all .i, j αi,j ≤ vi − vj

.

and E (b) For any permutation .σ of .[m] we have that . m i=1 αi,σ (i) ≤ 0. Proof Clearly (a) implies (b), by summing over the pairs .(i, σ (i)), as explained above: E E . αi,σ (i) ≤ vi − vσ (i) = 0. i∈J

i∈J

For the other direction, we shall use induction. Note that we are given a set of at most .m(m − 1) inequalities and we would like E to show they have a joint solution. Without loss of generality we may assume . vi = 0, so in fact these .m(m − 1) inequalities are on a vector which is essentially in .Rm−1 . By Helly’s theorem, it is enough to make sure that any m of these inequalities have a joint solution (the existence of a solution amounts to the intersection of the corresponding half-spaces being non-empty).

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Using induction on m, we may assume that the theorem is true for .(m − 1). In particular, given a subset P of m pairs .{(i, j )}, if there is an index within .{1, . . . , m} which does appear in any of these, then by induction we know that the intersection of the corresponding half-spaces in not empty. We may thus assume that the family of m inequalities which we are trying to satisfy simultaneously include each of the integers in .{1, . . . , m} at least once. Further, we argue that if one of these integers, say k, appears only once, say then the constraint on .vk is one sided. In particular, we may consider the vector v without its kth coordinate, solve the system on inequalities using the induction assumption, and then solve the single remaining inequality for .vk (since a single inequality in a single variable always has a solution). Moreover, the same argument applies if one of the integers, call it k again, appears (as many times as it likes) only as the first (resp. only as the second) in any pair .(i, j ) ∈ P . We may thus assume, without loss of generality, that each of the indices in .{1, . . . , m} appears at least once as a first index and at least once as a second index, in the set of inequalities P . Altogether in P there are precisely m appearances as a first index and m as a second, which means that each integer appears exactly once as a left index and once as a right. In other words, the pairs in P form a permutation of .{1, . . . , m}. We may decompose the permutation into its non interacting components. If there are more than one, then by induction we have a solution again. If not, then we have a full permutation .σ on .{1, . . . , m} and we want to find a joint solution to the inequalities αi,σ (i) ≤ vi − vσ (i) .

.

We fix .v1 arbitrarily and let .vσ (1) = v1 −α1,σ (1) , .vσ (σ (1)) = vσ (1) −ασ (1),σ (σ (1)) and inductively, having defined .vj , let .vσ (j ) = vj − αj,σ (j ) . After m steps we will have defined .vσ m−1 (1) , say .σ m−1 (1) = k. Clearly .σ (k) = 1. All inequalities are satisfied (as equalities, in fact) except possibly the last one: .αk,1 ≤ vk − v1 . Since we have defined everything explicitly, we can write this inequality explicitly as well: αk,1 ≤ vk − v1 = vσ −1 (k) − ασ −1 (k),k − v1

.

= · · · = vσ −i (k) −

i−1 E

ασ −(j +1) (k),σ −j (k) − v1

j =0

= · · · = v1 −

m−1 E

ασ −(j +1) (k),σ −j (k) − v1 .

j =0

Canceling .v1 and rewriting the inequality, it is precisely m E .

i=1

αi,σ (i) ≤ 0

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which holds by our assumption (b). We thus showed that any m inequalities can be satisfied simultaneously (the condition that the sum is 0 can be satisfied simply be adding a constant to all coordinates of the vector), and conclude, using Helly’s theorem and the fact that the problem is essentially .(m − 1) dimensional, that if (b) holds then the system of inequalities in (a) has a solution. u n To move from finite sets of inequalities to an infinite one, we use compactness in a standard fashion. Proposition 4.48 Let .αi,j ∈ R for .i, j ∈ I with .αi,i = 0 and I is some index set. Consider the family of inequalities αi,j ≤ vi − vj .

.

Assume that for any finite subset .J ⊆ I , one may find a solution .(vj )j ∈J satisfying jointly all the inequalities for .i, j ∈ J . Then there exists a solution .(vi )i∈I solving jointly all the inequalities. Proof Recall Tychonoff’s theorem which states that the product of any collection of compact topological spaces is compact with respect to the product topology. We fix some .i0 ∈ I and let .vi0 = 0. The space || where we are searching || for solutions to the family of inequalities is now .X = i∈I \{i0 } [αi,i0 , −αi0 ,i ] = i∈I Xi , since among the inequalities we will have .αi,i0 ≤ vi − vi0 ≤ −αi0 ,i (which, by the way, we know is a non-empty interval by our assumption of solvability for any finite subset, in the case the subset .{i0 , i}). Note that X is compact with respect ||to the product topology, in which a basis for open sets is the Cartesian product . i∈I Ai where .Ai is the whole interval .Xi except finitely many indices in which .Ai is an open set. The set of elements .v ∈ X for which a certain inequality .α(i, j ) ≤ vi − vj is not satisfied is clearly an open set, as it is an open set in the two participating coordinates (product with the full intervals in all the other components). Assume there does not exist a solution to the (possibly infinite) system of inequalities. This means that every point in the product space lies in at least one of these open sets, namely for every choice of v at least one of the inequalities is not satisfied. We thus have a cover of the space X, and by compactness there is some finite sub-cover. As it is finite, a finite number of indices, say only the finite subset J , participate in the corresponding inequality. The fact that it is a cover means that not all these inequalities can be satisfied simultaneously within X. But, after adding the inequalities .αi,i0 ≤ vi − vi0 ≤ −αi0 ,i for the indices .i ∈ J , we know by our assumption that there is a solution vector .(vi )i∈J , and we may add to it any .vi ∈ Xi for .i /∈ J , getting a contradiction to the covering property of the finite sub-cover. We conclude that the original collection was not a cover, namely, there is a solution to the infinite family of inequalities. u n We have thus completed the new and simple proof for the Rockafellar–Rochet– Rüschendorf theorem for real valued costs.

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Corollary 4.49 Let .c : X × Y → R be a finitely valued cost function. Assume we are given a set .G = {(xi , yi )}i∈I which is .c-cyclically monotone. Then there exists a c-class function .ϕ : X → (−∞, ∞] such that .G ⊆ ∂ c ϕ, namely for any .i ∈ I we have .(xi , yi ) ∈ ∂ c ϕ. The upshot of having such a simple proof, is that it crystallizes what needs to be added in the case of general costs, which is the subject of the next section.

4.2.8 Rockafellar-Type Theorem, Non-Traditional Our motivation for providing the new proof above was that for non-traditional costs (costs that are allowed to assume .+∞, such as the polar cost) it may happen that a set is c-cyclically monotone but fails to have a potential .ϕ, even when c is continuous and the spaces considered are simply .Rn . The proof above, however, crystallized what the “right” condition for the existence of a potential is, and we call it c-pathboundedness. Definition 4.50 Fix sets .X, Y and .c : X × Y → (−∞, ∞]. A subset .G ⊆ X × Y will be called c-path-bounded if .c(x, y) < ∞ for any .(x, y) ∈ G, and for any .(x, y) ∈ G and .(z, w) ∈ G, there exists a constant .M = M((x, y), (z, w)) ∈ R such that the following holds: For any .m ∈ N and any .{(xi , yi ) : 2 ≤ i ≤ m − 1} ⊆ G, denoting .(x1 , y1 ) = (x, y) and .(xm , ym ) = (z, w), we have m−1 E .

( ) c(xi , yi ) − c(xi+1 , yi ) ≤ M.

i=1

It is not hard to see that a c-path-bounded set must be c-cyclically monotone (indeed, if .(x, y) = (z, w) then if there is some path for which the sum is positive, one can duplicate it many times to get paths with arbitrarily large sums). It is also not hard to check that c-path-boundedness is a necessary condition for the existence of a potential. Our main theorem is that the condition of c-path-boundedness is in fact equivalent to the existence of a potential. Theorem 4.51 Let .X, Y be two arbitrary sets and .c : X × Y → (−∞, ∞] be an arbitrary cost function. For a given subset .G ⊆ X ×Y there exists a c-class function c .ϕ : X → [−∞, ∞] such that .G ⊆ ∂ ϕ if and only if G is c-path-bounded. Note that in Theorem 4.45 we did allow the cost to assume infinite values, so that we may use it to once again reformulate the problem of finding a potential for a given set .G ⊆ X × Y as a question regarding the existence of a solution to a linear system of inequalities: c(x, y) − c(z, y) ≤ a(x) − a(z),

.

(4.11)

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indexed by .(x, y), (z, w) ∈ G. The main difference between this system and the one given in (4.9) is that here some of the inequalities are in fact of the form .−∞ ≤ a(x) − a(z), which are satisfied automatically. While this seems at first sight to be a helpful feature, it ruins the compactness argument we used in Proposition 4.48. Indeed, one may easily construct a set of inequalities such that any finite subset of inequalities has a solution but the infinite system does not (here is an example: .v1 ≤ vi and .vi ≤ v2 − i for .i = 3, 4, . . .). This is precisely the reason we need a somewhat stronger condition here than we did in the classical Rockafellar–Rochet– Rüschendorf Theorem (c-path boundedness rather than c-cyclic monotonicity). Our main theorem is a direct consequence of the next theorem regarding systems of linear inequalities, using Theorem 4.45. Theorem 4.52 Let .{αi,j }i,j ∈I ∈ [−∞, ∞), where I is some arbitrary index set, and with .αi,i = 0. The system of inequalities αi,j ≤ xi − xj ,

.

i, j ∈ I

(4.12)

has a solution if and only if for any .i, j ∈ I there exists some constant .M(i, j ) such Em−1that for any m and any .i2 , · · · , im−1 , letting .i = i1 and .j = im one has that . k=1 αik ,ik+1 ≤ M(i, j ). Instead of proving Theorem 4.52 directly, we shall prove the following theorem, which at first glance might seem weaker. Theorem 4.53 Let .{ai,j }i,j ∈I ∈ [−∞, ∞), where I is some arbitrary index set. Assume that for any .m ≥ 1 and any .i1 , i2 , · · · , im it holds that .ai1 ,im ≥ Em−1 k=1 aik ,ik+1 . Then the system of inequalities ai,j ≤ xi − xj ,

.

i, j ∈ I

has a solution. Clearly, Theorem 4.52 implies Theorem 4.53. In fact, the reverse implication holds as well. We will show this implication first, namely that Theorem 4.52 follows from Theorem 4.53. Proof that Theorem 4.53 implies Theorem 4.52 The “only if” part of Theorem 4.52 is easy and does not require Theorem 4.53. Indeed, let .{αi,j }i,j ∈I ∈ [−∞, ∞), where I is some arbitrary index set, and with .αi,i = 0. Assume that the system of inequalities αi,j ≤ xi − xj ,

.

i, j ∈ I

has a solution, .(xi )i∈I . Summing the relevant inequalities we see that .M(i, j ) = xi − xj provides the required bound.

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For the opposite direction, we will use Theorem 4.53. Assume that for any .i, j ∈ m−1 I there exists some .M(i, j ) such that for any m and any .{ik }k=2 , letting .i1 = i and .im = j it holds that m−1 E .

αik ,ik+1 ≤ M(i, j ).

k=1

Define new constants .ai,j ∈ [−∞, ∞) as follows: ai,j = sup{

m−1 E

.

αik ,ik+1 : m ∈ N, m ≥ 2, i2 , . . . , im−1 ∈ I }.

k=1

By the above condition, the right hand side is bounded from above and so the supremum is not .+∞. We first claim that the system of inequalities .ai,j ≤ xi − xj , satisfies the conditions of Theorem 4.53.EAssume we are given .i1 , i2 , · · · , im−1 , im , and we m−1 want to prove that .ai1 ,im ≥ k=1 aik ,ik+1 . Fix .ε > 0. For each .k ∈ [m] use the (k) (k) (k) definition of .aik ,ik+1 to pick some .mk and .i2 , . . . , imk −1 such that, letting .i1 = ik (k)

and .imk = ik+1 , we have aik ,ik+1 ≤

mE k −1

.

l=1

αi (k) ,i (k) + ε/m. l

l+1

We have thus identified some finite set of indices in I , the set (k)

(k)

J = {ik , i2 , . . . , imk −1 : k ∈ [m − 1]} ∪ {im },

.

which is naturally arranged as a path from .i1 to .im . Using again the definition of ai,j , the path thus defined participates in the supremum, and we have that

.

ai1 ,im ≥

.

m m E ) (E (aik ,ik+1 − ε/m) = aik ,ik+1 − ε. k=1

k=1

As this holds for any .ε, we get the inequality in the condition of Theorem 4.53. Applying Theorem 4.53, we see that the system of inequalities ai,j ≤ xi − xj ,

.

(4.13)

admits a solution. Moreover, since .ai,j ≥ αi,j by definition, the resulting vector x u n is also a solution of the original system of inequalities.

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Having made the reduction from Theorem 4.52 to Theorem 4.53, we proceed by proving the latter. Proof of Theorem 4.53 We use Zorn’s Lemma. Consider the partially ordered set of pairs .(J, fJ ) where .J ⊆ I and .fJ : J → R are such that for any .i, j ∈ J we have .fJ (i) − fJ (j ) ≥ ai,j . We know the set is non-empty because it contains pairs .({i0 }, 0). The partial order we consider is .(J, fJ ) ≤ (K, fK ) if .J ⊆ K and .fK |J = fJ . First let us notice that every chain has an upper bound. Assume .(Jα , fJα )α∈A is a chain (namely any two elements are comparable). Consider .J = ∪α Jα and .fJ = ∪α fJα . This function is well defined because of the chain property (at a point .i ∈ J it is defined as .fJα (i) for any .α with .i ∈ Jα ). The pair .(J, fJ ) is in our set because if .i, j ∈ J then for some .α we have .i, j ∈ Jα , so .f |Jα satisfies the inequality on .fJ (i) − fJ (j ) ≥ ai,j and so does .fJ . Finally, .(J, fJ ) is clearly an upper bound for the chain. So, we have shown that every chain has an upper bound, and we may use Zorn’s lemma to find a maximal element. Denote the maximal element by .(J0 , fJ0 ). Assume towards a contradiction that .J0 /= I , that is, there exists some element .i0 ∈ I such that .i0 /∈ J0 . If we are able to extend .fJ0 to be defined on .{i0 } in such a way that all inequalities with indices of the form .(i0 , j ) and .(j, i0 ) with .j ∈ J0 still hold, we will contradict maximality and complete the proof. Note that the inequalities that need to be satisfied in order to extend the function are ai0 ,j ≤ f (i0 ) − f (j ) and

.

aj,i0 ≤ f (j ) − f (i0 ).

That is, .f (i0 ) is to be defined in such a way that .

) ( ( ) sup ai0 ,j + fJ0 (j ) ≤ f (i0 ) ≤ inf fJ0 (j ) − aj,i0 . j ∈J0

j ∈J0

For there to exist such an element, .fJ0 must satisfy that .

( ( ) ) sup ai0 ,j + fJ0 (j ) ≤ inf fJ0 (j ) − aj,i0 , j ∈J0

j ∈J0

(4.14)

or, in other words, that for any .j, k ∈ J0 ai0 ,j + fJ0 (j ) ≤ fJ0 (k) − ak,i0 .

.

We can rewrite the condition as .ai0 ,j + ak,i0 ≤ fJ0 (k) − fJ0 (j ). Recall that under our assumptions .ak,j ≥ ak,i0 + ai0 ,j . Since .fJ0 already satisfies the inequality .ak,j ≤ fJ0 (k) − fJ0 (j ), we know the above inequality holds for any .j, k, and so the inequality (4.14) holds and we may extend the function .fJ0 . This is a contradiction to the maximality, and we conclude .J0 = I , so that we have found a solution to the u n full system of inequalities.

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We are now ready to prove our new Rockafellar-type theorem for non-traditional costs. Proof of Theorem 4.51 One direction is immediate: assume .G ⊆ X × Y satisfies that for some c-class .ϕ : X → [−∞, ∞], .G ⊆ ∂ c ϕ. Then c(x, y) − c(z, y) ≤ ϕ(x) − ϕ(z)

.

holds for all .(x, y), (z, w) ∈ G. So, given a pair .(x, y), (z, w) ∈ G we set M = M((x, y), (z, w)) = ϕ(x) − ϕ(z). Any sum, as in the definition of c-pathboundedness, will be bounded by the corresponding sum of differences .ϕ(xi ) − ϕ(xi+1 ), which make for a telescopic sum adding up to M. So we conclude that G is c-path-bounded. Assume, in the other direction, that .G ⊆ X × Y is c-path-bounded. By Theorem 4.45 we needed to show that the family of inequalities

.

c(x, y) − c(z, y) ≤ ϕ(x) − ϕ(z),

.

where .(x, y), (z, w) ∈ G, has a solution. By Theorem 4.52, as G is c-path-bounded, a solution exists. u n

4.3 Lecture III: Concentration of Measure We discussed in Lecture I that usually one cannot find exact extremizers for the volume of a t-extension of a set of fixed measure, and the fact that finding such estimates is desirable since when these are strong, in the sense that the t-extension has large measure (namely, when we have “concentration of measure”), this is a great tool for proving many beautiful results. In this lecture we shall see other ways for obtaining concentration, without knowing exact extremizers, some of which are intimately connected with the transportation of measure results from Lecture II. We will not be discussing the extremely rich array of examples for applications of concentration of measure, which started with the proof of Dvoretzky’s theorem by V. Milman and these days spans a vast amount of literature. Many examples and insights into this topic can be found in the books [6, 8, 21] and in the references therein. In this lecture, we will mainly concentrate (!) on methods to obtain concentration. As mentioned, concentration of measure is a phenomenon of high dimensions which is responsible to many counter-intuitive (until intuition changes and these become intuitive) results. Let us approach it, as an introduction to this last lecture, from a somewhat non-standard angle. We recall the beautiful formula for computing the surface area of a convex body using its projections.

