Geometry of Isotropic Convex Bodies

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Mathematical Surveys and Monographs Volume 196

Silouanos Brazitikos Apostolos Giannopoulos Petros Valettas Beatrice-Helen Vritsiou

American Mathematical Society Geometry of Isotropic Convex Bodies

http://dx.doi.org/10.1090/surv/196

Mathematical Surveys and Monographs Volume 196

Geometry of Isotropic Convex Bodies

Silouanos Brazitikos Apostolos Giannopoulos Petros Valettas Beatrice-Helen Vritsiou

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer

2010 Mathematics Subject Classiﬁcation. Primary 52Axx, 46Bxx, 60Dxx, 28Axx.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-196

Library of Congress Cataloging-in-Publication Data Brazitikos, Silouanos, 1990– author. Geometry of isotropic convex bodies / Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, Beatrice-Helen Vritsiou. pages cm. – (Mathematical surveys and monographs ; volume 196) Includes bibliographical references and indexes. ISBN 978-1-4704-1456-6 (alk. paper) 1. Convex geometry. 2. Banach lattices. I. Giannopoulos, Apostolos, 1963– author. II. Valettas, Petros, 1982– author. III. Vritsiou, Beatrice-Helen, author. IV. Title.

QA331.5.B73 2014 516.362–dc23 2013041914

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Preface ix

Chapter 1. Background from asymptotic convex geometry 1 1.1. Convex bodies 1 1.2. Brunn–Minkowski inequality 4 1.3. Applications of the Brunn-Minkowski inequality 8 1.4. Mixed volumes 12 1.5. Classical positions of convex bodies 16 1.6. Brascamp-Lieb inequality and its reverse form 22 1.7. Concentration of measure 25 1.8. Entropy estimates 34 1.9. Gaussian and sub-Gaussian processes 38 1.10. Dvoretzky type theorems 43 1.11. The -position and Pisier’s inequality 50 1.12. Milman’s low M ∗-estimate and the quotient of subspace theorem 52 1.13. Bourgain-Milman inequality and the M-position 55 1.14. Notes and references 58

Chapter 2. Isotropic log-concave measures 63 2.1. Log-concave probability measures 63 2.2. Inequalities for log-concave functions 66 2.3. Isotropic log-concave measures 72 2.4. ψα-estimates 78 2.5. Convex bodies associated with log-concave functions 84 2.6. Further reading 94 2.7. Notes and references 100

Chapter 3. Hyperplane conjecture and Bourgain’s upper bound 103 3.1. Hyperplane conjecture 104 3.2. Geometry of isotropic convex bodies 108 3.3. Bourgain’s upper bound for the isotropic constant 116 3.4. The ψ2-case 123 3.5. Further reading 128 3.6. Notes and references 134

Chapter 4. Partial answers 139 4.1. Unconditional convex bodies 139 4.2. Classes with uniformly bounded isotropic constant 144 4.3. The isotropic constant of Schatten classes 150 4.4. Bodies with few vertices or few facets 155

v vi CONTENTS

4.5. Further reading 161 4.6. Notes and references 170

Chapter 5. Lq-centroid bodies and concentration of mass 173 5.1. Lq-centroid bodies 174 5.2. Paouris’ inequality 182 5.3. Small ball probability estimates 190 5.4. A short proof of Paouris’ deviation inequality 197 5.5. Further reading 202 5.6. Notes and references 209

Chapter 6. Bodies with maximal isotropic constant 213 6.1. Symmetrization of isotropic convex bodies 214 6.2. Reduction to bounded volume ratio 223 6.3. Regular isotropic convex bodies 227 6.4. Reduction to negative moments 231 ◦ 6.5. Reduction to I1(K, Zq (K)) 234 6.6. Further reading 239 6.7. Notes and references 242

Chapter 7. Logarithmic Laplace transform and the isomorphic slicing problem 243 7.1. Klartag’s ﬁrst approach to the isomorphic slicing problem 244 7.2. Logarithmic Laplace transform and convex perturbations 249 7.3. Klartag’s solution to the isomorphic slicing problem 251 7.4. Isotropic position and the reverse Santal´o inequality 254 7.5. Volume radius of the centroid bodies 256 7.6. Notes and references 270

Chapter 8. Tail estimates for linear functionals 271 8.1. Covering numbers of the centroid bodies 273 8.2. Volume radius of the ψ2-body 284 8.3. Distribution of the ψ2-norm 292 8.4. Super-Gaussian directions 298 8.5. ψα-estimates for marginals of isotropic log-concave measures 301 8.6. Further reading 304 8.7. Notes and references 310

Chapter 9. M and M ∗-estimates 313 9.1. Mean width in the isotropic case 313 9.2. Estimates for M(K) in the isotropic case 322 9.3. Further reading 330 9.4. Notes and references 332

Chapter 10. Approximating the covariance matrix 333 10.1. Optimal estimate 334 10.2. Further reading 349 10.3. Notes and references 354

Chapter 11. Random polytopes in isotropic convex bodies 357 11.1. Lower bound for the expected volume radius 358 CONTENTS vii

11.2. Linear number of points 363 11.3. Asymptotic shape 367 11.4. Isotropic constant 377 11.5. Further reading 381 11.6. Notes and references 387 Chapter 12. Central limit problem and the thin shell conjecture 389 12.1. From the thin shell estimate to Gaussian marginals 391 12.2. The log-concave case 397 12.3. The thin shell conjecture 402 12.4. The thin shell conjecture in the unconditional case 407 12.5. Thin shell conjecture and the hyperplane conjecture 415 12.6. Notes and references 422 Chapter 13. The thin shell estimate 425 13.1. The method of proof and Fleury’s estimate 427 13.2. The thin shell estimate of Guédon and E. Milman 436 13.3. Notes and references 458 Chapter 14. Kannan-Lovász-Simonovits conjecture 461 14.1. Isoperimetric constants for log-concave probability measures 462 14.2. Equivalence of the isoperimetric constants 473 14.3. Stability of the Cheeger constant 477 14.4. The conjecture and the first lower bounds 480 14.5. Poincaré constant in the unconditional case 485 14.6. KLS-conjecture and the thin shell conjecture 486 14.7. Further reading 505 14.8. Notes and references 509 Chapter 15. Infimum convolution inequalities and concentration 511 15.1. Property (τ) 512 15.2. Infimum convolution conjecture 523 15.3. Concentration inequalities 527 15.4. Comparison of weak and strong moments 533 15.5. Further reading 535 15.6. Notes and references 546 Chapter 16. Information theory and the hyperplane conjecture 549 16.1. Entropy gap and the isotropic constant 550 16.2. Entropy jumps for log-concave random vectors with spectral gap 552 16.3. Further reading 559 16.4. Notes and references 562

Bibliography 565 Subject Index 585 Author Index 591

Preface

Asymptotic convex geometry may be described as the study of convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension. This theory stands at the intersection of classical convex geometry and the local theory of Banach spaces, but it is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. The aim of this book is to introduce a number of basic questions regarding the distribution of volume in high-dimensional convex bodies and to provide an up to date account of the progress that has been made in the last fifteen years. It is now understood that the convexity assumption forces most of the volume of a body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. One such normalization, that in many cases facilitates the study of volume distribution, is the isotropic position. A convex body K in Rn is called isotropic if it has volume 1, barycenter at the origin, and its inertia matrix is a multiple of the identity: there exists a constant LK > 0 such that 2 2 x, θ dx = LK K for every θ in the Euclidean unit sphere Sn−1. It is easily verified that the affine class of any convex body K contains a unique, up to orthogonal transformations, isotropic convex body; this is the isotropic position of K. A first example of the role and significance of the isotropic position may be given through the hyperplane conjecture (or slicing problem), which is one of the main problems in the asymptotic theory of convex bodies, and asks if there exists an absolute constant c>0such ⊥ n that maxθ∈Sn−1 |K ∩ θ | c for every convex body K of volume 1 in R that has barycenter at the origin. This question was posed by Bourgain [99], who was interested in finding Lp-bounds for maximal operators defined in terms of arbitrary convex bodies. It is not so hard to check that answering his question affirmatively is equivalent to the following statement: Isotropic constant conjecture. There exists an absolute constant C>0 such that n Ln := max{LK : K is isotropic in R } C. This problem became well-known due to an article of V. Milman and Pajor which remains a classical reference on the subject. Around the same time, K. Ball showed in his PhD Thesis that the notion of the isotropic constant and the conjecture can be reformulated in the language of logarithmically-concave (or log-concave

ix xPREFACE for short) measures; however, without the problem becoming essentially more general. Let us note here that a finite Borel measure μ on Rn is called log-concave if, for any λ ∈ (0, 1) and any compact subsets A, B of Rn,wehave μ(λA +(1− λ)B) μ(A)λμ(B)1−λ; note also that the indicator function of a convex body is the density (with respect to the Lebesgue measure) of a compactly supported log-concave measure, but that not all log-concave measures are compactly supported. Isotropic convex bodies now form a genuine subclass of isotropic log-concave measures, but several properties and results that (may) hold for this subclass, including the boundedness or not of the isotropic constants, immediately translate in the setting of log-concave√ mea- 4 sures. Around 1990, Bourgain obtained the upper bound√ Ln c n log n and, in 4 2006, this estimate was improved by Klartag to Ln c n. The problem remains open and has become the starting point for many other questions and challenging conjectures in high-dimensional geometry, one of those being the central limit problem. The latter in the asymptotic theory of convex bodies means the task to identify those high-dimensional distributions which have approximately Gaussian marginals. It is a question inspired by a general fact that has appeared more than once in the literature and states that, if μ is an isotropic probability measure on Rn which satisfies the thin shell condition √ μ x2 − n ε ε n−1 for some ε ∈√(0, 1), then, for all directions θ in a subset A of S with σ(A) 1 − exp(−c1 n), one has −α |μ ({x : x, θ t}) − Φ(t)| c2(ε + n ) for all t ∈ R, where Φ(t) is the standard Gaussian distribution function and c1,c2,α > 0are absolute constants. Thus, the central limit problem is reduced to the question of identifying those high-dimensional distributions that satisfy a thin shell condition. It was the work of Anttila, Ball and Perissinaki that made this type of statement widely known in the context of isotropic convex bodies or, more generally, log- concave distributions. One of the main results in this area, first proved by Klartag in a breakthrough work, states that the assumption of log-concavity guarantees a thin shell bound, and hence an affirmative answer to the central limit problem. In fact, the following quantitative conjecture has been proposed. Thin shell conjecture. There exists an absolute constant C>0 such that, for any n 1 and any isotropic log-concave measure μ on Rn, one has √ 2 − 2 2 σμ := x 2 n dμ(x) C . Rn

A third conjecture concerns the Cheeger constant Isμ of an isotropic log-concave measure μ which is deﬁned as the best constant κ 0 such that μ+(A) κ min{μ(A), 1 − μ(A)} for every Borel subset A of Rn,andwhere μ(A ) − μ(A) μ+(A) := lim inf ε ε→0+ ε is the Minkowski content of A (also, Aε := {x :dist(x, A) <ε} is the ε-extension of A). PREFACE xi

Kannan-Lov´asz-Simonovits conjecture. There exists an absolute constant c> 0 such that n Isn := min{Isμ : μ is isotropic log-concave measure on R } c.

Another way to formulate this conjecture is to ask if there exists an absolute constant c>0 such that, for every isotropic log-concave measure μ on Rn and for every smooth function ϕ with Rn ϕdμ= 0, one has 2 ∇ 2 c ϕ dμ ϕ 2dμ. Rn Rn We then say that μ satisﬁes the Poincar´einequalitywith constant c>0. The equivalence of the two formulations can be seen by checking that ∇ϕ2dμ Is2 inf inf 2 . μ μ ϕ ϕ2dμ

In this book we discuss these three conjectures and what is currently known about them, as well as other problems that are related to and arise from them. We now give a brief account of the contents of every chapter; more details can be found in the introduction of each individual chapter. In Chapters 2–4, we present the hyperplane conjecture and the first attempts to an answer. This presentation is given in the more general setting of logarithmically concave probability measures, which are introduced in Chapter 2 along with their main concentration properties. Some of these properties follow immediately from the Brunn-Minkowski inequality (more precisely, from Borell’s lemma) and can be expressed in the form of reverse Hölder inequalities for seminorms: if f : Rn → R is a seminorm, then, for every Rn log-concave probability measure μ on , one has f ψ1(μ) c f L1(μ),where | | α f ψα(μ) =inf t>0: exp(( f /t) ) dμ 2 is the Orlicz ψα-norm of f with respect to μ, α ∈ [1, 2]. Isotropic log-concave measures are the log-concave probability measures μ that have barycenter at the origin and satisfy the isotropic condition x, θ2dμ(x)=1 Rn for every θ ∈ Sn−1. The isotropic constant of a measure μ in this class is defined as 1/n 1/n Lμ := sup f(x) (f(0)) , x∈Rn where f is the log-concave density of μ. K. Ball introduced a family of convex bodies Kp(μ), p 1, that can be associated with a given log-concave measure μ and showed that these bodies allow us to reduce the study of log-concave measures to that of convex bodies, but also enable us to use tools from the broader class of measures to tackle problems that have naturally, or merely initially, been formulated for bodies. A first example of their use, as mentioned above, is the fact that studying the magnitude of the isotropic constant of log-concave measures is completely equivalent to the respective task inside the more restricted class of convex bodies. xii PREFACE