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Lemma 4.54 (Cauchy Formula) Let .K ⊂ Rn be a convex body with non-empty interior. Then f κn .Voln−1 (∂K) = n Voln−1 (Pu⊥ K)dσ (u). κn−1 S n−1 Sometimes one uses integration on the sphere with respect to usual Lebesgue measure, the relation being .dσ (u) = du/Voln−1 (S n−1 ) = du/(nκn ) rewriting the formula as f 1 .Voln−1 (∂K) = Voln−1 (Pu⊥ K)du. κn−1 S n−1 Proof We work with a polytope P first. For a generic .u ∈ S n−1 , each facet .Fi of P has some angle .θi between its normal and u which is not .π/2. When such a facet is projected onto .u⊥ , its area when projected is .| cos(θ )| times its original area. Clearly the projection .Pu⊥ (P ) is covered twice by the projections of the facets of P . We get that 1E | cos(θi )|Voln−1 (Pi ). 2 m

Voln−1 (Pu⊥ (P )) =

.

i=1

When integrated over the sphere, we get f .

S n−1

1E Voln−1 (Pi ) 2 m

Voln−1 (Pu⊥ (P ))dσ (u) =

i=1

f S n−1

| cos(θ (ni , u))|dσ (u).

By rotation invariance, the latter integral does not depend on .ni and in simply a constant depending on the dimension (for example, let .n = (1, 0, . . . , 0), so that .cos(θ ) = u1 ). We have thus shown that there exists some constant .cn such that for any polytope P we have f .

S n−1

Voln−1 (Pu⊥ (P ))dσ (u) = cn Voln−1 (∂P ).

Since both sides are well-defined for convex bodies and are clearly monotone, and since we have equality for all polytopes, we have equality for bodies as well. Thus, for the same .cn (which is half the integral of .|u1 | over .u ∈ S n−1 ) we get for all convex K that f . Voln−1 (Pu⊥ (K))dσ (u) = cn Voln−1 (∂K). S n−1

To find the value of .cn , we plug in .K = B2n .

u n

216

S. Artstein-Avidan

Looking at the proof above, we see that (along with proving a nice formula for surface area) we computed f .

S n−1

|x1 |dσ (x) =

2 Voln−1 (∂K)

f S n−1

Voln−1 (Pu K)dσ (u) =

2κn−1 . nκn

Since we know what is the asymptotic behavior of .κn we may use it to compute: n−1

2 n + 1 1/2 / 2κn−1 2 π 2 r( n2 + 1) 1 ) = 2/π √ . = ( . = √ n nκn n π 2 r( n−1 + 1) 2 n π n 2 So, the average of .|u1 | on the sphere is of the order .n−1/2 , which is quite small. If one considers the median of .|u1 | instead of average, and up to factor 2 the median of a positive quantity is less than the average, this means that about .1/2 of the measure of √ the whole sphere is concentrated near the hyperplane .x1 = 0, in a strip of width .1/ n. Since the sphere is rotation invariant, this applies to any hyperplane through the origin. Concentration is not just about half the volume, but about the majority of volume. Indeed, one may compute .σ {x ∈ S n−1 : |x1 | < r}, to see how it behaves with respect to r. The above computation shows that this integral will be .1/2 when r is of the order .n−1/2 . One may write out this integral and estimate it. However, to avoid these computations, we can estimate it rather well (say for .r < 1/2) by the method explained in Sect. 4.1.3 to get σ {u : u1 > r} =

.

Voln ((1 − r 2 )1/2 B2n ) 1 2 Voln (A) ≤ = (1 − r 2 )n/2 ≤ e− 2 nr , n n Voln (B2 ) Voln (B2 )

which is not sharp, but suffices for many applications.

4.3.1 Borell’s Lemma Our first example for a concentration type inequality will be a direct application of the Brunn–Minkowski inequality, which implies a useful form of concentration. Here the neighborhood is not of euclidean distance, but captured in the form of the volume belonging to a dilate of the set considered. Borell’s lemma from [17] describes some form concentration of volume in convex bodies in .Rn : if .A ∩ K captures more than half of the volume of K, then the percentage of K that stays outside tA, when .t > 1, decreases exponentially with respect to t as .t → ∞, with a bound that does not depend at all on the body K or the dimension n.

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Theorem 4.55 (Borell’s Lemma) Let K be a convex body in .Rn with volume .Voln (K) = 1, and let A be a closed, convex and centrally symmetric set such that 1 .Voln (K ∩ A) = δ > . Then, for every .t > 1 we have 2 ( Voln (K ∩ (R \ tA)) ≤ δ n

.

1−δ δ

) t+1 2

.

Remark 4.56 The same is true when .Vol(A ∩ K) is replaced by any log-concave probability measure .μ (see Definition 4.5). Such measures serve as a natural habitat for geometric inequalities, as we discussed in the section on functional forms of geometric inequalities, Sect. 4.1.4. Proof We first show that (Rn \ A) ⊇

.

2 t −1 (Rn \ tA) + A. t +1 t +1

If this were not so, we could write some .a ∈ A in the form a=

.

t −1 2 y+ a1 , t +1 t +1

for some .a1 ∈ A and .y ∈ / tA. But then we would have .

t +1 t −1 1 y= a+ (−a1 ) ∈ A, t 2t 2t

because of the convexity and symmetry of A. This means that .y ∈ tA, which is a contradiction. Since K is convex, we have (Rn \ A) ∩ K ⊇

.

) ) t − 1( 2 ( n (R \ tA) ∩ K + A∩K . t +1 t +1

An application of the Brunn–Minkowski inequality (Theorem 4.1.1) yields 2

t−1

1 − δ = Voln ((Rn \ A) ∩ K) ≥ Voln ((Rn \ tA) ∩ K) t+1 Voln (A ∩ K) t+1

.

2

t−1

= Voln ((Rn \ tA) ∩ K) t+1 δ t+1 . This completes the proof.

u n

There is no magic to the number .1/2 appearing in the theorem, and it is a good exercise to check that a similar statement will hold when it is replaced by some other constant, say 1/3.

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S. Artstein-Avidan

To emphasize once again why concentration-type inequalities are helpful, let us show how Borell’s Lemma implies a useful Khinchine-type inequality, which allows to compare the .Lp average of a norm on a convex body (or with respect to a logconcave measure) for different p’s. Proposition 4.57 Let f be a norm on .Rn and let .μ be a log-concave probability measure. Let .1 ≤ p ≤ q. Then (f (f )1/p (f )1/q )1/p q . ≤ ≤c , f p dμ f p dμ f q dμ p where c is a universal constant independent on .f, p, q and n. The idea of the proof, which the readers can easily work out on their own, is to use f ∞ f q . f dμ = qs q−1 μ(f > s)ds Rn

0

and Borell’s inequality for the set f A = {x : f (x) ≤ 3( f p dμ)1/p },

.

which by Markov’s inequality has measure at least .2/3. Splitting the integral for q ||f ||q to two parts, and making some simple estimates, will help the interested reader complete the proof. (For details see [6, Section 1.5.2].)

.

4.3.2 Prékopa–Leindler Implying Gaussian Concentration As mentioned in Sect. 4.1.3, the fact that for a set .A ⊂ Rn with .γn (A) = 1/2, we 2 have that .γn (At ) ≥ 1−e−t /2 follows from comparison to half-spaces which are the isoperimetric extremizers. However, let us demonstrate another proof for the same result (with a slightly worse estimate) which avoids using exact extremizers, as such a method generalizes to other spaces as well. Theorem 4.58 Let .A ⊂ Rn with .γn (A) = 1/2, then for any .t > 0, .γn (At ) ≥ 2 1 − 2e−t /4 . Proof Consider the following three functions: .f (x) = exp(d(A, x)2 /4)e−|x| /2 , −|y|2 /2 and .h(z) = e−|z|2 /2 , and fix .λ = 1/2. These satisfy the .g(y) = 1A (y)e conditions of the Prékopa–Leindler inequality since 2

f (x)1/2 g(y)1/2 = e−|x|

.

≤e

2 /4−|y|2 /4

ed(A,x)

−|x|2 /4−|y|2 /4

e

2 /8

1A (y)

|x−y|2 /8

= e−|x+y|

2 /8

= h((x + y)/2).

4 Dualities, Measure Concentration and Transportation

219

f f f Therefore, . h ≥ ( f g)1/2 andf the same is true whenf both sides are normalized by .(2π )−n/2 . Note that .(2π )−n/2 h = 1 and .(2π )−n/2 g = γn (A) = 1/2, so the inequality implies f .

ed(x,A)

2 /4

dγn ≤ 2.

This type of inequality (if it is given for some probability measure, possibly with other constants) implies a concentration inequality by Markov’s inequality f 1 − γ (Ar ) = γ ({x : d(x, a) ≥ r}) ≤

.

ed(x,A)

2 /4

2 er /4

dx

≤ 2e−r

2 /4

. u n

This completes the proof.

4.3.3 The Cost-Santaló Inequality Let .(X, μ) be a probability space and let .c : X × X → R+ be some a cost function. For any measurable function .ϕ : X → [−∞, ∞], we have already defined (see Definition 4.33) ϕ c (y) = inf (c(x, y) − ϕ(x)) .

.

x

We mention that if .c(x, x) = 0 then .ϕ c ≤ −ϕ, and we note that for every bounded measurable function .ϕ : X → R we have f f eϕ dμ · e−ϕ dμ ≥ 1 . by Hölder’s inequality. The Cost-Santaló inequality is to do with the possibility of reversing this Hölder inequality when .−ϕ is replaced by .ϕ c . Definition 4.59 (Cost-Santaló Inequality) Let .c : X × X → R. We say that (X, μ) satisfies a Cost-Santaló inequality with respect to the cost function c if for any bounded measurable function .ϕ : X → R,

.

f .

c

eϕ dμ ·

f eϕ dμ ≤ 1.

(4.15)

A few remarks are in order. Recall from the discussion the c-transform f f about cc (after Definition 4.33) that .ϕ cc ≥ ϕ. In particular, . eϕ ≤ eϕ . At the same time, c = ϕ ccc . This means that when checking whether Definition 4.59 applies to a .ϕ

220

S. Artstein-Avidan

given probability measure and cost, we need only consider pairs .ϕ, ϕ c where .ϕ is in the c-class. A good example to keep in mind is that of the quadratic cost, .c(x, y) = |x−y|2 /2 on .Rn × Rn . In this case the c-class consists of functions .|x|2 /2 − φ(x) where .φ is convex lower semi continuous, and the c-transform of this function is simply 2 .|y| /2 − Lφ(y). Using this together with the Gaussian measure, a Cost-Santaló inequality is precisely the inequality f .

e−φ(x) dx

f

e−Lφ(y) dy ≤ (2π )n

(where the normalizing factor for the Gaussian measure was moved to the right). This is nothing other than the functional Blaschke–Santaló inequality, which we have encountered in Sect. 4.1.4. We discussed its geometric counterpart n 2 ◦ .Vol(K)Vol(K ) ≤ Vol(B ) in Sect. 4.1.4. It is important to note here that this 2 inequality is not true. As we shall see shortly, the Gaussian measure does satisfy a Cost-Santaló inequality but with respect to the cost .|x − y|2 /4. Still, the above inequality is almost true, in particular it is true when .ϕ is an even function, and it is true if some centroid is the origin. Our current aim is to show two claims. The first is that a certain class of measures, which include Gaussian but are more general, satisfy the Cost-Santaló inequality with the quadratic cost (properly normalized). Second, that satisfying a Cost-Santaló inequality with respect to some cost function, implies a form of concentration where the t-extension of a set is measured with respect to the cost function. When the cost is quadratic, the extension is the standard Euclidean one. Let us start with the latter. We start with the following easy statement. Proposition 4.60 Let .(X, μ) be a probability space. Assume that .μ satisfies the Cost-Santaló inequality (4.15) with respect to some cost function c. Then, for any measurable .A ⊆ X and .t > 0, ({ }) 1 − μ x : inf c(x, y) < t ≤

.

y∈A

1 −t e . μ(A)

Proof Consider the function .ϕ which is a .{0, −∞} indicator of the set A, namely is equal to 0 on the set A and .−∞ outside. (If one insists on real valued .ϕ, let it be .{0, −N} valued and eventually take .N → ∞). In this case ϕ c (y) = inf c(x, y) − ϕ(x) = inf c(x, y).

.

x∈A

f

Clearly . eϕ dμ = μ(A), and so the Cost-Santaló inequality assumption implies f

f .

e

infx∈A c(x,y)

dμ(y) =

c

eϕ dμ ≤ 1/μ(A).

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Using Markov’s inequality we see that μ(y : inf c(x, y) ≥ t) = μ(y : einfx∈A c(x,y) ≥ et )

.

x∈A

≤ e−t

f

einfx∈A c(x,y) dμ(y) ≤ e−t /μ(A), u n

as claimed.

In particular we get that if .μ satisfies the Cost-Santaló inequality with respect to c(x, y) = κ4 d(x, y)2 then for a set A of measure .1/2

.

μ(At ) = μ(y : inf c(x, y) < κt 2 /2) ≥ 1 − 2e−κt

.

2 /4

x∈A

.

As mentioned above, in our model example of Gaussian space, the Cost-Santaló inequality with respect to .|x −y|2 /4 is satisfied. This is due to the Prékopa–Leindler inequality, similarly to the way it was used above to show Gaussian concentration, and we next give a proof of this in a slightly more general setting. Theorem 4.61 Consider .X = Rn and the measure .μ with density .dμ = e−U where U is a twice differentiable convex function satisfying .∇ 2 U ≥ κI for some .κ > 0, or, more generally, which satisfies U (x) + U (y) − 2U (

.

x+y κ ) ≥ |x − y|2 . 2 4

Then .μ satisfies the Cost-Santaló inequality with respect to the cost function c(x, y) = κ4 |x − y|2 .

.

Let us prove this in the more general setting of a cost function. Theorem 4.62 Consider .X = Rn with some cost function .c : X × X → R, and the measure .μ with density .dμ = e−U and assume U satisfies U (x) + U (y) − 2U (

.

x+y ) ≥ c(x, y). 2

Then .μ satisfies the Cost-Santaló inequality with respect to the cost function c. Proof We show that the conditions in the Prékopa–Leindler inequality hold for .λ = 1/2 and the triplet of functions f = eϕ

.

c −U

, g = eϕ−U

and h = e−U .

Indeed, this amounts to showing U (x) + U (y) − ϕ c (x) − ϕ(y) ≥ 2U ((x + y)/2),

.

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S. Artstein-Avidan

and the assumptions on U imply it suffices to show c(x, y) ≥ ϕ c (x) + ϕ(y)

.

which is immediate from the definition of the cost transform. The inequality which the Prékopa–Leindler theorem implies is precisely the desired inequality, completing the proof. u n

4.3.4 The Weak Cost-Santaló Inequality It turns out that one can weaken the inequality and still obtain a similar, but not identical, form of concentration. Definition 4.63 (Weak Cost-Santaló Inequality) We say that .(X, μ) satisfies a Weak Cost-Santaló inequality with respect to the cost function c if for any bounded measurable function .ϕ : X → R, f f c . (4.16) eϕ dμ · e ϕdμ ≤ 1. f f Note that by Jensen’s inequality . eϕ dμ ≥ e ϕdμ , so that the new condition is indeed weaker (namely, any measure satisfying the Cost-Santaló inequality will automatically satisfy the Weak Cost-Santaló inequality). Nevertheless, satisfying a Weak Cost-Santaló inequality also implies a form of concentration, as the following claim indicates.

Proposition 4.64 Let .(X, μ) be a probability space. Assume that .μ satisfies the Weak Cost-Santaló inequality (4.16) with respect to some cost function .c ≥ 0. Then, for any measurable .A ⊆ X and .t > 0, ({ }) 1 − μ x : inf c(x, y) < t ≤ e−μ(A)t .

.

y∈A

Proof We shall plug into the inequality the function .ϕ which is 0 on A and .−t on X \ A. Note that if .infy∈A c(x, y) < t then .ϕ c (x) = infy∈X (c(x, y) − ϕ(y)) < t. On the other hand, if .infy∈A c(x, y) ≥ t then .ϕ c (x) = infy∈X (c(x, y) − ϕ(y)) ≥ t as this inequality is true both for .y ∈ A and .y /∈ A (one of them due to the cost, and the other due to .ϕ being t outside A). Therefore .μ(x : infy∈A c(x, y) ≥ t) = μ (x : ϕ c (x)f ≥ t). Clearly . ϕdμ = −tμ(X \ A) = −(1 − μ(A))t and by applying (4.16) to the function .ϕ we get that

.

f .

c

eϕ dμ ≤ et (1−μ(A)) .

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223

Using Markov’s inequality we write ( ) ( ) c μ(x : inf c(x, y) ≥ t) = μ x : ϕ c (x) ≥ t = μ x : eϕ (x) ≥ et

.

y∈A

≤ e−t

f

eϕ dμ ≤ e−t et (1−μ(A)) , c

u n

as claimed.

It turns out that the Weak Cost-Santaló inequality is completely equivalent to an inequality between cost and entropy. This type of inequality concerns, again, a fixed cost and some base measure .μ, and relates the Wasserstein distance from .μ of any .ν which is absolutely continuous with respect to .μ with the relative entropy of .ν with respect to .μ (we define these notions below). To approach this topic we start by defining entropy and relative entropy, and showing this connection. We will get more acquainted with inequalities about entropy and concentration in the section after the next one.

4.3.5 Entropy Definition 4.65 (Entropy) Let (X, μ) be a probability space. For every nonnegative integrable function f : X → R, the entropy of f with respect to μ is the quantity f .

f

Entμ (f ) =

f ln f dμ − X

f dμ · ln

(f

X

) f dμ ∈ [0, +∞].