The isotropic constant conjecture is discussed in detail in Chapter 3; it reads that there exists an absolute constant C>0 such that

Lμ C for every n 1 and every log-concave measure μ on Rn. In order to understand its equivalence to the hyperplane conjecture we formulated above, we recall that

n max{LK : K is an isotropic convex body in R } n sup{Lμ : μ is an isotropic log-concave measure on R }, and then we have to explain the relation of the moments of inertia of a centered convex body to the volume of its hyperplane sections passing through the origin. In particular, in Section 3.1.2 we show that, if K is an isotropic convex body in Rn, then for every θ ∈ Sn−1 we have c c 1 |K ∩ θ⊥| 2 , LK LK where c1,c2 > 0 are absolute constants, and, thus, all hyperplane sections through the barycenter of K have approximately the same volume, this volume being large enough if and only if LK is small enough. The hyperplane conjecture is also equivalent to the asymptotic versions of several classical problems in convex geometry. We discuss two of them: Sylvester’s problem on the expected volume of a random simplex contained in a convex body and the Busemann-Petty√ prob- 4 lem. In Sections 3.3 and 3.4 we discuss Bourgain’s upper bound LK C n log n for the isotropic constant of convex bodies K in Rn. We describe two proofs of Bourgain’s result. A key observation is that, if K is an isotropic convex body in Rn, then, as we saw above for every log-concave probability measure μ, one has · · ∈ n−1 ,θ ψ1(K) C ,θ L1(K) CLK for all θ S ,whereC>0 is an absolute constant. In fact, Alesker’s theorem shows that one has a stronger√ ψ2-estimate for the function f(x)= x 2: one has f ψ2(K) C f L2(K) C nLK .Markov’s inequality then implies exponential concentration of the mass√ of K in a strip of width CLK and normal concentration in a ball of radius C nLK . Chapter 4 is devoted to some partial affirmative answers to the hyperplane conjecture that were obtained soon after the problem became known. In order to make this statement more precise, we say that a class C of centered convex bodies satisfies the hyperplane conjecture uniformly if there exists a positive constant C such that LK C for all K ∈C. The hyperplane conjecture has been verified for several important classes of convex bodies. A first example is the class of unconditional convex bodies; these are the centrally symmetric convex bodies K in Rn that have a position that is symmetric with respect to the standard coordinate subspaces, namely they have a position K˜ such that, if (x1,...,xn) belongs to K˜ , n then ( 1x1,..., nxn) also belongs to K˜ for every ( 1,..., n) ∈{−1, 1} .Theclass of unconditional convex bodies will appear often in this book, mainly as a model for results or conjectures regarding the general cases. In this chapter, we also describe uniform bounds for the isotropic constants of some other classes of convex bodies and we give simple geometric proofs of the best known estimates for the isotropic constants of polytopes with N vertices or polyhedra with N facets, estimates that are logarithmic in N. In Chapters 5–7, we discuss more recent approaches to the slicing problem and some very useful tools that have been developed for these approaches as well as for PREFACE xiii related problems in the theory. Bourgain’s approach exploited the ψ1-information we have for the behavior of the linear functionals x →x, θ on an isotropic convex body. The aim to understand the distribution of linear functionals in an isotropic convex body or, more precisely, the behavior of their Lq-norms with respect to the uniform measure on the body, has been furthered by the introduction of the family of Lq-centroid bodies of a convex body K of volume 1 or, more generally, of a log-concave probability measure μ. For every q 1, the Lq-centroid body Zq(K)of K or, respectively, the Lq-centroid body Zq(μ)ofμ is defined through its support function, which is given by 1/q · | |q hZq (K)(y):= ,y Lq(K) = x, y dx , K 1/q · | |q or by hZq (μ)(y):= ,y Lq(μ) = x, y dμ(x) , respectively, for every vector y. Note that, according to our normalization, a convex n body K of volume 1 in R is isotropic if and only if it is centered and Z2(K)= n Rn LK B2 and, respectively, a log-concave probability measure μ on is isotropic n ifandonlyifitiscenteredandZ2(μ)=B2 . The development of an asymptotic theory for this family of bodies, and for their behavior as q increases from 2 up to the dimension n, was initiated by Paouris and has proved to be a very fruitful idea. In Chapter 5 we present the basic properties of the family {Lq(μ):q 2} of the centroid bodies of a centered log-concave probability measure μ on Rn and prove some fundamental formulas. The first main application of this theory is a striking and very useful deviation inequality of Paouris: for every isotropic log-concave probability measure μ on Rn one has √ √ n μ({x ∈ R : x2 ct n}) exp −t n for every t 1, where c>0 is an absolute constant. This is a consequence of the following statement: there exists an absolute constant C1 > 0 such that, if μ is an isotropic log-concave measure on Rn,then

(1) Iq(μ) C1I2(μ) √ for every q n,whereIq(μ) is deﬁned by

1/q q Iq(μ)= x 2dμ(x) Rn for all 0 = q>−n. Paouris has, moreover, proved an extension to this theorem which we also present: there exists an absolute constant c2 √such that, if μ is an n isotropic log-concave measure on R , then for any 1 q c2 n one has

I−q(μ) Iq(μ). √ In particular, this shows that, for all 1 q c2 n, one has Iq(μ) CI2(μ), where C>0 is an absolute constant. Using the extended result one can derive a small ball probability estimate: for every isotropic log-concave measure μ on Rn and for any 0 <ε<ε0, one has √ √ n c3 n μ({x ∈ R : x2 <ε n}) ε , xiv PREFACE where ε0,c3 > 0 are absolute constants. In a few words, the main results of Paouris imply that for any isotropic log-concave measure one has √ √ √ μ({x : c n x2 C n}) 1 − exp(− n). This is a rough version of the thin shell estimate, that is often enough for the applications. In fact, as we will explain in Chapter 13, a way to obtain a thin shell estimate is to prove a more precise√ version of (1), with the constant C1 being, for example,√ of the form 1 + cq/ n for some absolute constant c>0 and for as large q ∈ [1, n] as possible. In Chapters 6 and 7 we discuss some recent approaches to and reductions of the hyperplane conjecture. Chapter 6 deals with properties that bodies with maximal isotropic constant have, namely bodies whose isotropic constant is equal to or very close to Ln. It turns out that the isotropic position of such bodies is closely related to their M-position and this enables one to establish several interesting facts: for example, a reduction of the hyperplane conjecture, due to Bourgain, Klartag and V. Milman, to the question of boundedness of the isotropic constant of a restricted class of convex bodies, those that have volume ratio bounded by an absolute constant. Next, we give two more reductions of the conjecture to the study of parameters that can be associated with any isotropic convex body. The proofs of these reductions rely heavily on the existence of convex bodies with maximal isotropic constant whose isotropic position is not only closely related to their M- position, but is also compatible with regular covering estimates. The first of these reductions is a continuation of the work of Paouris on the behavior of the negative n moments of the Euclidean norm with respect to an isotropic measure√ μ on R .As√ we mentioned above, we already know that I−q(μ) I2(μ)= n√for 0 n is not known at all and, in fact, our current knowledge does not exclude the possibility that the moments stay constant for all positive q up to n − 1. Dafnis and Paouris have actually proved that this question is equivalent to the hyperplane conjecture: they introduce a parameter that, for each δ 1, is given by √ −1 −1 q−c(μ, δ):=max 1 q n − 1:I−q(μ) δ I2(μ)=δ n , and they establish that n 2 en Ln Cδsup log μ q−c(μ, δ) q−c(μ, δ) for every δ 1; additionally, they show that, if the hyperplane conjecture is correct, then we must have q−c μ, δ0 = n − 1forsomeδ0 1, for every isotropic log- concave measure μ on Rn. The other reduction is a work of Giannopoulos, Paouris and Vritsiou, based on the study of the parameter ◦ · I1(K, Zq (K)) = hZq (K)(x)dx = ,x Lq(K)dx, K K and can be viewed as a continuation√ of Bourgain’s initial approach that led to 4 the upper bound LK c n log n. Roughly speaking, this last reduction can be 1 formulated as follows: given q 2and 2 s 1, an upper bound of the form √ ◦ s 2 I1(K, Zq (K)) C1q nLK for all bodies K in isotropic position PREFACE xv leads to the estimate √ 4 2 C2 n log n Ln 1−s . q 2 Bourgain’s estimate is (almost) recovered by choosing q = 2, however, the behavior ◦ of I1(K, Zq (K)) may allow one to use s<1 along with large values of q to obtain improved bounds if possible. In Chapter 7 we first discuss Klartag’s solution to the isomorphic slicing problem, an isomorphic variation of the hyperplane conjecture that asks whether, given any convex body, we can find another convex body, with absolutely bounded isotropic constant, that is geometrically close to the first body. Klartag’s method relies on properties of the logarithmic Laplace transform of the uniform measure on a convex body. In general, given a finite Borel measure μ on Rn, the logarithmic Laplace transform of μ is given by 1 ξ,x Λμ(ξ):=log n e dμ(x) . μ(R ) Rn Klartag proved that, if K isaconvexbodyinRn, then, for every ε ∈ (0, 1), we ⊂ Rn ∈ Rn 1 ⊆ can find a centered convex body T and a point x such that 1+ε T K + x ⊆ (1 + ε)T and √ LT C/ ε, where C>0 is an absolute constant. Most remarkably, by combining this fact with the deviation inequality of Paouris, one may also deduce the currently best known upper bound for the isotropic constant, which is that 1/4 Lμ C n for every isotropic log-concave measure μ on Rn. The logarithmic Laplace transform is another important tool of the theory that, since it was first employed in the setting of isotropic convex bodies and log-concave measures, has proved to be ex- tremely useful given its various and interesting applications; these include Klartag’s solution to the isomorphic slicing problem, that we already mentioned, as well as an alternative approach of Klartag and E. Milman that combines the advantages of both the logarithmic Laplace transform and the extensive theory of the Lq-centroid bodies, and occupies the second part of Chapter 7. Klartag and E. Milman looked for lower bounds for the volume radius of the Lq-centroidbodiesofanisotropic log-concave measure μ. Through a delicate analysis of the logarithmic Laplace transform of μ, they showed that 1/n (2) Zq(μ) c1 q/n √ for all q n,wherec1 > 0 is an absolute constant. Apart√ from being interesting 4 on its own, this result leads again to the estimate Lμ c2 n. It is also plausible that (2) can hold for larger values of q ∈ [1,n] as well; this is the content of a recent work of Vritsiou that is also discussed in the chapter. She showed that (2) holds for every q up to a variant of the parameter q−c(μ, δ) of Dafnis and Paouris, which, as we previously mentioned, could be of the order of n (in fact, recall that the hyperplane conjecture is correct if and only if q−c(μ, δ0)isoftheorderofn for some n δ0 1 and every isotropic log-concave measure μ on R ). However, even a small improvement to the estimates we currently have for q−c(μ, δ) and its variant could permit one to extend the range of q with which the method of Klartag and Milman can be applied, and also improve on the currently known bounds for the isotropic xvi PREFACE constant problem. Other applications of the logarithmic Laplace transform are discussed in some of the following chapters, the most important of these appearing in Chapters 12 and 15. In Chapters 8–11, we deviate a little from those lines of results that are directly related to the hyperplane conjecture and the other two main conjectures of the theory so as to look at different applications of the tools that were developed in the previous part. Chapters 8 and 9 are devoted to some open questions, whose study so far has already shed more light on various geometric properties of convex bodies and log-concave measures. The first question was originally posed by V. Milman in the framework of convex bodies: it asks if there exists an absolute constant C>0 such that every centered convex body K of volume 1 has at least one sub-Gaussian direction with constant C. Following some positive results for special classes of convex bodies, Klartag was the first to prove the existence of “almost sub-Gaussian” directions for any isotropic convex body. More precisely, using again properties of the logarithmic Laplace transform, he proved that for every log-concave probability measure μ on Rn there exists θ ∈ Sn−1 such that

− t2 μ ({x : |x, θ| ct·,θ }) e (log(t+1))2α , √ 2 for all 1 t n logα n,whereα = 3. We describe the best known estimate, due to Giannopoulos, Paouris and Valettas, according to which one can always have α =1/2. The main idea is to deﬁne the symmetric convex set Ψ2(μ) whose support · function is hΨ2(μ)(θ)= ,θ ψ2 and to estimate its volume. One can show that for every centered log-concave probability measure μ in Rn one has 1/n |Ψ2(μ)| c1 c2 log n, |Z2(μ)| where c1,c2 > 0 are absolute constants. An immediate consequence√ is the existence of at least one sub-Gaussian direction for μ with constant b = O( log n). The main n tool in the proof of this result is estimates for the covering numbers N(Zq(K),sB2 ). An even more interesting question is to determine the distribution of the function →· θ ,θ ψ2 on the unit sphere; that is, to understand whether most of the directions have ψ2-norm that is, say, logarithmic in the dimension. In Chapter 9 we discuss the questions of obtaining an upper bound for the mean width

w(K):= hK (x) dσ(x), Sn−1 that is, the L1-norm of the support function of K with respect to the Haar measure on the sphere, as well as the respective L -norm of the Minkowski functional of K, 1

M(K):= xK dσ(x), Sn−1 when K is an isotropic convex body. We present some non-trivial but non-optimal estimates. We also discuss the same questions for the Lq-centroid bodies of an isotropic log-concave measure. Answering these questions requires a deeper un- derstanding of the behavior of linear functionals and of the local structure of the centroid bodies; this would bring new insights to the reductions of the hyperplane conjecture that were discussed in the previous chapters.