X

Note that Entμ (f ) ≥ 0 by Jensen’s inequality for the convex function x ln x and that entropy is homogeneous of degree 1, that is, Entμ (λf ) = λ Entμ (f ) for all λ > 0. We mention that in the literature sometimes the sign of entropy is reversed. Also, occasionally the reference measure is Lebesgue and not a probability measure. dν Definition 4.66 (Relative Entropy) In the case where f = dμ for a probability measure ν, the entropy of f is called the relative entropy of μ with respect to ν. We define f .H (ν|μ) := Entμ (f ) = ln f dν. X

224

S. Artstein-Avidan

f

Remark 4.67 We mention that, when dν/dμ), .

lim

p→1+

X

f dμ = 1 (as is in the case where f =

p ln ||f ||p = p−1

f f ln f dμ, X

which gives us yet another interpretation of entropy. Another representation of entropy which will be very useful to us and which is in a sense a duality relation, is given by the next lemma. Lemma 4.68 For every f ≥ 0 defined on a probability space (X, μ), {f .

f g dμ :

Entμ (f ) = sup

}

f

e dμ ≤ 1 , g

and the supremum remains the same if we consider only g which is bounded from above and from below. f Proof Since the relation is homogeneous, we may assume that Eμ (f ) = f dμ = 1. Young’s inequality (which is due to the fact that the function u ln u − u on R+ is the Legendre dual of the function ev ) states that uv ≤ u ln u − u + ev , u ≥ 0, v ∈ R

.

and hence, if g is a function on X such that f

f f g dμ ≤

.

f

{f

f f dμ +

f eg dμ ≤

f ln f dμ

f dμ = 1. Taking the supremum we get f

f g dμ :

sup

eg dμ ≤ 1, we have

f f ln f dμ −

due to the assumption that .

f

} f eg dμ ≤ 1 ≤ f ln f dμ = Entμ (f ).

Finally, plugging in g = f ln f (for which f Young’s inequality becomes an equality), which is allowed since eln f dμ = f dμ = 1, we have that {f .

sup

f f g dμ :

} f eg dμ ≤ 1 ≥ f ln f dμ,

and the equality follows. For the last assertion of the lemma note that if we set fN = min{max{f, N1 }, N } and gN = ln(fN /Eμ (fN )) then gN is bounded from above and f f below, eln gN dμ = 1 for all N , and as N → ∞ we get f gN dμ → Entμ (f ). u n

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225

4.3.6 The Cost-Entropy Inequality With the notion of entropy and relative entropy, we may now discuss yet another inequality which a probability space might satisfy with respect to a given cost, and which in turn is connected with concentration. As we shall shortly see, this inequality is equivalent to one of the two previous ones, although at this point it is not necessarily very easy to see this. Definition 4.69 (Cost-Entropy Inequality) We say that .(X, μ) satisfies a CostEntropy inequality with respect to the cost function c if for any bounded measurable function .ϕ : X → R, Wc (μ, ν) ≤ H (ν|μ).

(4.17)

.

Here we are using the notation .Wc (μ, ν) to represent the total cost .C(μ, ν) (so that the role of the cost function c is more apparent.) The letter W corresponds to Wasserstein. Remark 4.70 This is a good time to take a short break and see what the total cost is for certain special cost functions, and whether we can better understand it. When the cost function is defined using a distance function .c : X×X → R, .c(x, y) = d(x, y), the c-class is the class of all 1-Lip. functions, and the transform is simply .ϕ |→ −ϕ. In particular, using Kantorovich Duality Theorem, f Wc (μ, ν) = sup{

.

f ϕdμ −

ϕdν : ϕ is 1 Lip.}

A special case is where .d(x, y) = d0 (x, y) is the trivial metric, satisfying .d0 (x, x) = 0 and .d0 (x, y) = 1 if .x /= y. In this case 1-Lip. functions are simply functions whose image is in a unit interval, and as f Wc (μ, ν) = sup{

.

f ϕdμ −

ϕdν : ϕ(X) ⊆ [0, 1]}

where we have used the fact that adding a constant to .ϕ does not change the integral difference. Since this is a linear function it is extremized on extremal point of this cone which are precisely function attaining values in .{0, 1}, so we get that Wc (μ, ν) = sup{μ(A) − ν(A) : A ⊆ X} = ||μ − ν||T V ,

.

is the total variation distance between the measures. In particular if .ν 0, ({ }) 1 − μ x : inf c(x, y) < t ≤ e−μ(A)t .

.

y∈A

{ } Proof Fix some A of a given measure, and let .B = x : infy∈A c(x, y) ≥ t . We would like to bound .μ(B) from above. To this end, consider the measure .ν defined as .μ restricted to B, and normalized. So, letting .m = μ(B), we have that −1 μ| . Given some transport plan .π ∈ ||(μ, ν), it clearly has to transport all .ν = m B the measure inside A to the support of .ν, namely to B. The cost .c(x, y) for .x ∈ A and .y ∈ B is at least t, and so Wc (μ, ν) ≥ tμ(A).

.

On the other hand, the entropy of .ν with respect to .μ is particularly easy to compute. Indeed, f .H (ν|μ) = m−1 ln(m−1 )dμ = μ(B)m−1 ln(m−1 ) = − ln μ(B). B

The Cost-Entropy inequality thus gives tμ(A) ≤ − ln μ(B),

.

This completes the proof.

so μ(B) ≤ e−tμ(A) . u n

Proposition 4.72 Let .(X, d) be a metric space, .μ ∈ P (X) and .c : X×X → [0, ∞] a lower semi continuous cost function. Then, .μ satisfies a Cost-Entropy inequality with respect to c if and only if it satisfies the Weak Cost-Santaló inequality with respect to c.

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227

Proof of Proposition 4.72 We will make heavy use of the Kantorovich Duality Theorem, which we recall allows us to represent f Wc (μ, ν) =

.

f ϕdμ +

sup

ψdν.

ϕ,ψ∈Cb (X), admissible

In fact, as we have discussed, the supremum can, instead, run over all .ϕ, ψ = ϕ c which are in the c-class. Assume first that .μ satisfies the Cost-Entropy inequality. To show that the Weak Cost-Santaló inequality holds, consider a bounded continuous function .ϕ : X → R. dν Together with .ϕ c they form an admissible pair. Letting .g = dμ , the Cost-Entropy inequality implies f

f

f ϕ c gdμ ≤ Wc (μ, ν) ≤ Ent(g) =

ϕdμ +

.

g ln gdμ.

c f c As we are free to chose .ν (namely g), we pick .g = eϕ / eϕ dμ. Plugging it into the inequality and canceling some terms we obtain

f ϕ dμ + ln

.

(f

X

) c eϕ dμ ≤ 0.

X

which is the Weak Cost-Santaló inequality. Conversely, if we assume the Weak Cost-Santaló inequality and are given some .ν with .g = dν/dμ then from Lemma 4.68 we know that {f H (ν | μ) = Entμ (g) = sup

f hg dμ :

.

} eh dμ ≤ 1 .

Given an admissible pair .ϕ, ψ for the cost c we recall that .ψ ≤ ϕ c and we let f c .h = ϕ + ϕdμ. Note that by the Weak Cost-Santaló inequality f eh dμ = e

.

f

f ϕdμ

c

eϕ dμ ≤ 1

so we may use h in the equivalent formulation for entropy above, and get f H (ν|μ) ≥

.

f gh =

f ϕ c gdμ +

f ϕdμ

f gdμ =

f ϕ c dν +

ϕdμ

and as this was true for any .ϕ, using Kantorovich Duality Theorem we complete the proof. u n

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4.3.7 Cost-Entropy for the Gaussian Measure As promised in the previous section, we will next show how Brenier’s theorem allows us to prove that the Gaussian measure satisfies a Weak Cost-Santaló inequality for the quadratic cost, in the usual normalization. These theorems are due to Marton and to Talagrand [36, 50]. In particular this show that these inequalities are not equivalent, since the Cost-Santaló inequality does hold for the Gaussian measure and quadratic costs .|x − y|2 /2 but only for .|x − y|2 /4. Proposition 4.73 (Marton-Talagrand) Consider .X = Rn with the standard Gaussian measure .γn , let .c(x, y) = |x − y|2 /2, and let .ν = f (x)dγn (x). Then Wc (γn , ν) ≤ H (ν|γn ).

.

Proof Using Theorem 4.25, we find a convex function .ϕ : Rn → (−∞, ∞] such that .T = ∇ϕ maps .ν onto .γn , and 1 .Wc (ν, γn ) = 2

f Rn

|x − T (x)|2 dγn (x).

The condition of T being a transport map translates to the differential relation dγn (x) = dν(T x) det(DT (x)),

.

i.e.

e−|x|

2 /2

= e−|T x|

2 /2

f (T x) det(DT (x)).

Computing the relative entropy, using that .DT (x) = ∇ 2 ϕ > 0 and the differential relation we get that f H (ν|γn ) =

.

=

f (y) ln(f (y))dγn (y) 1 (2π )n/2

f

f (T x) ln(f (T x))e−|T x|

2 /2

det(DT (x))dx

( ) f 2 2 e|T x| /2−|x| /2 −|x|2 /2 1 dx ln = e det(DT (x)) (2π )n/2 ) f ( |T x|2 |x|2 − − ln(det(DT (x))) dγn (x). = 2 2 f We claim that the term in the last integral is greater than . |T (x) − x|2 /2dγn (x). If we show this, we are done, as this precisely was the expression for .Wc we got using the Brenier map. We are thus left, after rearrangement, with showing that f

f .

ln(det(DT (x)))dγn (x) ≤

dγn (x).

4 Dualities, Measure Concentration and Transportation

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To show this we use that .DT = ∇ 2 ϕ is diagonalizable, and its diagonal entries, in the relevant basis, are positive, namely some .(ti (x))ni=1 . Note that here .ti (x) = ∂2 ϕ(x) ∂xi2

or, letting .T = ∇ϕ = (T1 , . . . , Tn ), we have that .ti (x) = ∂x∂ i Ti (x). So, En .ln(det(DT (x))) = i=1 ln ti (x). We use the inequality .ln(ti (x)) ≤ ti (x) − 1 and get f .

ln(det(DT (x)))dγn (x) ≤

f E n

(ti (x) − 1)dγn (x) =

i=1

n f E

(ti (x) − 1)dγn (x)

i=1

= (2π )−n/2

n f E

e−|x|

2 /2

e−|x|

2 /2

(ti (x) − 1)dx

i=1

= (2π )−n/2

n f E i=1

∂(Ti − Ii ) (x)dx. ∂xi

Here we used the notation .Ii (x) = xi , namely the ith coordinates of the identity operator I . We next use integration by parts, justified as the Gaussian measure decreases rapidly, to get that n f E .

e−|x|

i=1

2 /2

E ∂(Ti − Ii ) (x)dx = − ∂xi n

i=1

=

n f E

f (Ti − Ii )(x)

∂ −|x|2 /2 e dx ∂xi

((T x)i − xi )xi e−|x|

2 /2

dx

i=1

Putting the two together we arrive at f f . ln(det(DT (x)))dγn (x) ≤ dγn (x), as claimed. This completes the proof.

u n

Acknowledgments S. Artstein-Avidan was supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 770127), and in part by Israel Science Foundation (ISF) grant 784/20.

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Chapter 5

Symmetrizations Gabriele Bianchi and Paolo Gronchi

Abstract In this chapter we present the operation of symmetrization of sets and briefly touch on the strictly connected operation of rearrangement of functions. We define the main known symmetrizations and, for each of them, we describe its main properties. We also define an abstract setting for dealing with symmetrizations and we present some characterizations of Minkowski and Steiner symmetrizations and of polarization in terms of their properties. We also present shadow systems, and the topic of convergence of successive symmetrals.

5.1 Introduction The idea of replacing an object by one that retains some of its features but is in some sense more symmetrical has been extremely fruitful over the years. The object may be a set or a function, for example, and the process is then often called symmetrization or rearrangement, respectively. Steiner symmetrization, introduced by Jakob Steiner around 1836 in his attempt to prove the isoperimetric inequality, is still today a potent tool for establishing crucial inequalities in geometry. The influence of such inequalities, which often have analytical versions, extends far beyond geometry to other areas such as analysis and PDEs, and even outside mathematics, to economics and finance. The topic received a huge boost in 1951 from the classic text of Pólya and Szeg˝o [72]. By this time, many other types of symmetrization had been introduced, with similar applications. The general idea is to find a symmetrization that preserves one physical quantity, while not increasing (or sometimes not reducing) another. As well as volume, surface area, and mean width, the book by Pólya and Szeg˝o considers electrostatic capacity, principal frequency (the first eigenvalue of the Laplacian), and torsional rigidity, thereby extending the scope to mathematical physics.

G. Bianchi (O) · P. Gronchi Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Firenze, Italy e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6_5

233

234

G. Bianchi and P. Gronchi

A beautiful and quite recent survey on rearrangements by Talenti [84] contains a comprehensive bibliography, conveniently divided between the main periods of development. In Sect. 5.3 we introduce the abstract setting used to deal with symmetrizations and present in detail all the known ones, describing for each of them the main properties. Section 5.4 is devoted to the notion of shadow systems, a very fruitful way of seeing Steiner symmetrization very effective in proving inequalities. In Sect. 5.5 we present new expressions for Steiner and Minkowski symmetrizations that bring to light the dual relationship between them. We also present some containment relations between different symmetrals. Section 5.6 presents characterizations of Steiner and Minkowski symmetrizations and of polarization. Section 5.7 deals with the convergence of sequences of iterated symmetrizations of a set to a ball. We summarize what is known up to date, which questions remain unanswered and how certain properties can be generalized to different symmetrization processes and to the class of compact sets. The last section, Sect. 5.8, first briefly present the connections between rearrangements and symmetrization and then it describes a recent unifying proof of the Pólya–Szeg˝o inequality valid for many rearrangements.

5.2 Preliminaries Throughout this chapter we use the notation and many of the notions which have been introduced in Chap. 1, but we need to introduce some more notation. Let .D n be the open unit ball in .Rn . If .x, y ∈ Rn we write .[x, y] for the line segment with endpoints x and y. If X is a set, we denote by .conv X, .clo X, and .dim X the convex hull, closure, and dimension (that is, the dimension of the affine hull) of X, respectively. Throughout the paper, the term subspace means a linear subspace. The Grassmannian of k-dimensional subspaces in .Rn is denoted by .G(n, k). If H is a subspace of .Rn , then .X|H is the (orthogonal) projection of X on H and .x|H is the projection of a vector .x ∈ Rn on H . Moreover, .X† denotes the reflection of X in H , i.e., the image of X under the map that takes .x ∈ Rn to .2(x|H ) − x. If .x ∈ Rn \ {o}, then ⊥ is the .(n − 1)-dimensional subspace orthogonal to x. .x If .X† = X, we say X is H-symmetric. If .H = {o}, we instead write .−X = (−1)X for the reflection of X in the origin and o-symmetric for .{o}-symmetric. A set X is called rotationally symmetric with respect to the i-dimensional subspace H if for all .x ∈ H , .X∩(H ⊥ +x) is a union of .(n−i−1)-dimensional spheres, each with center at x. If .dim H = n − 1, then a compact convex set is rotationally symmetric with respect to H if and only if it is H -symmetric. The term H-symmetric spherical cylinder will always mean a set of the form .Dr (x) + s(B n ∩ H ⊥ ) = Dr (x) × s(B n ∩ H ⊥ ), where .s > 0 and .Dr (x) ⊂ H is the ball with .dim D = dim H , center x, and radius .r > 0. Of course, H -symmetric spherical cylinders are rotationally symmetric with respect to both H and .H ⊥ . The phrase translate orthogonal to H means translate by a vector in .H ⊥ .

5 Symmetrizations

235

We write .Hk for k-dimensional Hausdorff measure in .Rn , where .k ∈ {1, . . . , n}. We denote by .Cn , .Mn , and .Ln the class of non-empty compact sets, .Hn -measurable sets, and .Hn -measurable sets of finite .Hn -measure, respectively, in .Rn . Let .Kn be the class of convex bodies, i.e. non-empty compact convex subsets of .Rn , and let .Knn be the class of members of .Kn with interior points. For .K ∈ Kn , .S(K) denotes its surface area, defined in Sect. 1.9. If .K ∈ Knn then .S(K) = Hn−1 (∂K). By .κn we denote the volume .Hn (B n ) of the unit ball in .Rn . The Blaschke addition .K # L of .K, L ∈ Knn is a convex body whose surface area measure is Sn−1 (K # L, ·) = Sn−1 (K, ·) + Sn−1 (L, ·).

.

The existence of this body is a consequence of Minkowski’s existence theorem, Theorem 1.46. The body .K # L is determined up to a translation. Let .H ∈ G(n, i), .i ∈ {0, . . . , n}. If .p ∈ Rn , write .p = (x, y), where .x ∈ H and .y ∈ H ⊥ satisfy .p = x + y. Suppose that .s, t ∈ R and .K, L ∈ Kn . The fiber combination .(s ◦ K) nH (t ◦ L) of K and L relative to H , defined by (s ◦ K) nH (t ◦ L) = {(x, sy + tz) : (x, y) ∈ K, (x, z) ∈ L},

.

(5.1)

was introduced by McMullen [63], who noted that .(s ◦ K) nH (t ◦ L) ∈ Kn , .(s ◦ K) nH (t ◦ L) = sK + tL if .i = 0, and .K nH L = K ∩ L if .i = n. (We have adapted the definition in [63] to suit our purposes.) We recall the Brunn–Minkowski inequality for intrinsic volumes. Theorem 5.1 If K, .L ∈ Kn and .i ∈ {1, . . . , n} then 1

1

1

Vi (K + L) i ≥ Vi (K) i + Vi (L) i

.

(5.2)

When .i = n this is the Brunn–Minkowski inequality, Theorem 3.23. When .i /= n this is a consequence of the fact that .Vi (K) equals, up to a positive multiplicative constant, the mixed volume .V (K[i], B n [n−i]) and that the .1/ i-power of this mixed volume is concave with respect to K, as stated in the Generalized Brunn–Minkowski inequality, Theorem 3.24. When dealing with relationships between sets in .Rn or functions on .Rn , the term essentially means up to a set of .Hn -measure zero. If .A ∈ Mn , ( ) Hn A ∩ (x + rD n ) , .0(A, x) = lim Hn (x + rD n ) r→0+ is the density of A at x, provided the limit exists. Moreover, we define A∗ = {x ∈ Rn : 0(A, x) = 1}.

.

(5.3)

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Elements of .A∗ are called Lebesgue density points, or simply density points, of A. Note that .A∗ = A, essentially, by the Lebesgue density theorem. Given .A ∈ ∗ M(Rn ) and .C ∈ Knn containing o in its interior, let .MC (A) and .M∗C (A) denote, respectively, its upper and lower anisotropic outer Minkowski content with respect to C, i.e., ∗

MC (A) = lim sup ε→0+

.