Chapters 10 and 11 contain applications of the theory of Lq-centroid bodies and of the main inequalities of Paouris to random matrices and random polytopes. PREFACE xvii

In Chapter 10 we discuss a question of Kannan, Lov´asz and Simonovits on the approximation of the covariance matrix of a log-concave measure. If K is an isotropic convex body in Rn, then one has 1 ⊗ I = 2 x xdx, LK K where I is the identity operator. Given ε ∈ (0, 1), the question is to ﬁnd N0,assmall as possible, for which the following holds true: if N N0,thenN independent random points x1,...,xN that are uniformly distributed in K must have, with probability greater than 1 − ε, the property that

1 N (1 − ε)L2 x ,θ2 (1 + ε)L2 K N i K i=1 for every θ ∈ Sn−1 . The question had its origin in the problem of finding a fast algorithm for the computation of the volume of a given convex body, and 2 Kannan, Lovász and Simonovits proved that one can take N0 = C(ε)n for some 3 constant C(ε) > 0 depending only on ε. This was improved to N0 = C(ε)n(log n) 2 by Bourgain and to N0 = C(ε)n(log n) by Rudelson. It was finally proved by Adamczak, Litvak, Pajor and Tomczak-Jaegermann that the best estimate for N0 is C(ε)n. We describe the history and the solution of the problem. In Chapter 11 we discuss the asymptotic shape of the random polytope KN := conv{±x1,...,±xN } that is spanned by N independent random points x1,...,xN uniformly distributed in an isotropic convex body K in Rn. The literature on the approximation of convex bodies by random polytopes is very rich, but the main point here is that N is fixed in the range [n, en] and we are interested in estimates which do not depend on the affine class of a convex body K. Some basic tasks in this spirit are: to determine the asymptotic behavior of the volume radius |K|1/n, to understand the typical “asymptotic shape” of KN and to estimate the isotropic constant of KN . The same questions can be formulated and studied more generally if we assume that we have N independent copies X1,...,XN of an isotropic log- concave random vector X. A general, and rather precise, description was obtained by Dafnis, Giannopoulos and Tsolomitis: given any isotropic log-concave measure n μ on R and any n N exp(n), the random polytope KN defined by N i.i.d. random points X1,...,XN which are distributed according to μ satisfies, with high probability, the next two conditions: (i) KN ⊇ cZlog(N/n)(μ) and (ii) for every α>1andq 1, E { } −q σ( θ : hKN (θ) αhZq (μ)(θ) ) Nα .

Using this description of the shape of KN and the theory of centroid bodies which was developed in the previous chapters, one can determine the volume radius√ and the quermassintegrals of a random KN , at least in the range n N exp( n). A question concerning the isotropic constant of KN can be made precise in the following way: one would like to show that, with probability tending to 1 as n → ∞, the isotropic constant of the random polytope KN := conv{±x1,...,±xN } is bounded by CLK where C>0 is a constant independent of K, n and N.We describe a method that was initiated by Klartag and Kozma when dealing with the class of Gaussian random polytopes. Variants of the method also work in the cases that the vertices xj of KN are distributed according to the uniform measure on an xviii PREFACE isotropic convex body which is either ψ2 (with constant b) or unconditional. The general case remains open. Chapters 12–14 provide an exposition of our state of knowledge on the thin shell and Kannan-Lovász-Simonovits (or KLS for short) conjectures. Historical and other information about the thin shell conjecture and its connections with the central limit problem is given in Chapter 12. We present the work of Anttila, Ball and Perissinaki and various central limit theorems for isotropic convex bodies which would follow from thin shell estimates. This question has been studied by many authors and has been verified in some special cases. Klartag was the first to give a positive answer in full generality. In fact, aside from the immediate consequence of a general thin shell estimate that, as we mentioned again earlier in the Introduction, is that most one-dimensional marginals are close to Gaussian distributions, Klartag also established normal approximation for multidimensional marginal distributions. In Section 12.4 we give an account of Klartag’s positive answer to the thin shell conjecture for the class of unconditional isotropic log-concave random vectors, which is one of the special cases for which this question was fully verified. Klartag proved that if K is an unconditional isotropic convex body in Rn,then √ 2 E − 2 2 σK := μK x 2 n C , where C 4 is an absolute positive constant. We also describe a result of Eldan and Klartag which shows that the thin shell conjecture is stronger than the hyperplane conjecture and implies it; more precisely, they prove that Ln Cσn where n σn := max{σμ : μ is isotropic log-concave measure on R }, and, hence, any estimate one establishes for the former conjecture immediately holds for the latter too. Chapter 13 is then devoted to a complete proof of the 1/3 currently best known estimate for the thin shell conjecture, σn Cn ,whichis due to Guédon and E. Milman. Chapter 14 is devoted to the Kannan-Lovász-Simonovits conjecture. We first introduce various isoperimetric constants which provide information on the inter- play between a log-concave probability measure μ and the underlying Euclidean metric (the Cheeger constant Isμ, the Poincaré constant Poinμ, the exponential concentration constant Expμ and the first moment concentration constant FMμ) and we discuss their relation. Complementing classical results of Maz’ya, Cheeger, Gromov, V. Milman, Buser, Ledoux and others, E. Milman established the equivalence of all four constants in the log-concave setting: one has