Hn (A + εC) − Hn (A) , ε

Hn (A + εC) − Hn (A) . M∗C (A) = lim inf ε→0+ ε

(5.4)

We observe that the limits are unchanged if C is replaced by .int C. When .C = B n they are called upper and lower outer Minkowski content. When the two limits coincide we denote them by .MC (A). When .A ∈ Kn these limits coincide and .MC (A) coincides with the surface area of A and with its perimeter. In Chap. 3.3.1 essentially the same concept is considered, where the .lim sup and the .lim inf are denoted as the upper and lower (outer) relative surface area of A with respect to C. Let .M(Rn ) (or .M+ (Rn )) denote the set of real-valued (or non-negative, respectively) measurable functions on .Rn and let .S(Rn ) denote the set of functions f in n n n .M(R ) such that .H ({x : f (x) > t}) < ∞ for .t > ess inf f . By .V(R ), we denote n n the set of functions f in .M+ (R ) such that .H ({x : f (x) > t}) < ∞ for .t > 0. Members of .S(Rn ) have been called symmetrizable and those of .V(Rn ) are often said to vanish at infinity. If .f ∈ M(Rn ), we denote its graph by .Gf and define its subgraph .Kf ⊂ Rn+1 by Kf = {(x, t) ∈ Rn × R : f (x) ≥ t}.

.

(5.5)

5.3 i-Symmetrization: Properties and Examples Let .i ∈ {0, . . . , n − 1} and let .H ∈ G(n, i) be fixed. Let .B ⊂ Cn be a class of nonempty compact sets in .Rn and let .BH denote the subclass of members of .B that are H -symmetric. We call a map .♦ : B → BH an i-symmetrization on .B (with respect to H ). If .K ∈ B, the corresponding set .♦K is called a symmetral. We consider the following properties, where it is assumed that the class .B is appropriate for the properties concerned and that they hold for all .K, L ∈ B. Recall that .K † is the reflection of K in H . 1. (Monotonicity or strict monotonicity) .K ⊂ L ⇒ ♦K ⊂ ♦L (or .♦K ⊂ ♦L and .K /= L ⇒ ♦K /= ♦L, respectively). 2. (F-preserving) .F (♦K) = F (K), where .F : B → [0, ∞) is a set function. In particular, we can take .F = Vj , .j = 1, . . . , n, the j th intrinsic volume, though we generally prefer to write mean width preserving, surface area preserving, and volume (or measure) preserving when .j = 1, .n − 1, and n, respectively.

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237

3. (Idempotent) .♦2 K = ♦(♦K) = ♦K. 4. (Invariance on H-symmetric sets) .K † = K ⇒ ♦K = K. 5. (Invariance on H-symmetric spherical cylinders) If .K = Dr (x) + s(B n ∩ H ⊥ ), where .s > 0 and .Dr (x) ⊂ H is the i-dimensional ball with center x and radius .r > 0, then .♦K = K. 6. (Projection invariance) .(♦K)|H = K|H . 7. (Invariance under translations orthogonal to H of H-symmetric sets) If K is H -symmetric and .y ∈ H ⊥ , then .♦(K + y) = ♦K. In this chapter sometimes we will also consider sets in .Ln , since some of the symmetrizations that we deal with can be defined also in this class. When n .B ⊂ L the property of being monotonic has to be intended up to sets of measure zero: .♦ is monotonic if .K ⊂ L, essentially, implies .♦K ⊂ ♦L, essentially. With the application to symmetrizations of sets in .Ln in mind we introduce another property of i-symmetrizations. 8. (Smoothing) For each .d > 0 and bounded .K ∈ Ln , (♦∗ K) + dB n ⊂ ♦∗ (K + dB n ) ⊂ ♦(K + dB n ),

.

(5.6)

essentially, where, for .A ∈ Ln , .♦∗ A is defined by ♦∗ A = (♦A)∗ .

.

We remark that .(♦∗ K) + dB n = (♦∗ K) + dD n is open (see [12, Lemma 2.1] for a proof). Lemma 4.4 in [12] proves that the definition of smoothing can be rephrased in other ways. Lemma 5.2 The following statements are equivalent. i) .♦ is smoothing (in the sense of (5.6)). ii) For each .d > 0 and bounded .K ∈ Ln , we have (♦∗ K) + dD n ⊂ ♦∗ (K + dD n ).

.

(5.7)

iii) For each .d > 0 and bounded .K ∈ Ln , (5.7) holds essentially. Sometimes when dealing with symmetrizations defined on .B ⊂ Cn or .Kn , we refer to smoothing as satisfying, for .d > 0 and .K ∈ B, (♦K) + dB n ⊂ ♦(K + dB n ).

.

(5.8)

Remark If .♦ is measure preserving and smoothing then .Hn ((♦∗ K) + εB n ) ≤ Hn (♦∗ (K + εB n )) = Hn (K + εB n ) and .Hn (♦∗ K) = Hn (K), for .K ∈ Ln and .ε > 0. This implies .

Hn (K + εB n ) − Hn (K) Hn ((♦∗ K) + εB n ) − Hn (♦∗ K) ≤ , ε ε

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and that .♦ reduces the upper and lower outer Minkowski content (and the perimeter, when K and .♦∗ K are set whose outer and lower Minkowski content coincide with the perimeter). Invariance under translations orthogonal to H of H -symmetric sets, as well as being idempotent, are satisfied by most natural symmetrizations, so in discussing examples we shall only mention these properties when they do not hold. Two special cases are of particular importance: .i = 0 and .i = n−1. If .i = 0, then .H = {o} and 0-symmetrization is the same as the o-symmetrization. One example of 0-symmetrization is central symmetrization, given for .K ∈ Kn by ♦K = AK =

.

1 1 K + (−K). 2 2

(5.9)

The central symmetral .AK differs from the ubiquitous difference body .DK = K + (−K) only by a dilatation factor of .1/2. It is a particular instance of Minkowski symmetrizations that we will define in a moment. Other examples of 0-symmetrizations are the pth central symmetrization, given for .K ∈ Kn and .p ≥ 1 by ( ( ) ) ♦K = Ap K = 2−1/p K +p 2−1/p (−K)

.

(here .+p denotes the general .Lp addition introduced in Sect. 3.4.1) and the Msymmetrization. For its definition and for more on 0-symmetrizations we refer the reader to Gardner et al. [38] and to Bianchi, Gardner, and Gronchi [8, Section 4]. The other case of particular importance is .i = n − 1, to which we will devote more attention.

5.3.1 Steiner Symmetrization The prime example of an .(n − 1)-symmetrization is Steiner symmetrization. If K ∈ Cn , the Steiner symmetral of K with respect to .H ∈ G(n, n − 1) is the set .SH K such that for each line G orthogonal to H and meeting K, the set .G ∩ SH K is a (possibly degenerate) closed line segment with midpoint in H and .H1 -measure equal to that of .G ∩ K. An extension of this definition to Lebesgue measurable subsets of .Rn is possible and is presented below in the more general setting of Schwarz symmetrization. We list some of its properties. .

(a) If .K ∈ Cn , then .SH K ∈ Cn , and if .K ∈ Kn , then .SH K ∈ Kn . The first claim is elementary and we prove the second one. Let .u ∈ Sn−1 be orthogonal to H . There are two functions .f, g : K|H → R such that we can describe .K ∈ Kn as K = {x + tu ∈ Rn : g(x) ≤ t ≤ f (x)},

.

x ∈ K|H.

5 Symmetrizations

239

Since K is convex, the function g is convex and f is concave. We have SH K = {x + tu ∈ Rn : −(f (x) − g(x))/2 ≤ t ≤ (f (x) − g(x))/2},

.

and this shows that .SH K is convex. (b) On .Kn , Steiner symmetrization is strictly monotonic, invariant on H -symmetric sets, volume preserving (by Fubini’s theorem), and projection invariant. (c) On .Kn and for .j ∈ {1, . . . , n − 1}, the j th intrinsic volume .Vj is generally reduced (meaning not increased and not always preserved) by .SH . In particular, Steiner symmetrization generally reduces the surface area. See Sect. 5.4 and Theorem 5.10 for a proof. (d) On .Cn , Steiner symmetrization is monotonic but not strictly monotonic (if .H = {x1 = 0}, then .B n  B n ∪ {(2, 0, . . . , 0), (−2, 0, . . . , 0)} but .SH (B n ) = B n = SH (B n ∪{(2, 0, . . . , 0), (−2, 0, . . . , 0)})) and it is not invariant on H -symmetric sets (same example). (e) On .Kn , .SH is smoothing, in the sense of (5.8). This is a consequence of the inclusion proved in the next theorem. Choosing .L = dB n and using .SH dB n = dB n , (5.10) yields (5.8). Theorem 5.3 If .K, L ∈ Cn , then SH K + SH L ⊂ SH (K + L).

.

(5.10)

Proof Let G be a line orthogonal to H . It is enough to prove (SH K + SH L) ∩ G ⊂ SH (K + L) ∩ G .

.

For .y ∈ H let .Gy = G + y. We can write (SH K + SH L) ∩ G =

||

.

(SH K ∩ Gy ) + (SH L ∩ G−y ) .

y∈H

Since all the segments .(SH K ∩Gy )+(SH L∩G−y ) are contained in G and centered in H , their union equals the largest of its elements, .(SH K ∩ Gy¯ ) + (SH L ∩ G−y¯ ). Its length equals the sum of the lengths of the two segments H1 (SH K ∩ Gy¯ ) + H1 (SH L ∩ G−y¯ ).

.

Now, .SH (K + L) ∩ G is a segment whose length is not smaller than ) ( H1 (K ∩ Gy¯ ) + (L ∩ G−y¯ ) ≥ H1 (K ∩ Gy¯ ) + H1 (L ∩ G−y¯ )

.

= H1 (SH K ∩ Gy¯ ) + H1 (SH L ∩ G−y¯ ) . This concludes the proof.

u n

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Section 5.4 presents many other properties of the Steiner symmetrization. It also contains a proof of (5.10) valid for convex bodies and which uses shadow systems.

5.3.2 Schwarz Symmetrization Let .i ∈ {0, . . . , n − 1} and .K ∈ Cn . The Schwarz symmetral of K with respect to .H ∈ G(n, i) is the set .SH K such that for each .(n − i)-dimensional plane G orthogonal to H and meeting K, the set .G ∩ SH K is a (possibly degenerate) .(n − i)-dimensional closed ball with center in H and .Hn−i -measure equal to that of .G ∩ K. See [37, p. 62] and also [43, p. 178] (where the process is referred to as Schwarz rounding). When .i = n − 1 it coincides with the Steiner symmetrization. It is convenient to use the same notation for Steiner and Schwarz symmetrizations. An extension of this definition to measurable subsets A of .Rn is possible. Let G be a .(n − i)-dimensional plane G orthogonal to H and meeting A. If .G ∩ A is not n−i .H -measurable then .G ∩ SH A = ∅. If .Hn−i (G ∩ A) = ∞ then .G ∩ SH A = G. n−i If .H (G ∩ A) < ∞ then .G ∩ SH A is defined as in the case of compact sets, i.e., it is a (possibly degenerate) .(n − i)-dimensional closed ball with center in H and n−i .H -measure equal to that of .G ∩ A. See [37, p. 62], [83, p. 106] and [2, p. 182]. In the literature on rearrangements of functions this symmetrization is often indicated as the .(n − i, n)-Steiner symmetrization. It is at the heart of the definition of symmetric decreasing rearrangement of a function. Indeed, the symmetric decreasing rearrangement of a function .f ∈ S(Rn ) is the function whose subgraph is the Schwarz symmetrization, with respect to the .xn+1 -axis, of the subgraph of u. We list some of its properties. (a) .SH K is rotationally symmetric with respect to H . (b) If .K ∈ Cn , then .SH K ∈ Cn . (c) On .Kn and on .Cn , Schwarz symmetrization is monotonic, volume preserving (again by Fubini’s theorem), and projection invariant. (d) On .Kn , Schwarz symmetrization is invariant on H -symmetric spherical cylinders, but it is not invariant on H -symmetric sets. (e) On .Knn , Schwarz symmetrization is strictly monotonic, but on .Cn and on .Kn it is not (for instance, when H is the .xn -axis then .SH (B n ∩ {x1 = 0}) = SH (B n ∩ {x1 = 0, x2 = 0})). (f) The Schwarz symmetrization, for .i ∈ {0, . . . , n − 2}, can be viewed as a limit, in the Hausdorff distance, of a sequence of Steiner symmetrizations. This issue will be treated properly in Sect. 5.7 but we anticipate that there exist sequences of hyperplanes .(Hm ) containing H such that, for each .K ∈ Cn , the sequence .(SHm . . . SH2 SH1 K) of iterated Steiner symmetrizations of K converges to .SH K, i.e. .

lim δ(SHm . . . SH2 SH1 K, SH K) = 0.

m→∞

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(g) The existence of the approximating sequence presented in item (f) allows to prove that properties valid for Steiner symmetrization and which are maintained in the passage to the limit, are also valid for Schwarz symmetrization. For instance, it can be used to prove that if .K ∈ Kn then .SH K ∈ Kn . Indeed, if n .(Hm ) is as in item (f), then, for .m ∈ N, .SHm . . . SH2 SH1 K ∈ K , since Steiner symmetrization maintains convexity. Since the limit of a sequence of convex bodies is a convex body, so is .SH K. (h) The same idea can be used to prove that Schwarz symmetrization generally reduces the j th intrinsic volume .Vj for .j ∈ {1, . . . , n − 1}. Moreover, together with the fact that Steiner symmetrization satisfies (5.10) and is monotonic, the same method gives that Theorem 5.3 is valid also for Schwarz symmetrization and that .SH is smoothing on .Cn , in the sense of (5.8). (i) Schwarz symmetrization is smoothing as a map on .Ln . The result is valid for each i, and in particular, for Steiner symmetrization. Theorem 5.4 Let .i ∈ {0, . . . , n − 1}. The map .SH : Ln → LnH is smoothing. Proof Let .A ∈ Ln be bounded and .d > 0. Step 1.

Let .D n−i = D n ∩ H ⊥ . We prove that, for .x ∈ H and .r > 0,

( ( ) ) SH A ∩ (x + H ⊥ ) + rD n−i ⊂ SH (A ∩ (x + H ⊥ )) + rD n−i .

.

(5.11)

The set .(A ∩ (x + H ⊥ )) + rD n−i is .Hn−i -measurable, because it is open in the relative topology on .x + H ⊥ . If .A ∩ (x + H ⊥ ) is not .Hn−i -measurable, the set on the left-hand side in set and the inclusion holds true. ( (5.11) is the empty ) If it is measurable, .Hn−i A ∩ (x + H ⊥ ) < ∞, since A is bounded. Both sets in (5.11) are .(n − i)-dimensional balls in .x + H ⊥ with center in x. The one on the left has radius ( ) 1 1 n−i n−i + d, r1 = Hn−i A ∩ (x + H ⊥ ) /κn−i

.

while the one on the right has radius ( ) 1 1 n−i n−i . /κn−i r2 = Hn−i A ∩ (x + H ⊥ ) + rD n−i

.

The Brunn–Minkowski inequality in .Rn−i implies .r1 ≤ r2 and (5.11). Step 2. For .x ∈ H , denote by .||x the orthogonal projection onto .x + H ⊥ . If L is any set in .Rn , then (L + dD n ) ∩ (x + H ⊥ ) =

||

.

( ) ||x (L ∩ (y + H ⊥ )) + ry D n−i ,

y∈H,|y−x| 5 the problem is still open. i (K) the maximal i-th intrinsic volume of a polytope If we denote by .Mm contained in K with at most m vertices, then we define a similar functional: .

i (K) Mm i

,

Vn (K) n where the power on the volume of K is chosen so that the functional is scaling invariant and convex along shadow systems. However, such a functional is not affine invariant and has no upper bound.

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Remark If a functional F on .Kn is convex along shadow systems and upper bounded then it is also .SL(n) invariant. Indeed, let H be a hyperplane, let .v /= o a vector in .H ⊥ and let .α be a linear function defined on H . We define, for .t ∈ R, the shadow system Kt = conv({x + tα(x|H )v : x ∈ K}).

.

For each t, .Kt is an affine image of K. If a functional F is convex along such a shadow system and bounded from above, then .F (Kt ) has to be constant and this implies that F is invariant with respect to shears. Since shears generate the whole group .SL(n), the statement is proved. If F is also scaling and reflection invariant, then it is affine invariant. Theorem 5.11 implies the following result. Theorem 5.16 For every .K ∈ Knn and .m > n .

i (K) Mm i

Vn (K) n

≥

i (B n ) Mm i

Vn (B n ) n

.

5.4.2 Sylvester’s Functional Instead of taking the maximal volume of a polytope contained in K with at most m vertices, we can focus on its average. Clearly we have to declare how to evaluate the average. The most natural way is to define Sn (K; m) =

.

1 Vn (K)m+1

f

f . . . Vn (conv[x1 , x2 , . . . , xm ]) dx1 . . . dxm , K

K

which is the expected volume of a random polytope from K divided by .Vn (K) (to ensure scaling invariance), where the vertices are selected uniformly and independently in K. If we consider the shadow system .{Kt }t∈[0,2] defined in (5.20), then the volume of .Kt is constant and the functional can be rewritten as Sn (Kt ; m) =

.

1 Vn (K)m+1

f

f m || . . . Vn (conv[ xj + tα(xj |H )v]) dx1 . . . dxm .