Isμ Poinμ Expμ FMμ for every log-concave probability measure, where a b means that c1a b c2b for some absolute constants c1,c2 > 0. As an application, E. Milman obtained stability results for the Cheeger constant of convex bodies. Loosely speaking, if K n and T are two convex bodies in R and if |K| |T | |K ∩ T |,thenIsK IsT .We introduce the KLS-conjecture in Section 14.4 and we present the first general lower bounds for Isμ in the isotropic log-concave case.√ From the work of Kannan, Lovász and Simonovits and Bobkov one has that nIsμ c,wherec>0 is an absolute constant. Actually, Bobkov proved that √ √ 4 n σμIsμ c; PREFACE xix this provides a direct link between the KLS-conjecture and the thin shell conjecture: combined with the thin shell estimate of Guédon and E. Milman his result leads 5/12 to the bound n Isμ c. In Section 14.5 we describe Klartag’s logarithmic in the dimension lower bound for the Poincaré constant√ PoinK of an unconditional Rn c isotropic convex body K in ; one has IsK PoinK log n ,wherec>0isan absolute positive constant. We close this discussion with a result of Eldan which, again, connects the thin shell conjecture with the KLS-conjecture: there exists an absolute constant C>0 such that 1 n σ2 C log n k . Is2 k n k=1 Taking into account the result of Guédon and E. Milman, one gets the currently −1 1/3 best known bound for Isn:Isn Cn log n. In the last two chapters of the book we are concerned with two more approaches to the main questions in this theory. Chapter 15 is devoted to a probabilistic approach and related conjectures of Latala and Wojtaszczyk on the geometry of log-concave measures. The starting point is an infimum convolution inequality which was first introduced by Maurey when he gave a simple proof of Talagrand’s two level concentration inequality for the product exponential measure. In general, if μ is a probability measure and ϕ is a non-negative measurable function on Rn, one says that the pair (μ, ϕ)hasproperty (τ) if, for every bounded measurable function f on Rn, efϕdμ e−f dμ 1, Rn Rn where fϕ is the infimum convolution of f and ϕ, defined by (fϕ)(x) = inf {f(x − y)+ϕ(y): y ∈ Rn} . That the property (τ) is satisfied by a pair (μ, ϕ) is directly related to concentration properties of the measure μ since the former property implies that, for every measurable A ⊆ Rn and every t>0, we have −1 −t μ (x/∈ A + Bϕ(t)) (μ(A)) e , where Bϕ(t)={ϕ t}. Therefore, given a measure μ it makes sense to ask for the optimal cost function ϕ for which we have that (μ, ϕ) has property (τ). The first main observation is that, if we restrict ourselves to even probability measures μ and convex cost functions ϕ, then the (pointwise) largest candidate for a cost function ∗ is the Cramer transform Λμ of μ; this is the Legendre transform of the logarithmic Laplace transform of μ. In the setting of log-concave probability measures, ∗ the conjecture Latala and Wojtaszczyk formulate is that the pair (μ, Λμ) always has property (τ). A detailed analysis shows that this conjecture would imply an affirmative answer to most of the conjectures addressed in this book: among them, the thin shell conjecture as well as the hyperplane conjecture. The problems that are raised through this approach are very interesting and challenging. An affirmative answer has been given for some rather restricted classes of measures: even n log-concave product measures, uniform distributions on p -balls and rotationally invariant log-concave measures. In the last chapter we give an account of K. Ball’s information theoretic ap- − proach, which is based on the study of the Shannon entropy Ent(X)= Rn f log f of an isotropic random vector X with density f. Itisknownthat,amongall xx PREFACE isotropic random vectors, the standard Gaussian random vector G has the largest entropy, and the main observation is that comparing the entropy gap Ent X√+Y − 2 Ent(X)(withY being an independent copy of X)toEnt(G) − Ent(X)providesa link between the KLS-conjecture and the hyperplane conjecture. A first result of this type was obtained by Ball, Barthe and Naor for one-dimensional distributions. The main result of this chapter is a recent high-dimensional analogue for isotropic log-concave random vectors, which is due to Ball and Nguyen: if X is an isotropic log-concave random vector in Rn and its density f satisfies the Poincaré inequality with constant κ>0, then X + Y κ Ent √ − Ent(X) (Ent(G) − Ent(X)) , 2 4(1 + κ) where G is a standard Gaussian random vector in Rn. In addition, Ball and Nguyen 17/κ show that this implies LX e . Thus, for each individual isotropic log-concave distribution X, a lower bound for the Poincaré constant implies a bound for the isotropic constant.

The book is primarily addressed to readers who are familiar with the basic theory of convex bodies and the asymptotic theory of finite dimensional normed spaces as these are developed in the books of Milman and Schechtman and of Pisier. Nevertheless, we have included an introductory chapter where all the prerequisites are described; short proofs are also provided for the most important results that areusedinthesequel. This book grew out of our working seminar in the last fifteen years. Among the main topics that were discussed in our meetings were the developments on the basic questions addressed in the text. A large part of the material forms the basis of PhD and MSc theses that were written at the University of Athens and the University of Crete. We are grateful to Nikos Dafnis, Dimitris Gatzouras, Mari- anna Hartzoulaki, Labrini Hioni, Lefteris Markessinis, Nikos Markoulakis, Grigoris Paouris, Eirini Perissinaki, Pantelis Stavrakakis and Antonis Tsolomitis for their active participation in our seminar, for collaborating with us at various stages, for numerous discussions around the subject of this book and for their friendship over the years. We are very grateful to Sergei Gelfand for many kind reminders regarding this project and for believing that we would be able to complete it. We are also grateful to Christine Thivierge and Luann Cole for their precious help in the prepa- ration of this book. Finally, we would like to acknowledge partial support from the ARISTEIA II programme of the General Secretariat of Research and Technology of Greece during the final stage of this project.

Athens, February 2014

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Subject Index

(T,E)-symmetrization, 214 q∗-parameter, 186 H (κ, τ)-regular body, 234 r -parameter, 264 (τ)-property, 512 s-concave 0 − 1-polytope, 383 measure, 66 B-theorem, 190 Iq(K, C), 234 Alexandrov Iq(μ), 182 inequalities for quermassintegrals, 15 Kp(f) uniqueness theorem, 16 Ball’s bodies, 84 Alexandrov-Fenchel inequality, 15 convexity of, 87 asymptotic shape, 357 volume, 90 axis of inertia, 215 Ln monotonicity, 220 Ball Lq-Rogers-Shephard inequality, 179 normalized version of the Brascamp-Lieb Lq-aﬃne isoperimetric inequality, 175 inequality, 24 L -centroid body, 174 q reverse isoperimetric inequality, 24 volume, 181 Barthe M-ellipsoid, 56 reverse Brascamp-Lieb inequality, 22 M-position, 57 barycenter, 2 of order α,57 of a function, 64 MM∗-estimate, 52 of a measure, 64 QS(X), 54 basis Z+(f), 437 p of inertia axes, 215 Zq-projection formula, 179 Binet ellipsoid, 107 Λp(μ), 257 Blaschke α-regular measure, 529 formula, 129 β-center, 384 selection theorem, 3 -norm, 50 Blaschke-Santal´o inequality, 11 γ-concave Borell lemma, 10, 80 function, 66 Bourgain φ(K) upper bound for ψ2-bodies, 123 functional, 133 upper bound for the isotropic constant, ψ2-body, 284 117 ψα Bourgain-Milman inequality, 55 body, 114 Brascamp-Lieb inequality, 22 estimate, 79 Brunn concavity principle, 4 measure, 79 Brunn-Minkowski inequality, 4 ψα-norm, 78 Busemann formula, 131 ε-concentration hypothesis, 394 Busemann-Petty problem, 132 d∗-parameter, 193 j-th area measure, 14 Cauchy formula, 16 p-median, 273 Cauchy-Binet formula, 157 q−c-parameter, 231 central limit problem, 389