K

K

j =1

Therefore, the functional is convex with respect to t. Furthermore, it is continuous with respect to the Hausdorff distance, reflection invariant and affine invariant. As a consequence of Theorem 5.11 we deduce Blaschke-Groemer inequality (see [17] and [42])

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Theorem 5.17 (Blaschke–Groemer Inequality) For every .K ∈ Knn and .m > n Sn (K; m) ≥ Sn (B n ; m) ,

.

where equality holds if and only if K is an ellipsoid. Proof We already sketched a proof of the inequality, but here the discussion of the equality cases is a bit more involved than the one for intrinsic volumes in Theorem 5.12. Indeed, .Sn (K; m) is affine invariant, and so the strict convexity along every shadow system cannot be proved. Blaschke and Groemer based their arguments on the characterization of ellipsoids as the only convex bodies which are affine images of their Steiner symmetrals along every direction. Such a characterization is due to Brunn [23] in an equivalent form: If all midpoints of every family of parallel chords of K are contained in a hyperplane, then K is an ellipsoid. Hence, if K is not an ellipsoid, then we can choose a direction v and m of the midpoints of chords of K parallel to v so that their convex hull is a polytope of positive volume. Following such a polytope along the shadow system (5.20) give rise to a shadow system .{Pt }t∈[0,2] with .Vn (Pt ) strictly convex. A continuity argument leads to the conclusion. u n For .n = 2, .Sn (K; m) is closely related to the so-called Sylvester’s four-point problem, which appeared in the Educational Times of 1864, question 1491: Show that the chance of four points forming the apices of a reentrant quadrilateral is .1/4 if they be taken at random in an indefinite plane, but .1/4 + e2 + x 2 , where e is a finite constant and x a variable quantity, if they be limited by an area of any magnitude and of any form.

Sylvester’s four-point problem captured the attention of many mathematicians who proposed as many different solutions. For a historical overview of the problem and its developments, we refer to the interesting article by Pfiefer [71]. Here, we just mention that Groemer introduced a power .p ≥ 1 of the volume in the functional and observed that this does not affect minimizers. Schöpf [81] extended to all .p > 0. Nowadays, the search of maximizers of 1 .Sn (K; m, p) = Vn (K)m+p

f

f . . . Vn (conv[x1 , x2 , . . . , xm ])p dx1 . . . dxm , K

K

is called Sylvester’s problem and it is solved only in the plane. For .p = 1, Dalla and Larman [34] proved that triangles are maximizers and Giannopoulos [39] proved they are the only ones. Campi et al. [27] showed that parallelograms are maximizers among centrally symmetric figures and Saroglou [77] proved the uniqueness of such maximizers. In higher dimensions the main result is due to Bárány and Buchta [3], who proved that for every .K ∈ Kn there exists .m, ¯ depending on K, such that .Sn (K; m) ≤ Sn (T ; m), for all .m ≥ m, ¯ where T is a simplex.

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Hartzoulaki and Paouris in [45] replaced the volume in Sylvester’s functional with intrinsic volumes and the p-power with an arbitrary increasing function. They proved that f .

f . . . f (Vi (conv[x1 , x2 , . . . , xm ])) dx1 . . . dxm ,

K

K

1 ≤ i ≤ n − 1, is minimal when K is a ball, among bodies of given volume. They further proved that the ball is the only minimizer if f is convex and strictly increasing.

.

5.4.3 Busemann’s Functional A slight modification of Sylvester’s functional appears in the Busemann formula, which expresses the volume of a convex body in terms of the areas of its central sections. Assume .K ∈ Knn contains the origin in its interior: The Busemann intersection formula (see [26]) states that Vn (K)n−1 =

.

f n! Vn−1 (K ∩ u⊥ )n+1 Bn−1 (K ∩ u⊥ ; n − 1, 1) du , 2 Sn−1

where, for .m ≥ n, 1 .Bn (K; m, p) = Vn (K)m+p

f

f . . . Vn (conv[o, x1 , . . . , xm ])p dx1 . . . dxm . K

K

The functional .Bn (K; m, p) is called Busemann’s functional and differs from Sylvester’s in having fixed a point in the origin. It is no more translation invariant, but it clearly remains .GL(n) invariant, continuous with respect to Hausdorff distance, and convex along shadow systems. In [26] Busemann proved that .Bn (K; n, 1) attains its minimum if and only if K is an origin symmetric ellipsoid. Such a result is known as Busemann random simplex inequality. Theorem 5.18 (Busemann Random Simplex Inequality) For every .m ≥ n, .p ≥ 1 and .K ∈ Knn Bn (K; m, p) ≥ Bn (B n ; m, p) ,

.

where equality holds if and only if K is an origin symmetric ellipsoid. The proof of the equality case is similar to the one for Theorem 5.17

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When .m = n Busemann’s functional may be represented in a different way, highlighting a Minkowski sum of segments. Indeed, ) ( n E 1 Vn .Vn (conv[o, x1 , . . . , xn ]) = [o, xi ] . n! i=1

This fact suggests a different extension that we shall meet in next paragraph.

5.4.4 Zonotopes Associated to K Bourgain et al. [21], in connection with some comparisons between norms in the local theory of Banach spaces, considered the functional 1 .I (K; m, p) = Vn (K)m+p

f

)p ( m f E . . . Vn [0, xi ] dx1 . . . dxm ,

K

K

i=1

for .m ≥ n, and the more general version I (K1 , K2 , . . . , Km ; p)

.

=

f

1 (Vn (K1 ) . . . Vn (Km ))

m+p m

)p f (E m . . . Vn [0, xi ] dx1 . . . dxm ,

K1 Km

i=1

where each point is chosen from a different convex body in .Knn . As we have already observed, if we consider the shadow system .{Kt }t∈[0,2] defined in (5.20) and leave every point in K move solidly to the chord on which it lays, then the volume of .Kt is constant and each segment .[o, xi ] is a shadow system. Since the Minkowski sum of shadow systems is a shadow system, .I (Kt ; m, p) is a convex function of t for all .p ≥ 1. Standard arguments provide the continuity with respect to Hausdorff distance and .GL(n) invariance. Hence, Theorem 5.11 recovers a result proved in [21]. Theorem 5.19 (BMMP Zonotope Inequality) For every .m ≥ n, .p ≥ 1 and .K ∈ Knn I (K; m, p) ≥ I (B n ; m, p) ,

.

where equality holds if and only if K is an origin symmetric ellipsoid. The equality condition was proved by Campi and Gronchi [30].

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In fact, Bourgain et al. [21] proved the inequality for all .p ≥ 0 and also for the functional involving more bodies: I (K1 , . . . , Km ; p) ≥ I (B1 , . . . , Bm ; p) ,

.

where .Bi is the ball with the same volume as .Ki centered at the origin. The Minkowski sum of a finite number of segments is called a zonotope. The simplest zonotope is a parallelotope, the sum of n affinely independent segments, that is an affine image of the n-dimensional cube. By increasing the number of segments, zonotopes can approximate the unit ball. A set which is the limit, in the Hausdorff metric, of a sequence of zonotopes is called a zonoid. Zonoids play a basic role in the Brunn–Minkowski theory of convex bodies and appear in different contexts of the mathematical literature. We refer to [80] for an exhaustive review on this topic.

5.4.5 The Lp Busemann–Petty Inequality An easy way to associate a zonoid to a body K with positive volume is to sum all segments joining the origin to a point in K. Due to the link between Minkowski addition and sums of support functions (see Proposition 1.28), we present the zonoid .rK by means of its support function: hrK (x) =

.

2π κn κn+1 Vn (K)

f || dz , K

where the constant in front of the integral is such that .r(λK) = λrK, for all .λ > 0, and .rB n = B n . This body (usually with a different normalization) is known in the literature as the centroid body of K. Centroid bodies were first defined and investigated by Petty [69], but the concept had previously appeared in work of Dupin, in connection with problems for floating bodies (see Gardner [37, Chap. 9] and Schneider [79, Sect. 7.4] for references). When K is an origin symmetric body, the boundary of .rK is, up to a dilatation, the locus of the centroids of all the halves of K obtained by cutting K with hyperplanes through the origin. One of the basic results obtained by Petty [69] is an integral representation of the volume of .rK by means of Busemann’s functional .Bn (K; n, 1). Using the Busemann random simplex inequality, Petty proved the well known Busemann– Petty centroid inequality: For .K ∈ Knn , Vn (rK) ≥ Vn (K) ,

.

where equality holds if and only if K is an origin symmetric ellipsoid.

(5.23)

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Petty [70] proved that the Busemann–Petty centroid inequality implies the Petty projection inequality: Vn (K)n−1 Vn (||◦ K) ≤ κnn ,

.

(5.24)

where equality holds if and only if K is an ellipsoid. Here, .||◦ K is the polar projection body of K, i.e., the polar body of the projection body .||K of K, that can be defined by h||K (x) = Vn−1 (K|x ⊥ ) .

.

A short way to show that (5.23) implies (5.24) was obtained by Lutwak in [54]. Zhang [91] proved a reverse form of (5.24), known as Zhang projection inequality: ( ) 1 2n n−1 , .Vn (K) Vn (||◦ K) ≥ n n n with equality if and only if K is a simplex. The .Lp extension of the classical Brunn–Minkowski theory for convex bodies was initiated by Lutwak [55], in which the idea of Firey [35] of the p-Minkowski addition for sets is widely developed. As already seen in Sect. 3.4, if .p ≥ 1 and K, L are convex bodies containing the origin in their interior, the p-sum of K and L is the convex body .K +p L defined by hK+p L (x)p = hK (x)p + hL (x)p .

.

Bianchini and Colesanti [14] observed that the p-sum of shadow systems is again a shadow system, since the projection of a p-sum is the p-sum of the projections. Notice that the p-sum is the Minkowski sum for .p = 1 and tends to the convex hull as p tends to infinity. Taking into account the p-sum of segments we can define f κ2 κp−1 κn p ||p dz , (5.25) .hrp K (x) = κn+p Vn (K) K where the constant is so that .rp (λK) = λrp K, for all .λ > 0, and .rp B n = B n . The body .rp K is known as the .Lp centroid body of K. For .p = 1, .r1 K is the centroid body of K, while for .p = 2, the body defined by (5.25) is also well known. Indeed, up to a constant, .r2 K is the ellipsoid of inertia (or Legendre ellipsoid) of K, i.e., the ellipsoid having the same moments of inertia as K about every axis. Many results concerning this body, which is fundamental in classical mechanics, can be found in the literature (see, e.g., Milman and Pajor [68] and Lindenstrauss and Milman [50] for references). In 1918 Blaschke [18] proved that, for .n = 3, Vn (r2 K) ≥ Vn (K) ,

.

(5.26)

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where equality holds if and only if K is an origin symmetric ellipsoid. In 1937 this result was extended by John [46] in all dimensions; other proofs were given by Petty [69] and by Lutwak, Yang, and Zhang [56]. Inequalities (5.23) and (5.26) are special instances of the more recent .Lp Busemann–Petty centroid inequality. Theorem 5.20 (.Lp Busemann–Petty Centroid Inequality) For .p ≥ 1 and .K ∈ Knn Vn (rp K) ≥ Vn (K) ,

.

where equality holds if and only if K is an origin symmetric ellipsoid. Theorem 5.20 was first proved by Lutwak et al. [57], while Campi and Gronchi [28] presented a proof based on shadow systems. If we consider the shadow system .{Kt }t∈[0,2] defined in (5.20), each segment joining o to a moving point .x ∈ K is a shadow system, and their p-sum (better their p-integral) .rp Kt is a shadow system too. Therefore, not only the volume of .rp Kt is convex but also its intrinsic volumes, its diameter, its Sylvester’s functional, etc., are convex. Focusing on Steiner symmetrization, this implies .F (SH K) ≤ F (K) for many functionals F . In addition to this, the use of shadow systems allows to recover arguments scattered over the years and in the literature to the search of maximizers (when the functional is bounded from above). Such arguments are confined, at least until now, to the plane (in the symmetric or non-symmetric case) or to the case of polyhedra with few vertices or to zonotopes. See, for example, Campi and Gronchi [29].

5.4.6 The Blaschke–Santaló Inequality As in Chap. 1, the polar body .K ◦ of .K ∈ Knn is the convex body defined by K ◦ = {x ∈ Rn | ≤ 1, ∀y ∈ K} .

.

Notice that the polar body of K strongly depends on the location of the origin. If K is an origin-symmetric convex body, then the product Vn (K)Vn (K ◦ )

.

is called the volume product of K, and it is invariant under linear transformations. For a general convex body K, the volume product is defined as the minimum, 1 for .x ∈ K, of .Vn (K)Vn ((K − x)◦ ). Aleksandrov [1] proved that .Vn ((K − x)◦ )− n is a strictly concave function of x. The unique point .s(K) where this minimum is

5 Symmetrizations

263

attained is called the Santaló point of K and is characterized by the fact that .(K −z)◦ has its centroid at the origin if and only if .z = s(K). A sharp upper bound for the volume product of a convex body .K ∈ Kn with centroid at the origin is given by the Blaschke–Santaló inequality. Theorem 5.21 (Blaschke–Santaló Inequality) For .K ∈ Knn , Vn (K)Vn (K ◦ ) ≤ κn2 ,

.

where equality holds if and only if K is an ellipsoid centered at the origin. It was proved by Blaschke [15] for .n ≤ 3 and by Santaló for all n. A sharpening of this inequality was proved by Meyer and Pajor [65]. A different proof of the Blaschke–Santaló inequality relies on the following result. Theorem 5.22 If .Kt , .t ∈ [0, 1], is a shadow system of convex bodies in .Knn , then ◦ −1 is a convex function of t. .Vn (Kt ) This was proved by Campi and Gronchi [31] for centrally symmetric bodies and by Meyer and Reisner [66] in full generality. We sketch here a proof in the symmetric case. One of the main ingredients in the proof of Theorem 5.22 is a consequence of the Borell–Brascamp–Lieb inequality, which deals with p-means of functions and their integrals. It can be interpreted as an inverse Hölder inequality, and its links with other well-known inequalities are widely described in the survey article by Gardner [36]. Theorem 5.23 (Borell–Brascamp–Lieb Inequality) If .0 < λ < 1, .−1/n ≤ p ≤ ∞, and f , g, h are non-negative integrable functions on .Rn satisfying h((1 − λ)x + λy) ≥ [(1 − λ)f (x)p + λg(y)p ]1/p ,

.

for all x, .y ∈ Rn , then [

f .

Rn

h(x) dx ≥ (1 − λ)

(f

) Rn

f (x) dx

p np+1

(f +λ

Rn

) g(x) dx

p np+1

] np+1 p .

We shall use a corollary that expresses the concavity of an integral in terms of that of the integrand and the dimension of the space of integration. Let us recall the definition of p-concave function. Definition 5.24 Let .p /= 0. A non-negative function f on .Rn is called p-concave on a convex set L if ] [ p p 1/p .f ((1 − λ)x + λy) ≥ (1 − λ)f (x) + λf (y) for all x, .y ∈ L and .0 < λ < 1.

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Note that if .p < 0, then f is p-concave if and only if .f p is convex. The above definition can be extended to the case .p = 0 by continuity. Corollary 5.25 Let .F (x, y) be a non-negative p-concave function on .Rn × Rm , m .p ≥ −1/n. If, for every y in .R , the integral f .

Rn

F (x, y) dx

p exists, then it is a . np+1 -concave function of y.

Proof Take .y0 , .y1 ∈ Rm and fix .λ ∈ (0, 1). Let .yλ = (1 − λ)y0 + λy1 , and f (x) = F (x, y0 ) , g(x) = F (x, y1 ) , h(x) = F (x, yλ ) .

.

For every .x0 , .x1 ∈ Rn , we have that ( )1/p h((1 − λ)x0 + λx1 ) = F ((1 − λ)x0 + λx1 , yλ ) ≥ (1 − λ)f p (x0 ) + λg p (x1 ) ,

.

where we used the p-concavity of F . The Borell–Brascamp–Lieb inequality now gives the desired conclusion.

u n

We are now ready to prove Theorem 5.22. Proof of Theorem 5.22 Let .{Kt }t∈[0,1] be a shadow system along the direction v, with speed function .α on .K0 , and originated by the .(n + 1)-dimensional convex ˜ such that body .K, Kt = conv{x + α(x)t v : x ∈ K0 }

.

⊥ . can be thought of as the projection along the direction .en+1 − tv of .K˜ onto .en+1 The support functions .hKt , .t ∈ [0, 1], and that of .K˜ are clearly related. Precisely, ⊥ , we have for .u ∈ en+1

hKt (u) = hK˜ (u + ten+1 ) .

.

(5.27)

We know that Vn (Kt◦ ) =

.

f 1 h−n (z) dz . n Sn−1 Kt

Let .D n−1 = {x / ∈ v ⊥ : |x| ≤ 1}; thus .Sn−1 + = {z ∈ Sn−1 : ≥ 0} is the graph of the function . 1 − |x|2 , .x ∈ D n−1 . Consequently, f .

Sn−1

h−n Kt (z) dz = 2

f D n−1

/ h−n (x + 1 − |x|2 v) Kt / dx , 1 − |x|2

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where we took into account that .Kt is origin-symmetric. By (5.27), hKt (x +

/

.

1 − |x|2 v) = hK˜ (x +

/

/ 1 − |x|2 v + t 1 − |x|2 en+1 ) .

Therefore, we obtain that 2 ◦ .Vn (Kt ) = n

( / ) x/ 1 − |x|2 + v + ten+1 h−n ˜

f

K

D n−1

(1 − |x|2 )

n+1 2

dx ,

where we used also the homogeneity / of the support function. By the change of variable .y = x/ 1 − |x|2 in the latter integral (with Jacobian 2 −(n+1)/2 ), we conclude that .(1 − |x| ) Vn (Kt◦ ) =

.

f 2 h−n (y + v + ten+1 ) dy . n Rn−1 K˜

Since the function .hK˜ is convex in .Rn+1 , by Corollary 5.25, we infer that .Vn (Kt◦ ) is p-concave, with respect to t, with .p = (−1/n)/(1 − (n − 1)/n) = −1. u n In [58] Lutwak and Zhang dealt with the functional Gp (K) = Vn (rp◦ K)Vn (K) ,

.