585 586 SUBJECT INDEX centroid bodies super-Gaussian, 272 covering numbers, 273 discrete cube, 26 centroid body, 174 distance normalized, 279 Banach-Mazur, 3 Cheeger constant, 462 geometric, 3 Cheeger inequality, 462 Hausdorff, 3 comparison Wasserstein, 410 weak and strong moments, 533 dual concentration norm, 3 exponential, 27 space, 3 first moment, 473 duality of entropy normal, 26 theorem, 37 of measure, 25 Dudley-Fernique decomposition, 121 concentration function, 26 Dvoretzky theorem, 43 concentration inequality, 531 Dvoretzky-Rogers lemma, 20 conjecture ellipsoid, 17 Gaussian correlation, 159 M-ellipsoid, 56 hyperplane, 104 Binet, 107 infimum convolution, 527 maximal volume, 17 isotropic constant, 104 minimal volume, 17 Kannan-Lovász-Simonovits, 480 empirical process Mahler, 55 boundedness, 544 random simplex, 128 entropy thin shell, 403 duality conjecture, 37 weak and strong moments, 534 duality theorem, 37 constant monotonicity, 559 Cheeger, 462 Euclidean exponential concentration, 471 norm, 1 isoperimetric, 462 unit ball, 1 isotropic, 75 unit sphere, 1 log-Lipschitz, 443 exponential Poincaré, 465 inequality, 471 super-Gaussian, 272 exponential concentration contact point, 17 constant, 471 convex body, 2 first moment concentration, 473 set, 2 Fisher information, 552 convex body, 2 formula 2-convex, 147 Zq-projection, 179 almost isotropic, 92 Blashke, 129 barycenter, 2 Busemann, 131 centered, 2 Cauchy, 16 isotropic, 72 Cauchy-Binet, 157 mean width, 12 Fourier inversion, 170 mixed width, 185 Holmstedt, 368 polar, 2 Kubota, 13 position, 16 Saint-Raymond, 151 small diameter, 117 Steiner, 13 symmetric, 2 Fourier unconditional, 139 inversion formula, 170 width, 2 transform, 163 convolution, 275 Fradelizi inequality, 67 covariance matrix, 76 function covering number, 34 γ-concave, 66 Cramer transform, 523 ψK , 292 barycenter, 64 difference body, 8 centered, 64 direction concentration, 26 SUBJECT INDEX 587

indicator, 7 Urysohn, 12 isotropic, 76 Vaaler, 161 Lipschitz, 27 Young, 22 logarithmically concave, 64 inertia radial, 2 axis, 215 support, 2 moments, 105 Walsh, 51 inertia matrix, 76 functional infimum convolution, 512 φ(K), 133 property, 524 infimum convolution conjecture, 527 Gauss space, 26 information Gaussian Fisher, 552 correlation conjecture, 159 isomorphic slicing problem, 244 isoperimetric inequality, 31 isoperimetric measure, 26 coefficient, 480 Grünbaum lemma, 71 constant, 462 Grassmann manifold, 2 problem, 29 isoperimetric inequality Haar measure, 2 discrete cube, 32 Hausdorff for convex bodies, 10 limit, 16 in Gauss space, 31 Hausdorff metric, 3 spherical, 29 hyperplane conjecture, 104 isotropic hyperplane sections, 105 condition, 73 hypothesis constant of a convex body, 75 ε-concentration, 394 constant of a measure, 76 variance, 402 convex body, 72 indicator function, 7 function, 76 inequality measure, 17, 75 position, 73 Lq-Rogers-Shephard, 179 Alexandrov-Fenchel, 15 random vector, 78 Barthe, 22 isotropic constant Blaschke-Santaló, 11 monotonicity, 220 Bobkov-Nazarov, 202 stability, 117 Borell, 69 isotropic convex body Brascamp-Lieb, 22 (κ, τ)-regular, 234 Brunn-Minkowski, 4 circumradius, 108 Busemann, 398 covering numbers, 109 Cheeger, 462, 522 inradius, 108 concentration, 531 with small diameter, 284 dual Sudakov, 35 John Dudley, 40 representation of the identity, 18 exponential, 471 theorem, 19 Fradelizi, 67 Hadamard, 145 k(X) Kahane-Khintchine, 34 critical dimension, 45 Khintchine, 33 Kahane-Khintchine inequality, 34 log-Sobolev, 431 Kannan-Lovasz-Simonovits conjecture, 480 Loomis-Whitney, 140 Kashin decomposition, 50 Lyapunov, 69 Khintchine inequality, 33 MacLaurin, 157 Klartag Minkowski, 13 bound for the isotropic constant, 252 non-commutative Khintchine, 352 isomorphic slicing problem, 251 Poincaré, 192, 465 KLS-conjecture, 480 Prékopa-Leindler, 5 Knothe map, 7 Rogers-Shephard, 8 Kubota formula, 13 Sudakov, 35 Talagrand, 516 Löwner position, 20 588 SUBJECT INDEX