(5.28)

where .rp◦ K is the polar of the .Lp centroid body of K. Using Steiner symmetrization they proved the so-called .Lp Blaschke–Santaló inequality. Theorem 5.26 (.Lp Blaschke–Santaló Inequality) For all .p ≥ 1 and .K ∈ Knn Vn (rp◦ K)Vn (K) ≤ Vn (rp◦ B n )Vn (B n ),

.

and equality holds if and only if K is an ellipsoid centered at the origin. The name of this inequality comes from the fact that it implies the Blaschke– Santaló inequality for centrally symmetric bodies as p tends to infinity. One of the main questions still open in convex geometry is the problem of finding a sharp lower bound for the volume product of a convex body (see the survey article [54]). It was conjectured by Mahler [60] that the minimum of the volume product is attained when K is a simplex, that is Vn (K)Vn (K ◦ ) ≥

.

(n + 1)n+1 . (n!)2

(5.29)

In 1939 Mahler [61] proved the conjecture in the plane and in 1991 Meyer [64] showed that equality holds only for triangles.

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For centrally symmetric convex bodies the inequality Vn (K)Vn (K ◦ ) ≥

.

4d d!

(5.30)

is a conjecture as well, where the value on the right-hand side is the volume product of a parallelotope. It was proved in the plane by Mahler [61] and Reisner [74] characterized parallelograms as the only minimizers. Saint Raymond [76] showed that in higher dimension there are convex bodies, other than parallelotopes and their polars, giving equality in (5.30). He also proved that the conjecture holds true in all dimensions for the affine images of convex sets symmetric with respect to the coordinate hyperplanes (called unconditional bodies). Barthe and Fradelizi [4] generalized to all bodies whose hyperplanes of symmetries have a one-point intersection. Inequality (5.30) was proved by Reisner [73, 74] for all zonoids. Different proofs were presented by Gordon et al. [40] and by Campi and Gronchi [32] using shadow systems. Bourgain and Milman [22] proved that there exists a constant c, not depending on the dimension, such that Vn (K)Vn (K ◦ ) ≥ cn κn2 .

.

In [31] Campi and Gronchi dealt with the lower bound of (5.28). It is easy to check that .Gp is continuous, .(−1)-concave along shadow systems and .GL(n) invariant. Besides, .Gp (K) tends to zero as K moves away from the origin. If .cK denotes the centroid of K, Campi and Gronchi proved that in the two-dimensional case the functionals .

min Gp (K − x) ,

x∈K

maxn Gp (K − x) , Gp (K − cK )

x∈R

are minimized by triangles (or parallelograms, in the symmetric case).

5.4.7 The Affine Quermassintegrals Inequality The intrinsic volumes (or their close relatives, the quermassintegrals) of a convex body .K ∈ Knn are not invariant under volume preserving affine transformations. An affine invariant version was defined by Lutwak in [52] by replacing the .L1 norm in Kubota’s formula (5.22) by the .L−n norm: κn .0i (K) = κi

(f G(n,i)

Vi−n (K|E) dE

)− 1

n

The affine invariance of .0i (K) was shown by Grinberg [41].

.

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Conjectured by Lutwak [53] in 1988, the affine quermassintegrals inequality has been proved only recently by Milman and Yehudayoff [67] and it compares the affine quermassintegral of a convex body K with that of .BK , a ball with the same volume as K. Theorem 5.27 (Affine Quermassintegrals Inequality) For every .K ∈ Knn and .i = 1, 2, . . . , n − 1, 0i (BK ) ≤ 0i (K) ,

.

(5.31)

with equality for a given i if and only if K is an ellipsoid. For origin-symmetric convex bodies, .0−n 1 (K) is proportional to the volume of the polar body .K ◦ , and so the case .i = 1 of (5.31) amounts to the Blaschke—Santaló inequality. For general convex bodies the Blaschke–Santaló inequality is stronger. On the other extreme, .0−n n−1 (K) is proportional to the volume of the polar projection body .||◦ K and the case .i = n − 1 of (5.31) is equivalent to Petty projection inequality. The elegant proof of Milman and Yehudayoff goes through shadow systems to arrive to the familiar looking inequality 0i (SH K) ≤ 0i (K) ,

.

but they need the definition of a new body, the Projection Rolodex of K, (a subset of a vector bundle over a lower-dimensional Grassmannian) and appropriate measures on Grassmannians. Last but not least, they solved the equality cases with a ten pages proof. In short: The arguments they used are surely too involved to fit into these notes.

5.5 Duality Between Steiner and Minkowski Symmetrals The following theorem shows a duality between Steiner (or, more generally, fiber) and Minkowski symmetrization. We recall that .FH K = AK = MH K if .i = 0 and that, if .i = n − 1, Steiner and fiber coincide. It is proved in [8] and the proof is taken from there. Theorem 5.28 Let .H ∈ G(n, i), .i ∈ {0, . . . , n − 1}, and for .K ∈ Kn and .y ∈ H ⊥ , let Ky = K + y

and

.

Ky† = (Ky )† = K † − y.

(5.32)

Then for .K ∈ Kn , we have FH K =

||

.

y∈H ⊥

(Ky ∩ Ky† )

(5.33)

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G. Bianchi and P. Gronchi

and MH K =

n

.

conv(Ky ∪ Ky† ).

(5.34)

y∈H ⊥

Proof Let .z ∈ FH K. Then, using (5.14), we have .z = ((x + a)/2) + (x − b)/2, where .x ∈ H , .a, b ∈ H ⊥ , and .x + a, x + b ∈ K. Let .y = −(a + b)/2. Then † † .z = (x + a) + y ∈ K + y and .z = (x − b) − y ∈ K − y. Therefore .z ∈ Ky ∩ Ky , so .FH K is contained in the right-hand side of (5.33). For the reverse inclusion, note first that .Ky ∩ Ky† is H -symmetric. From the invariance of .FH on H -symmetric sets and the fact that .FH is monotonic and invariant on translations orthogonal to H of H -symmetric sets, we obtain Ky ∩ Ky† = FH (Ky ∩ Ky† ) ⊂ FH Ky = FH K

.

for all .y ∈ H ⊥ . This proves (5.33). To prove (5.34), let .y ∈ H ⊥ and let .Qy = conv(Ky ∪ Ky† ). Then since .Ky and † .Ky are contained in .Qy and the latter is convex, M H K = M H Ky =

.

1 1 Ky + Ky† ⊂ Qy , 2 2

so .MH K ⊂ ∩y∈H ⊥ Qy . To prove the reverse containment in (5.34), observe that if v ∈ Sn−1 , then by (5.32) and .hK±y (v) = hK (v) ± , we obtain

.

h∩y∈H ⊥ Qy (v) ≤ min hQy (v) = min max{hKy (v), hK † (v)}

.

y∈H ⊥

y∈H ⊥

y

{ } = min max hK (v) + , hK † (v) − y∈H ⊥

=

1 1 hK (v) + hK † (v) = hMH K (v), 2 2

(5.35)

as required, where the first equality in (5.35) results from observing that the minimum occurs when the two expressions are equal, i.e., when . = (hK † (v) − hK (v))/2. u n Corollary 5.29 ([8]) If .H ∈ G(n, i), .i ∈ {0, . . . , n − 1}, and .K ∈ Kn , then the fiber symmetral .FH K (and therefore the Steiner symmetral .SH K, if .i = n−1) is the union of all H -symmetric compact convex sets such that some translate orthogonal to H is contained in K, and the Minkowski symmetral .MH K is the intersection of all H -symmetric compact convex sets such that some translate orthogonal to H contains K. Proof Let us prove the claim regarding .FH K. For each .y ∈ H ⊥ , the set .Ky ∩ Ky† is H -symmetric and its translation by .−y is contained in K. On the other hand, if .M ∈

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269

† Kn is H -symmetric and .M + w ⊂ K for some .w ∈ H ⊥ , then .M ⊂ K−w ∩ K−w . u n The proof of the other claim is analogous.

5.5.1 Inclusions Between General Symmetrizations and Known Ones The containment results presented here have a crucial role in proving many of the results for general i-symmetrizations described in the next sections. All of them are proved in [8], which also contains a critical discussion of the necessity of each hypothesis. The first one is a corollary of Theorem 5.28. Corollary 5.30 Let .H ∈ G(n, i), .i ∈ {0, . . . , n − 1}, and let .B = Kn or .B = Knn . Suppose that .♦ : B → BH is monotonic, invariant on H -symmetric sets, and invariant under translations orthogonal to H of H -symmetric sets. Then FH K ⊂ ♦K ⊂ MH K

.

(5.36)

for all .K ∈ B. Proof Let .K ∈ B and let .y ∈ H ⊥ . The set .Ky ∩ Ky† is H -symmetric. Hence, using the monotonicity and invariance property of .♦, we have Ky ∩ Ky† = ♦(Ky ∩ Ky† ) ⊂ ♦Ky = ♦K.

.

(5.37)

This formula and (5.33) prove the inclusion on the left. The set .conv(Ky ∪ Ky† ) is H -symmetric and .K ⊂ conv(Ky ∪ Ky† ) − y. Again, using the monotonicity and invariance property of .♦, we have ( ) ♦K ⊂ ♦ conv(Ky ∪ Ky† ) − y = ♦ conv(Ky ∪ Ky† ) = conv(Ky ∪ Ky† ).

.

This formula and (5.34) prove the inclusion on the right.

(5.38) u n

We present, without proof, two more results. The first one proves for .i = n − 1 the same inclusions of Corollary 5.30 under assumptions of a different nature. Theorem 5.31 Let .H ∈ G(n, n − 1), let .B = Kn or .B = Knn , and let .F : B → [0, ∞) be a strictly increasing set function invariant under translations orthogonal to H of H -symmetric sets. If .♦ : B → BH is monotonic, F -preserving, and either invariant on H -symmetric spherical cylinders or projection invariant, then .♦ is invariant under translations orthogonal to H of H -symmetric sets and SH K ⊂ ♦K ⊂ MH K

.

for all .K ∈ B.

(5.39)

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G. Bianchi and P. Gronchi

The second result corresponds to Corollary 5.30 for symmetrizations whose symmetral is rotationally symmetric with respect to H , like .SH and .M H . Theorem 5.32 Let .H ∈ G(n, i), .i ∈ {1, . . . , n − 1}, let .B = Kn or .B = Knn , and let .♦ : B → BH be invariant on H -symmetric spherical cylinders and invariant under translations orthogonal to H of H -symmetric sets. Consider the expression IH K ⊂ ♦K ⊂ OH K.

.

(5.40)

The left-hand inclusion holds for all .K ∈ B if, in addition to the assumptions stated before (5.40), .B = Knn and .♦ is monotonic. The right-hand inclusion holds for all .K ∈ B if, in addition to the assumptions stated before (5.40), .♦ is strictly monotonic and idempotent.

5.6 Characterization of Minkowski and Steiner Symmetrizations and of Polarization In this section we present some characterizations proved in [8] and in [9]. We refer to these papers for a critical discussion of the necessity of each hypothesis.

5.6.1 Characterizations of Minkowski Symmetrization Theorem 5.33 Let H ∈ G(n, i), i ∈ {0, . . . , n − 1}, and let B = Kn or B = Knn . Suppose that ♦ : B → BH is monotonic. Assume in addition either that (i) i = n−1 and ♦ is mean width preserving and either invariant on H -symmetric spherical cylinders or projection invariant, or that (ii) i ∈ {1, . . . , n − 1} and ♦ is mean width preserving, invariant on H -symmetric sets, and invariant under translations orthogonal to H of H -symmetric sets, or that (iii) i = 0 and ♦ is invariant on o-symmetric sets and invariant under translations of o-symmetric sets. Then ♦ is Minkowski symmetrization with respect to H . Proof Let K ∈ B. Let us prove part (i). Theorem 5.31 with F = V1 proves ♦K ⊂ MH K.

.

Since both ♦ and MH preserve mean width we have V1 (♦K) = V1 (MH K) = V1 (K). Since V1 is strictly monotonic the previous inclusion is an equality. Let us prove part (ii). Corollary 5.30 proves again ♦K ⊂ MH K. The conclusion follows as before.

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Part (iii) is an immediate consequence of Corollary 5.30, since when i = 0 we have FH = MH . n u The next result exploits the linearity of MH with respect to Minkowski addition. Theorem 5.34 Let H ∈ G(n, i), i ∈ {0, . . . , n−1}. If ♦ : Kn → KnH is monotonic, invariant on H -symmetric sets, and linear (i.e., ♦(K + L) = ♦K + ♦L for all K, L ∈ Kn ), then ♦ is Minkowski symmetrization with respect to H . For the proof we refer to [8].

5.6.2 Characterizations of Steiner Symmetrization Here we present one characterization valid for .(n − 1)-symmetrizations defined in Cn and one valid for .(n − 1)-symmetrizations defined in .Knn .

.

Theorem 5.35 (i) Let .H ∈ G(n, n − 1). Suppose that .♦ : Cn → CnH is an .(n − 1)-symmetrization that is monotonic, volume preserving, and invariant on H -symmetric spherical cylinders. Then

.

( ( ) ) Hn−1 (♦K) ∩ (H ⊥ + x) = Hn−1 K ∩ (H ⊥ + x)

.

(5.41)

for all .K ∈ Cn and .Hn−1 -almost all .x ∈ H . ⊥ .(ii) Suppose that in addition to the assumptions in .(i), .(♦K) ∩ (x + H ) is n−1 a line segment for .H -almost all .x ∈ H . Then .♦ is essentially Steiner symmetrization on .Cn , in the sense that for all .K ∈ Cn , .(♦K)∩G = (SH K)∩G for .Hn−1 -almost all lines G orthogonal to H . Theorem 5.36 Let .H ∈ G(n, n − 1) and let .♦ : Knn → (Knn )H be an .(n − 1)-symmetrization. If .♦ is monotonic, volume preserving, and either invariant on H -symmetric spherical cylinders or projection invariant, then .♦ is Steiner symmetrization with respect to H . We refer to [8] for the detailed proofs. Here we only explain the main ideas. To prove Theorem 5.35 (i) we argue as follows. For .x ∈ H let .Dr (x) be the ⊥ ⊂ .(n−1)-dimensional ball in H with center x and radius r. Let s be such that .K|H ⊥ n [−s, s] (we identify .H with .R so that .[−s, s] is a shorthand for .s(B ∩ H ⊥ )). The set .Dr (x) + [−s, s] is an H -symmetric spherical cylinder which contains .K ∩ (Dr (x) + H ⊥ ). For .L ∈ Cn let ( ) mr,L (x) = Hn L ∩ (Dr (x) + [−s, s]) .

.

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G. Bianchi and P. Gronchi

The three assumption on .♦ in part (i) imply mr,♦K ≥ mr,K .

.

(5.42)

Indeed, monotonicity and invariance on H -symmetric spherical cylinders imply ( ) ♦ K ∩ (Dr (x) + [−s, s]) ⊂ (♦K) ∩ ♦(Dr (x) + [−s, s]) .

= (♦K) ∩ (Dr (x) + [−s, s]).

From this inclusion, using the fact that .♦ is volume preserving, we obtain ) ) ( ( mr,♦K (x) = Hn (♦K) ∩ (Dr (x) + [−s, s]) ≥ Hn ♦(K ∩ (Dr (x) + [−s, s])) ( ) = Hn K ∩ (Dr (x) + [−s, s]) = mr,K (x).

.

This proves (5.42). Dividing both sides in (5.42) by .Hn−1 (Dr (x)) and passing to the limit as .r → 0 we obtain (5.41) with the inequality (left-hand side .≥ righthand side), for .Hn−1 -almost all .x ∈ H . Integrating this inequality over H , using Fubini’s theorem and the volume invariance of .♦, we conclude that (5.41) holds with equality, for .Hn−1 -almost all .x ∈ H . The proof of part Theorem 5.35 (ii) follows directly from (5.41) and the definition of .SH K. Theorem 5.36, under the assumption that .♦ is invariant on H -symmetric spherical cylinders, is a corollary of Theorem 5.35, while under the assumption that .♦ is projection invariant it requires other ideas, which we do not describe here. We conclude this section with a characterization of Schwarz symmetrization from [11]. Theorem 5.37 Let .i ∈ {1, . . . , n − 2}, let .H ∈ G(n, i), and let .♦H be an i-symmetrization on .Knn . Suppose that .♦H is monotonic, volume preserving, rotationally symmetric, and invariant on H -symmetric cylinders. Then .♦H is Schwarz symmetrization with respect to H .

5.6.3 Characterizations of Polarization We consider the four maps .Id, .†, .PH , or .PH† = † ◦ PH , where .Id and .† denote the identity map and reflection in H , respectively. In this section we present two results from [9] which characterize these four maps among maps .♦ : E ⊂ Ln → Ln . Theorem 5.38 Let .H = u⊥ , .u ∈ Sn−1 , be oriented with .u ∈ H + , let .E = Cn or n n .L , and suppose that .♦ : E → L is monotonic, measure preserving, perimeter preserving on convex bodies, and invariant on H -symmetric unions of two disjoint balls. Then .♦ essentially equals .Id, .†, .♦PH , or .♦†PH .

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We say that .♦ is perimeter preserving on convex bodies if, for each .K ∈ Knn , .♦K is a set of finite perimeter such that .S(♦K) = S(K) = 2Vn−1 (K), where S denotes perimeter in the sense of De Giorgi. The condition of invariance on H -symmetric union of two disjoint balls may seem peculiar, but it is much weaker than the natural assumption that .♦ is invariant on all H -symmetric sets. For maps .♦ : Kn → Ln , invariance on H -symmetric unions of two disjoint balls is not available. The next result resorts to a different and rather strong condition; we say that .♦ is convexity preserving away from H if .♦K is essentially convex (that is, n n .♦K coincides with a convex set up to a set of .H -measure zero) for all .K ∈ K with .K ∩ H = ∅. Theorem 5.39 Let .H = u⊥ , .u ∈ Sn−1 , be oriented with .u ∈ H + and let n n .♦ : Kn → L be monotonic, measure preserving, invariant on H -symmetric sets, perimeter preserving on convex bodies, and convexity preserving away from H . Then .♦ essentially equals .Id, .†, .♦PH , or .♦†PH .

5.7 Convergence of Successive Symmetrals Around 1836, Jacob Steiner introduced the symmetrization process that today bears his name in an attempt to prove the isoperimetric inequality. As we have explained, many other inequalities have since been proved by the same method. Two are the main features that have allowed to prove that a certain functional is minimized (or maximized) by a ball: 1. The functional at hand is always decreased (or increased) by a symmetrization; 2. There are sequences of hyperplanes such that the corresponding sequence of successive symmetrals of a convex body converges to a ball of the same volume. In this section we focus on item 2. Given an i-symmetrization .♦ we focus on the convergence of successive applications of .♦ through a sequence of idimensional subspaces. (In this section convergence always means convergence in the Hausdorff metric.) We refer to this as a symmetrization process. We try to summarize what is known up to date, to explain how the answer depends on the specific symmetrization, whether it depends on the class, .Kn or .Cn , of the initial seed, to describe which questions still remain unanswered. It is well known that in the plane a sequence of successive Steiner symmetrals may converge to a ball or to a regular polygon, and convergence may sometimes seem simple or even obvious. In fact, in the literature there are few examples of Steiner symmetrization processes which do not converge (see [13, Section 1], [25, Lemma 6.3], [7, Examples 2.1], [86, Section 4]). These examples exhibit the same behavior and here we describe one in dimension two. We remark that in this example the directions orthogonal to the sequence of the lines of symmetrization form a dense subset of .S1 .

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G. Bianchi and P. Gronchi

Choose a sequence .(αm ) in .(0, π/2) with ∞ E

αm = ∞ ,

.

m=1

∞ ||

cos αm = r > 0 .

(5.43)

m=1

E Let .βk = km=1 αm and .vm = (cos βm , sin βm ). Besides, let .K ∈ K22 have area smaller than .π r 2 and contain a horizontal segment L, with L of length 1 and o as midpoint. The sequence .(Km ) defined as Km = Svm⊥ . . . Sv ⊥ K , m ∈ N,

.

1

(5.44)

does not converge. Indeed, let .Lm = Svm⊥ . . . Sv ⊥ L. Since .βm diverges, the segments 1 .Lm (which are contained in .Km ) spin in circles forever while their length decreases monotonically to r. Moreover, the closure of .∪m∈N Lm contains .rD n . If .(Km ) were converging, the limit would contain .rD n , but the area of .Km , which is equal to the area of K, is strictly less than the area of .rD n . Thus, a contradiction. Bianchi et al. [7] coined the term convergence in shape for the behaviour of this example; a sequence .(Hm ) converges in shape if there exists a sequence of isometries .(Im ) such that .Im SHm . . . SH1 C is a convergent sequence, for each compact set C. The same paper proves that the sequence above has indeed this property, and, furthermore, poses the following question: Do Steiner Processes Always Converge in Shape? Some partial answer is given in [7], but the question in full generality is still open. This question suggests that convergence of Steiner processes is not rare. To be more precise, let us introduce some definitions, following Coupier and Davydov [33] and widening the field of interest to all symmetrizations encountered so far. Let .♦ be an i-symmetrization on .B, where .B = Kn or .Cn . A sequence .(Hm ) of i-dimensional subspaces is called weakly .♦-universal for .B if, for any .k ∈ N, and .C ∈ B, the sequence ♦Hm ♦Hm−1 . . . ♦Hk C

.

(5.45)

converges, as m tends to infinity, to an origin-symmetric ball, which may depend on C and k. The sequence is called .♦-universal if it is weakly .♦-universal and the limit ball is independent of k. We remark that every weakly Steiner-universal sequence is in fact universal, since Steiner processes maintain the volume. In [86] Ulivelli introduces a more general concept. The sequence .(Hm ) is called .♦-stable for .B if, for any .k ∈ N and .C ∈ B, the sequence in (5.45) converges. (We do not prescribe the limit set in this case). The first example of a Steiner-universal sequence in .Kn , .n = 2, 3, goes back to Blaschke [19]. The sequence he chose is a periodic one. For .n = 3 he chose three planes .U1 , .U2 and .U3 , so that their normal vectors span the whole space and at least two of the angles between these vectors are irrational multiples of .π . He defined

5 Symmetrizations

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the process as a cyclical repetition of the three symmetrizations .SU1 , .SU2 and .SU3 , i.e. as .SU1 , .SU2 , .SU3 , .SU1 , .SU2 , .SU3 , . . . . The universality of the sequence was the right ingredient that Blaschke needed to present Schwarz symmetrization as a limit of sequences of Steiner symmetrizations, as explained in Sect. 5.3.2, and to prove the Brunn–Minkowski inequality. The argument used by Blaschke was extended by Klain [47, Theorem 5.1] to prove that every sequence of hyperplanes chosen from a finite set .F is Steinerstable in .Kn and the limit is symmetric under reflection in each hyperplane occurring infinitely often in the sequence. Klain [47, Corollary 5.4] uses this result to construct Steiner-universal sequences in .Kn , as described in the next theorem. We say that n n n .v1 , . . . , vn ∈ R form an irrational basis of .R if they span .R and the angle between any two of them is an irrational multiple of .π . Theorem 5.40 Let .(Hm ) be a sequence chosen from a finite set .F = {v1⊥ , . . . , vn⊥ } ⊂ G(n, n − 1). Assume that .v1 , . . . , vn form an irrational basis of .Rn and that, for each i, .vi⊥ appears infinitely many times in .(Hm ). Then .(Hm ) is Steiner-universal in .Kn . The main steps in the proofs by Blaschke and Klain are similar: 1. existence of convergent subsequences; 2. existence of a continuous functional F such that .F (SHm K) ≤ F (K), with equality if and only if .SHm K = K or .Vn (K) = 0; 3. the limit of a convergent subsequence is symmetric under reflection in each hyperplane occurring infinitely many times; 4. the hypothesis on the vectors implies that this limit is a ball. We try to explain these items one by one in a sketch of the proof of Theorem 5.40. Proof 1) The existence of convergent subsequences follows by (now) standard compactness arguments. The monotonicity of Steiner symmetrization and Blaschke selection theorem, Theorem 1.17, offer a simple conclusion. 2) Blaschke chose as F the surface area, while Klain uses the layering function f

∞

F (K) =

.

Vn (K ∩ rB n )e−r dr . 2

0

f Another possible choice is .F (K) = K |x|2 dx. 3) We follow Klain’s argument as interpreted by A. Burchard and presented in [7, Theorem 6.1] and [11, Theorem 5.6]. For simplicity, we present only the argument which assumes .K ∈ Knn . Let .Km = SHm . . . SH1 K. Since Steiner symmetrization preserves volume, we have Vn (Km ) = Vn (K).

.

(5.46)

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The main idea is to construct a subsequence along which the subspaces .vj⊥ ∈ F appear in a particular order. With each index m, we associate a permutation .πm of .{1, . . . , n} that indicates the order in which the subspaces .v1⊥ , . . . , vn⊥ appear for the first time among those .Hj with .j ≥ m. Since there are only finitely many permutations, we can pick a subsequence .(Hmp ) such that the permutation .πmp is the same for each p. By relabeling the subspaces, we may assume that this permutation is the identity. Passing to a further subsequence, we may assume that every subspace in .F appears in each segment .Hmp , Hmp +1 , . . . , Hmp+1 −1 . By Blaschke selection theorem, there is a subsequence (again denoted by n .(Kmp )) that converges in the Hausdorff metric to some .L ∈ Kn . We show by induction that L is .vj⊥ -symmetric for .j = 1, . . . , n. For .j = 1, observe that .Hmp = v1⊥ for each p. Therefore .Kmp is .v1⊥ -symmetric for each p and the same is true for L. Suppose that L is .vr⊥ -symmetric for .r = 1, . . . , j − 1. Let .m'p be the index where .vj⊥ appears for the first time after .Hmp . Then for .mp + 1 ≤ m ≤ m'p − 1, .Hm = vr⊥ for some .r = 1, . . . , j − 1. Steiner symmetrization does not increase the symmetric difference distance of two sets (because it does not change their volume and it does not decrease the volume of their intersection). Using this, the inductive hypothesis and the convergence of .(Kmp ) to L, one proves that .(Km' −1 ) too converges in the Hausdorff metric to p L as .p → ∞. Thus Sv ⊥ L = lim Sv ⊥ Km'p −1 = lim Km'p .

.

j

p→∞

p→∞

j

We also have F (Sv ⊥ L) = lim F (Km'p ) ≥ lim F (Kmp+1 ) = F (L) .

.

j

p→∞

p→∞

Therefore, .Sv ⊥ L = L, i.e. L is .vj⊥ -symmetric and this concludes the inductive j step. Once that the symmetry of L with respect to each .vj⊥ is proved, the fact that Steiner symmetrization does not increase the symmetric difference distance can be used again to prove that the entire sequence .(Km ) converges to L. 4) Blaschke and Klain used roughly the same hypothesis on the hyperplanes mutual position. Burchard, Chambers, and Dranovski [24] tackled the problem of characterizing the sets of reflections with respect to hyperplanes in .Rn that generate a dense subgroup of .O(n). They proved that the set of reflections in .vi⊥ , .i = 1, 2, . . . , n, generate a dense subgroup of .O(n) if the .vi ’s i) span .Rn , ii) cannot be partitioned into two mutually orthogonal non-empty subsets, and iii) at least two of them form an angle that is an irrational multiple of .π .

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Conditions i) and ii) are also necessary, but, when .n > 2, iii) is not, and [24] explains that iii) implies the right necessary and sufficient condition. i.e. that the group generated by the reflections is not a finite Coxeter group in .O(n). u n The extension of Klain’s result to compact sets was first obtained in [7]. Now it can also be seen as a consequence of Bianchi et al. [11, Theorem 7.3] which proves that a sequence of hyperplanes is Steiner-universal in .Cn if and only if it is Steineruniversal in .Knn , Volˇciˇc [90] extends Klain’s result to measurable sets. To complete the picture regarding Steiner symmetrization, we recall some results on the rate of convergence to a ball and on random sequences. In 1986 Mani-Levitska [62] was the first to deal with a sequence of hyperplanes uniformly, independently and randomly chosen. He proved that the sequence of related Steiner symmetrizations almost surely rounds every convex body with positive volume and conjectured that the same holds for compact sets. The conjecture was settled by van Schaftingen [87] and extended to measurable sets by Volˇciˇc [89]. The first result we know of on the rate of convergence (that is, on the determination of the least number of successive symmetrals required to transform a set K of volume .κn within a certain distance from .B n ) goes back to Hadwiger [44], even though the estimate was very rough. Such results often require very delicate analysis, as evidenced by the deep work of Bourgain, Klartag, Lindenstrauss, Milman, and others. (See [48, 49], and the references given there.) Klartag [49] proves that there exist m = [cn4 log2 (1/ε) + 1]

.

Steiner symmetrizations that transform .K ∈ Knn in a convex body with a Hausdorff distance less than .ε from .B n . (Here c is some numerical constant.) Bianchi and Gronchi [6] established a lower bound on the rate of convergence by constructing bodies “hard to be rounded”. In every dimension n and for each positive integer m, they constructed origin symmetric convex bodies whose Hausdorff distance from the ball centered in the origin of the same volume cannot be decreased by any sequence of m successive Steiner symmetrizations. Coupier and Davydov [33] complements and strengthens the results of [49, 62] and [89]. In particular, [33] gives an estimate on the speed of convergence to a sphere of random Steiner symmetrizations. Coupier and Davydov [33] also proves the following result, which maybe was one of the reasons for their choice of the term universal. Its proof is nice and elegant and we report it here. Theorem 5.41 A sequence .(Hk ) of hyperplanes is Steiner-universal in .Knn if and only if it is Minkowski-universal in .Knn .

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Proof Let .K ∈ Knn , and assume that .(Hk ) is a Steiner-universal sequence of hyperplanes. Since Minkowski symmetrization increases the volume and is monotonic, the sequence .Vn (MHk . . . MH1 K) is nondecreasing and bounded and, as .k → ∞, Vn (MHk . . . MH1 K) → V ,

.

(5.47)

for a suitable .V > 0. By the Blaschke selection Theorem, there exists a subsequence (λk ) such that, as .k → ∞,

.

MHλk . . . MH1 K → E,

.

for a suitable .E ∈ Knn with .Vn (E) = V . For any positive integer .k, m, .m > k, the inclusion between Steiner and Minkowski symmetrization (5.18) implies SHλm . . . SHλk +1 MHλk . . . MH1 K ⊂ MHλm . . . MHλk +1 MHλk . . . MH1 K

.

= MHλm . . . MH1 K. Passing to the limit with respect to m, and using the Steiner universality of the sequence .(Hk ), we obtain that a ball with volume .Vn (MHλk . . . MH1 K) is contained in E. The arbitrariness of k and (5.47) imply that E contains a ball of volume V . Since .V = Vn (E), a comparison of the volume forces E to be a ball of volume V . Therefore, any convergent subsequence has the same limit, that is .MHk . . . MH1 K converges to a ball. The same argument can be repeated for the sequence MHk . . . MHl K,

.

for any .l ∈ N. Observe that the limiting ball has the same mean width as K, since Minkowski symmetrization is mean width preserving, and therefore it does not depend on l. The reverse is completely analogous. Assume that .(Hk ) is a Minkowski-universal sequence of hyperplanes. Since Steiner symmetrization decreases the mean width, the sequence .V1 (SHk . . . SH1 K) is nonincreasing and positive and, as .k → ∞, V1 (SHk . . . SH1 K) → W

.

for a suitable W . There exists a subsequence of convex bodies converging, as .k → ∞, to a suitable .E ∈ Kn , with .V1 (E) = W . For any positive integer .k, m, .m > k, the inclusion between Steiner and Minkowski symmetrization implies MHλm . . . MHλk +1 SHλk . . . SH1 K ⊃ SHλm . . . SH1 K .

.

The Minkowski universality of the sequence .(Hk ) yields, letting .m → ∞, that a ball with the same mean width as that of .SHλk . . . SH1 K contains E. A comparison

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of mean widths forces E to be a ball with mean width W . Therefore, any convergent subsequence has the same limit, that is, .SHk . . . SH1 K converges to a ball with the same volume of K, since Steiner symmetrization is volume preserving. u n Bianchi et al. [11] studies some of the questions touched on in this section for other known symmetrizations and for general i-symmetrizations. It proves that Klain’s Theorem is valid in .Kn for fiber, Schwarz, and Minkowski–Blaschke symmetrizations, as well as for any i-symmetrization satisfying certain hypotheses, and it is valid in .Cn for Schwarz symmetrization (see [11, Section 5]). It also proves results in the spirit of Coupier and Davydov; for .i = 1, . . . , n − 1, a sequence of subspaces in .G(n, i) is Minkowski-universal in .Knn if and only if it is so in .Cn , and the same holds true for Steiner-universal and for Schwarz-universal sequences (see [11, Section 7]). These results, together with a study of the problem of which reflections with respect to i-dimensional subspaces generate a dense subgroup of n .O(n), carried out in [10], enable the authors to create universal sequences in .K and n in .C for many of the symmetrizations mentioned above (see [11, Section 6]). Ulivelli [85] further extends Klain’s Theorem by proving its validity in .Cn for Minkowski symmetrization (as a consequence of the proof that a Minkowski symmetrizations process with seed .C ∈ Cn converges if and only if the one with seed .conv(C) converges). The same author proves in [86] that the family .F of all i-symmetrizations .♦ which are monotonic, invariant on H -symmetric sets and invariant under translations orthogonal to H of H -symmetric sets share the same behavior with respect to symmetrizations processes. Since .F contains Steiner, Minkowski and fiber symmetrizations, this means that if a sequence .(Hk ) is Minkowski-universal or Minkowski-stable, then it is also .♦-universal or .♦-stable, for every .♦ ∈ F. A similar result is proved also for convergence in shape.

5.8 Rearrangements and a Proof of the Pólya–Szeg˝o Inequality for Smoothing Rearrangements A familiar version of the Pólya–Szeg˝o inequality states that if .0 : [0, ∞) → [0, ∞) is convex and .0(0) = 0 (i.e. a Young function) and .f ∈ V(Rn ) is Lipschitz, then # .f is Lipschitz and f .

f # 0(|∇f (x)|) dx ≤ 0(|∇f (x)|) dx; Rn Rn

(5.48)

see, e.g., [2]. Here .f # denotes the symmetric decreasing rearrangement of f , the function whose superlevel sets have the same .Hn -measure as those of f and such that, for .t > 0, .{x : f # (x) > t} is a ball centered at the origin o of .Rn . The subgraph of .f # is the Schwarz symmetrization of the subgraph of f with respect to the .xn+1 axis.

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The map that takes f to .f # is the primary example of a rearrangement. Other examples are the rearrangements associated to Steiner and Schwarz symmetrizations (often called .(k, n)-Steiner rearrangements), and polarization. The Pólya–Szeg˝o inequality (5.48) holds for each of the just-mentioned rearrangements. Diverse variants and applications of the Pólya–Szeg˝o inequality have generated a very substantial literature, surveyed by Talenti who in [84, p. 126] provides over fifty references. At the heart of the definition of the rearrangement Tf of a function f there is the formula {x : Tf (x) > t} = ♦T {x : f (x) > t}

.

(5.49)

where .♦T denotes a map from .Ln to itself which is monotonic and measure preserving. The superlevel set .{x : Tf (x) > t} depends only on .{x : f (x) > t} and this relation, the map .♦T , is the same for each t. A rearrangement is a map from function spaces and one may wonder which properties of this map make (5.49) valid. An answer is given in the following theorem, proved in [9], but before stating it let us define two properties. Let .X ⊂ M(Rn ) and .T : X → X. We say that: 1. T is equimeasurable if .Hn ({x : Tf (x) > t}) = Hn ({x : f (x) > t}) for .t ∈ R; 2. T is monotonic if .f, g ∈ X, .f ≤ g, essentially, implies .Tf ≤ T g. Theorem 5.42 Let .X = S(Rn ) or .V(Rn ), let .T : X → X be equimeasurable and monotonic. Then there exists a map .♦T : Ln → Ln for which (5.49) is valid. This map is defined for .A ∈ Ln by ♦T A = {x : T 1A (x) = 1},

.

and it is measure preserving and monotonic. Moreover T is defined, essentially, by ♦T .

.

We recall that the term essentially means up to a set of .Hn -measure zero. We can thus define the notion of rearrangement as follows. Let .X ⊂ M(Rn ). A map .T : X → X is called a rearrangement if it is equimeasurable and monotonic. Bianchi et al. [9] and [12] have studied rearrangements in an abstract setting, based on the properties that they satisfy, independent from the specific rearrangement. For the convenience of the reader, we now state five results proved in [9] as Lemmas 4.1, 4.5, 4.7, Theorem 4.8 and the remarks that follow it, and Theorem 4.9, respectively. Proposition 5.43 (i) If .T : S(Rn ) → S(Rn ) is equimeasurable, then .ess inf Tf = ess inf f for n .f ∈ S(R ).

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(ii) If .T : M(Rn ) → M(Rn ) is a rearrangement, then .ess inf Tf ≥ ess inf f for n n n .f ∈ M(R ). Hence, .T : S(R ) → S(R ). n n (iii) In either case, .T : V(R ) → V(R ) and T is essentially the identity on constant functions. Proposition 5.44 Let .X = M(Rn ), .M+ (Rn ), .S(Rn ), or .V(Rn ), and let .T : X → X be equimeasurable. (i) The induced map .♦T : Ln → Ln given by ♦T A = {x : T 1A (x) = 1}

.

for .A ∈ Ln is well defined and measure preserving. (ii) If .X = M+ (Rn ), .S(Rn ), or .V(Rn ), then T essentially maps characteristic functions of sets in .Ln to characteristic functions of sets in .Ln , in the sense that for each .A ∈ Ln , T 1A = 1♦T A ,

.

essentially. Proposition 5.45 Let .X = S(Rn ) or .V(Rn ) and let .T : X → X be a rearrangement. For .X = S(Rn ), .A ∈ Ln , and .α, β ∈ R with .α ≥ 0, we have T (α1A + β) = α T 1A + β,

.

essentially. When .X = V(Rn ), (5.45) holds, essentially, if .β = 0. Proposition 5.46 Let .X = M(Rn ), .M+ (Rn ), .S(Rn ), or .V(Rn ) and let .T : X → X be a rearrangement. (i) The map .♦T : Ln → Ln defined by (5.44) is monotonic. (ii) If .X = S(Rn ) or .V(Rn ) and .f ∈ X, then {x : Tf (x) ≥ t} =♦T {x : f (x) ≥ t} and

.

{x : Tf (x) > t} =♦T {x : f (x) > t}, essentially, for .t > ess inf f . Moreover, T is essentially determined by .♦T , since Tf (x) = max {sup{t ∈ Q, t > ess inf f : x ∈ ♦T {z : f (z) ≥ t}}, ess inf f } ,

.

essentially. Proposition 5.47 Let .T : S(Rn ) → S(Rn ) be a rearrangement and let .f ∈ S(Rn ). If .ϕ : R → R is right-continuous and increasing (i.e., non-decreasing), then .ϕ ◦ f ∈ S(Rn ) and ϕ(Tf ) = T (ϕ ◦ f ),

.

essentially.

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We recall that the symbol .A∗ , for a subset A of .Rn , denotes the set of points of .Rn of density 1 for A. We say that a rearrangement T is smoothing if the associated map .♦T is smoothing, i.e. if (♦∗T A) + dB n ⊂ ♦∗T (A + dB n ),

.

(5.50)

essentially, for each .d > 0 and bounded measurable set A, where .♦∗T A is defined by ♦∗T A = (♦T A)∗ .

.

We recall (see Lemma 5.2) that in this definition one can equivalently require the pointwise inclusion (♦∗T A) + dD n ⊂ ♦∗T (A + dD n ).

.

(5.51)

Bianchi et al. [12] prove that the notion of smoothing is equivalent to the rearrangement reducing the modulus of continuity. Theorem 5.48 Let .X = S(Rn ) or .V(Rn ). The rearrangement .T : X → X is smoothing if and only if T reduces the modulus of continuity, that is, T is such that .ωd (Tf ) ≤ ωd (f ) for .d > 0 and .f ∈ X, where ωd (f ) = ess sup |f (x) − f (y)|.

.

|x−y|≤d

All special rearrangements mentioned in the lines following (5.48) are smoothing, as we have proved in Sect. 5.3. Bianchi et al. [12] proves the Pólya–Szeg˝o inequality for all smoothing rearrangements. Theorem 5.49 Let .X = S(Rn ) or .V(Rn ), let .T : X → X be a rearrangement, and let .0 : [0, ∞) → [0, ∞) be convex with .0(0) = 0. If T is smoothing and .f ∈ X is Lipschitz then Tf coincides with a Lipschitz function .Hn -almost everywhere on n .R , and f f . 0 (|∇Tf (x)|) dx ≤ 0 (|∇f (x)|) dx (5.52) {x: Tf (x)≥a}

{x: f (x)≥a}

for each .a > ess inf f . Hence f f . 0 (x)|) dx ≤ 0 (|∇f (x)|) dx, (|∇Tf Rn Rn

(5.53)

where the integrals may be infinite. The analogous result is valid also for functions in the Orlicz space .W 1,0 (Rn ), the same function space where (5.48) is valid, and for Young functions .0 with values in .[0, +∞].

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Example We have observed that smoothing rearrangements reduce outer Minkowski content (and perimeter, when the two concepts coincide) and one may wonder if the Pólya–Szeg˝o inequality is valid for every rearrangement T such that .♦T has this property, but this is not the case. Let B be a ball containing o in its interior and with a center different from o. For .A ∈ Ln define .♦T A = λB where .λ is chosen so that .Hn (A) = Hn (λB). This map clearly does not increase the outer Minkowski content, by the isoperimetric inequality. On the other hand if, say, .f (x) = max{1 − |x|, 0} and .p > 1 then f .

f p |∇Tf (x)| dx > |∇f (x)|p dx. Rn Rn

(5.54)

This can be computed directly but it also comes from the study of the equality cases in (5.48). Indeed, if (5.54) were false then (5.54) would hold with the equality, because f coincides with its symmetric decreasing rearrangement .f # . This contradicts the fact that equality in (5.54), for that particular f , holds only for functions which, up to a translation, coincide with their symmetric decreasing rearrangement, and Tf does not satisfy this property.

5.8.1 Proof of the Pólya–Szeg˝o Inequality The method of proof of Theorem 5.49 is new and we present it here. We divide the proof in four steps and, for each step, we first describe the relevant ideas and then write and prove the relative lemmas (with one exception). Step 1. The function Tf coincides with a Lipschitz function .Hn -almost everywhere on .Rn . Assume that L is the Lipschitz constant for f . Then Tf is Lipschitz with a Lipschitz constant not larger than L, as T reduces the modulus of continuity, by Theorem 5.48. In the proof we use the following abbreviations: let .Kf,a = Kf ∩ {xn+1 ≥ a}, ∗ ∗ .KTf,a = KTf ∩ {xn+1 ≥ a} and .K Tf,a = (KTf,a ) . Step 2. Let .C ∈ Kn+1 n+1 be an o-symmetric convex body of revolution about the .xn+1 -axis, supported by the hyperplanes .{xn+1 = ±1}. This body in a later step is chosen to represent .0. We prove, for .d > 0, ( ∗ ( ) ) Hn+1 KTf,a + d int C ≤ Hn+1 Kf,a + d int C .

.

(5.55)

We prove this slice by slice, where by this we mean that we prove formula (5.58) below, for .t > ess inf f . Taking the .Hn -measures of both sides of (5.58), integrating with respect to t, and using Fubini’s theorem and the fact that .♦T is measure preserving, we obtain (5.55). In order to prove (5.58) we need .{x : f (x) ≥ a} bounded, and Lemma 5.51 proves that this is the case for any Lipschitz .f ∈ S(Rn ) and .a > ess inf f .

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In formula (5.58) we are slightly abusing notation by extending the action of .♦T to horizontal hyperplanes in .R n+1 . To make this rigorous, if E is a subset of the hyperplane .{xn+1 = t} = Rn + ten+1 in .Rn+1 such that .E|Rn ∈ Ln , we shall define ♦T E = (♦T (E | Rn )) + ten+1 .

.

(5.56)

The action of .♦∗T can be extended in a similar fashion. Note that we have, for .t > ess inf f , KTf ∩ {xn+1 = t} = ♦T (Kf ∩ {xn+1 = t}),

.

essentially, and .

)∗ ( KTf ∩ {xn+1 = t} = ♦∗T (Kf ∩ {xn+1 = t}),

(5.57)

where here and below, sets of Lebesgue density points are taken with respect to the appropriate horizontal hyperplane identified with .Rn . Lemma 5.50 Let .X = S(Rn ) or .V(Rn ), let .T : X → X be a rearrangement, and let .d > 0. Let .C ∈ Kn+1 n+1 be an o-symmetric convex body of revolution about the .xn+1 -axis, supported by the hyperplanes .{xn+1 = ±1}. If T is smoothing, .a > d + ess inf f , and .f ∈ S(Rn ) is such that .{x : f (x) ≥ a} is bounded, then ( ( ∗ ) ) ∗ . KTf,a + d int C ∩ {xn+1 = t} ⊂ ♦T (Kf,a + d int C) ∩ {xn+1 = t} (5.58) for .t > ess inf f . Proof Let .d > 0 and let .C = {(x, xn+1 ) ∈ Rn × R : |xn+1 | ≤ 1, |x| ≤ g(xn+1 )}, for a suitable concave function g defined on .[−1, 1]. For .t ∈ R, denote by .||t the orthogonal projection onto .{xn+1 = t}. If L is any set in .Rn+1 , then || ( ) .(L+d int C)∩{xn+1 = t} = ||t (L ∩ {xn+1 = s}) + rs D n , (5.59) t−d a}) = Hn ({y : f (y) > a}), the terms 1 in the integrands give the same contribution, they cancel each other, and (5.70) implies the Pólya– Szeg˝o inequality (5.52). If .Hn ({x : f (x) = a}) > 0 we argue by approximation. The set of values t such that .Hn ({x : f (x) = t}) = 0 is dense in .(ess inf f, ∞), so there is an increasing sequence .{am } contained in .(ess inf f, a) and converging to a such that .Hn ({x : f (x) = am }) = Hn ({x : Tf (x) = am }) = 0 for each m. The validity of (5.52) with .a = am , for each m, implies, in the limit, its validity for a. Finally, by Proposition 5.43, we have .ess inf Tf = ess inf f . Letting .a → ess inf f in (5.52), we arrive at (5.53). Lemma 5.53 Let .0 : [0, ∞) → [0, ∞) be convex with .0(0) = 0 and let .M > 0. Then there exist .b > 0 and an o-symmetric convex body .C ⊂ Rn+1 of revolution about the .xn+1 -axis, such that hC (y, 1) = 1 + b 0(|y|),

(5.71)

.

for .y ∈ Rn with .|y| ≤ M. In particular, C is supported by the hyperplanes .{xn+1 = ±1} and hence satisfies the conditions in Lemma 5.50. Proof Define { w(t) =

.

0(t),

if 0 ≤ t ≤ M,

mt + q,

if t ≥ M,

where .m > 0 and .q ≤ 0 are such that .w : [0, ∞) → [0, ∞) is convex. Then, for y ∈ Rn and .t ∈ R, define

.

g(y, t) =

.

{ |t| (1 + b w(|y|/|t|) , b m|y|,

if t /= 0, if t = 0,

.

(5.72)

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=

{ |t| (1 + b 0(|y|/|t|) , b m|y| + (1 + b q)|t|,

if |t| ≥ |y|/M, if |t| ≤ |y|/M,

(5.73)

where .b > 0. It is enough to show that b can be chosen so that .g = hC is the support function of a convex body C, since the origin symmetry and symmetry about the .xn+1 -axis of C, and (5.71), then follow directly from the definition of g. To this end, note that from (5.72), the positive homogeneity of g follows immediately, and the subadditivity of g for .t > 0 or for .t < 0 is a routine exercise using the triangle inequality and the convexity of .w. It is then enough to observe that if b is small enough to ensure that .1 + b q > 0, then the function .b m|y| + (1 + b q)|t| in (5.73) coincides with the support function of the o-symmetric spherical cylinder with the .xn+1 -axis as its axis, with height .2(1 + b q) and radius .b m. u n

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Index

affinely equivalent, 150 affine quermassintegrals inequality, 267 affine surface area, 26 .Lp , 28 Orlicz, 28 Aleksandrov body, 125 Aleksandrov–Fenchel inequality, 114 area measure, 96 .A-transform, 198

Banach–Mazur distance, 149 Barthe inequality, 139 Birkhoff polytope, 180 Blaschke addition, 235 Blaschke–Santaló inequality, 163, 263, 267 .Lp , 265 functional, 174 Borell–Brascamp–Lieb inequality, 263 Borell lemma, 216 Bourgain–Meyer–Milman–Pajor zonotope inequality, 259 Brascamp–Lieb inequality, 139, 150, 185 Brenier–McCann theorem, 182 Brunn–Minkowski deficit, 103 Brunn–Minkowski inequality, 101, 102, 160 Blaschke’s proof, 243 generalized, 120 for intrinsic volumes, 235 log, 125, 126 .Lp , 124 Brunn–Minkowski theory .Lp , 122

Busemann–Petty centroid inequality, 260 .Lp , 262 Busemann random simplex inequality, 258

canonical simplex decomposition, 34 capacity, 130, 131, 133 Cauchy formula, 215 c-basic functions, 196 c-class, 196 c-cyclic monotonicity, 201 centered Gaussian, 186 centroid body, 260 .Lp , 261 concentration of measure, 163 convex body, 2 unconditional, 127 convex function coercive, 54 geometric, 199 super-coercive, 63 convex hull, 3 Cost-Entropy inequality, 225 cost function, 179 Cost-Santaló inequality, 219 weak, 222 c-path-bounded, 210 cross polytope, 10, 141, 146, 155 c-sub-gradient, 199 c-transform, 196 cyclic monotonicity, 189, 201

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Colesanti, M. Ludwig (eds.), Convex Geometry, C.I.M.E. Foundation Subseries 2332, https://doi.org/10.1007/978-3-031-37883-6

293

294 Dehn invariant, 43 difference body, 52 discrete cube, 167 doubly stochastic matrix, 180 doubly stochastic measure, 180

entropy, 223 relative, 223 epi-additive, 71 epi-convergence, 54 epi-graph, 64, 169 epi-translation invariant, 64 extreme point, 87

fiber combination, 235 first Dirichlet eigenvalue of the Laplacian, 134, 137 function symmetrizable, 236 vanishing at infinity, 236 functional Blaschke–Santaló inequality, 174

gauge body, 146 gauge function, 10, 162 Gaussian isoperimetry, 166 Gaussian measure, 166 Gauss map, 15

Hadwiger theorem, 43 Hausdorff distance, 6 homogeneous decomposition, 39, 66 H -symmetric set, 234 H -symmetric spherical cylinder, 234

inf-convolution, 65, 169 intrinsic volume, 11, 92, 93 inversion formula, 90 isodiametric inequality, 111 isoperimetric inequality, 110 for intrinsic volumes, 253 isoperimetric ratio, 110, 148

John decomposition, 151 John ellipsoid, 150

Kantorovich duality theorem, 192 Khinchine inequality, 218

Index Legendre transform, 64, 169 Loewner ellipsoid, 150 logarithmic capacity, 130 log-concave measure, 168

mean width, 12, 93, 94 metric projection, 3, 4 Minkowski addition, 5, 160 log, 126 .Lp , 123 Minkowski additive, 41 Minkowski content, 12 anisotropic outer, 113, 236 Minkowski functional, 11 Minkowski inequality, 109 generalized, 120 quadratic, 114 second, 111 Minkowski problem, 16 Minkowski theorem, 184 mixed area measure, 96 mixed surface area measure, 96 mixed volume, 14, 88, 90, 184

non-traditional cost, 210

orthogonal cylinder, 36 orthogonal simplex, 36 outer normal vector, 3

parallel set, 6, 11, 85 permutation polytope, 181 Petty projection inequality, 261, 267 Poincaré inequality, 137 polar body, 10, 151, 162, 262 polarity transform, 198 Pólya–Szeg˝o inequality, 279, 282 Prékopa–Leindler inequality, 103, 172 principal kinematic formula, 47 projection body, 261 polar, 261, 267 Prokhorov theorem, 94, 189

quermassintegral, 12, 92

Rayleigh quotient, 137 rearrangement, 280 .(k, n)-Steiner, 280

Index polarization, 247, 280 reduction of modulus of continuity, 282 smoothing, 282 symmetric decreasing, 240, 279 reverse isoperimetric inequality, 155 reverse isoperimetric problem, 148, 150, 155 Rockafellar theorem, 203 rotationally symmetric set, 234 Sas–Macbeath inequality, 255 shadow system, 251 simple polytope, 117 spherical image, 15, 27 Steiner formula, 11, 92 strongly isomorphic, 117 sub-gradient, 183 support function, 7, 162 supporting hyperplane, 3 support set, 82 support triangle, 29 surface area, 12, 83, 84 lower (outer) relative, 113, 236 upper (outer) relative, 113, 236 surface area measure, 15, 96, 97 .Lp , 124 symmetric difference metric, 29, 149 symmetrization Blaschke, 245 central, 238 convergence in shape, 274 fiber, 245, 267 F -preserving, 236 idempotent, 237 inner rotational, 249 invariant on H -symmetric sets, 237 invariant on H -symmetric spherical cylinders, 237

295 invariant under translations, 237 i-symmetrization, 236 Minkowski, 243, 267, 270 Minkowski–Blaschke, 244 monotonic, 236 outer rotational, 249 polarization, 247, 272 projection invariant, 237 pth central, 238 Schwarz, 240, 272 smoothing, 237 Steiner, 238, 267, 271 two-point, 247

torsion, 134, 135 total cost, 180 total variation distance, 225 transport map, 178 transport plan, 178

universal and weakly universal sequence, 274, 277

valuation, 19, 49 volume, 2, 83, 84 volume product, 262 volume ratio, 150, 151

Weak Cost-Santaló inequality, 222 width, 12, 93 Wulff shape, 125

Zhang projection inequality, 261

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