Lévy family, 26 Milman Lévy mean, 27 M-ellipsoid, 56 Legendre transform, 523 isomorphic symmetrization, 55 lemma low M ∗-estimate, 52 Arias de Reyna-Ball-Villa, 30 quotient of subspace theorem, 54 Borell, 10, 80 reverse Brunn-Minkowski inequality, 57 Dvoretzky-Rogers, 20 version of Dvoretzky theorem, 44 Grünbaum, 71 minimal Lewis, 50 mean width, 21 Lovász-Simonovits, 94, 481 surface, 21 Lozanovskii, 146 surface invariant, 21 Sidák, 158 Minkowski Slepian, 40 content, 10, 29, 462 Lipschitz function, 27 existence theorem, 15 localization lemma, 94, 481 inequality for mixed volumes, 13 log-concave map, 16 density, 64 sum, 2 function, 64 theorem on mixed volumes, 12 measure, 64 mixed log-Lipschitz constant, 443 area measure, 14 logarithmic Laplace transform, 249 volume, 13 Loomis-Whitney inequality, 140 mixed width, 185 monotonicity Mahler conjecture, 55 of Ln, 220 majorizing measure theorem, 42 movement manifold type-i, 443 Grassmann, 2 Neumann Laplacian, 408 map norm Knothe, 7 ψ -norm, 78 Minkowski, 16 α Orlicz, 78 marginal, 178 Rademacher projection, 52 matrix trace dual, 50 covariance, 76 unconditional, 139 inertia, 76 unitarily invariant, 151 mean width, 12 measure order statistics, 536 α-regular, 529 Orlicz norm, 78 s-concave, 66 Ornstein-Uhlenbeck semigroup, 553 barycenter, 64 centered, 64 packing number, 34 concentration, 25 Paouris convolution, 275, 513 deviation inequality, 182 even, 64 small ball estimate, 190 Gaussian, 26 parameter Haar, 2 d∗, 193 isotropic, 17, 75 q−c, 231 H logarithmically concave, 64 q−c, 268 majorizing, 42 q∗, 186 marginal, 178 q,c, 267 H mixed area, 14 q , 267 H peakedness, 161 r , 264 product exponential, 516 Pisier surface area, 14 α-regular M-position, 57 symmetric, 64 norm of the Rademacher projection, 52 symmetric exponential, 515 Poincaré constant, 465 metric Poincaré inequality, 192, 465 Hausdorff, 3 polar body, 2 metric probability space, 25 polynomials SUBJECT INDEX 589

Khintchine type inequalities, 97 Rogers-Shephard ineqality, 8 polytope, 15 0 − 1, 383 Saint-Raymond random, 357 formula, 151 polytopes Schatten class, 150 with few facets, 160 semigroup with few vertices, 156 Ornstein-Uhlenbeck, 553 position separated set, 34 M-position, 57 set isotropic, 19, 73 convex, 2 John, 17 separated, 34 Löwner, 20 star-shaped, 2 Lewis, 149 Sidák lemma, 158 minimal mean width, 21 Slepian lemma, 40 minimal surface, 21 slicing problem, 104 ◦ of a convex body, 16 reduction to I1(K, Zq (K)), 234 Prékopa-Leindler inequality, 5 reduction to q−c(K, δ), 231 principle reduction to bounded volume ratio, 223 Brunn, 4 space problem normed, 3 Busemann-Petty, 132 spectral gap, 465 Sylvester, 128 spherical process cone, 46 empirical, 348 isoperimetric inequality, 29 Gaussian, 38 symmetrization, 30 sub-Gaussian, 38 star body, 2 projection body, 16 Steiner property (τ), 512 formula, 13 symmetrization, 4 quermassintegrals, 13 sub-Gausian Alexandrov inequalities, 15 direction, 271 normalized, 14 subindependence of coordinate slabs, 405 Rademacher Sudakov inequality, 35 functions, 33 super-Gaussian projection, 51 constant, 272 radial function, 2 direction, 272 Radon transform, 163 support function, 2 spherical, 163 surface area, 10 random measure, 14 polytope, 357 Sylvester’s problem, 128 random polytopes symmetrization asymptotic shape, 368 (T,E), 214 isotropic constant, 377 isomorphic, 55 volume radius, 375 spherical, 30 random vector Steiner, 4, 358 isotropic, 78 reverse Talagrand Brascamp-Lieb inequality, 22 comparison theorem, 42 Brunn-Minkowski inequality, 57 isoperimetric inequality for the discrete isoperimetric inequality, 24 cube, 32 Santaló inequality, 55, 254 majorizing measure theorem, 42 Urysohn inequality, 52 theorem Riemannian Adamczak-Litvak-Pajor-Tomczak, 334, manifold, 418 335, 347 metric, 417 Alesker, 115 package, 417 Alexandrov, 16 package, isomorphism, 417 Anttila-Ball-Perissinaki, 397 package, log-concave, 417 Artstein-Ball-Barthe-Naor, 559 590 SUBJECT INDEX

Artstein-Milman-Szarek, 37 Paouris, 179, 181–183, 187, 188, 190 Bakry-Ledoux, 469 Petty, 21 Ball, 24, 87, 91, 166, 169, 550 Pisier, 52, 57, 227 Ball-Nguyen, 552 Rudelson, 352 Ball-Perissinaki, 405 Szarek-Tomczak, 48 Blaschke, 3 Talagrand, 32, 42 Bobkov, 391, 395, 506 thin shell Bobkov-Nazarov, 142, 202, 203, 306, 307 conjecture, 403 Borell, 64, 66 transform Bourgain, 97, 118, 121, 123, 350 Cramer, 523 Bourgain-Klartag-Milman, 215, 220, 223 Fourier, 163 Bourgain-Milman, 55 Legendre, 523 Busemann, 398 Radon, 163 Carl-Pajor, 158 type-i movement, 443 Cordero-Fradelizi-Maurey, 190 unconditional convex body, 139 Dafnis-Giannopoulos-Guédon, 381 unitarily invariant norm, 151 Dafnis-Giannopoulos-Tsolomitis, 368, Urysohn inequality, 12 375 Dafnis-Paouris, 227, 232, 233 variance hypothesis, 402 Dudley-Sudakov, 39 volume, 1 Dvoretzky, 43 mixed, 13 Dyer-Füredi-McDiarmid, 384 of Lq-centroid bodies, 181 Eldan, 487 radius, 9 Eldan-Klartag, 415 ratio, 17 Figiel-Tomczak, 51 sections of the cube, 166 Fleury, 433 volume product, 11 Fradelizi-Guédon, 94 volume ratio, 47 Gatzouras-Giannopoulos, 386 outer, 144 Gatzouras-Giannopoulos-Markoulakis, theorem, 48 384, 386 theorem, global form, 49 Giannopoulos-Milman, 21 uniformly bounded, 47 Giannopoulos-Paouris-Valettas, 273, 284 Giannopoulos-Paouris-Vritsiou, 234 Walsh functions, 51 Gluskin, 158 Wasserstein distance, 410 Gromov-V. Milman, 471 weak and strong moments comparison, 533 Guédon-E. Milman, 436, 443 Wiener process, 489 John, 18, 19 zonoid, 16 König-Meyer-Pajor, 155 Löwner position, 149 König-Milman, 37 Lewis position, 149 Kannan-Lovász-Simonovits, 480 minimal mean width position, 150 Kanter, 161 Kashin, 48 Klartag, 91, 207, 244, 251, 407, 485 Klartag-E. Milman, 260, 262, 453 Klartag-Kozma, 380 Klartag-Vershynin, 194, 205 Koldobsky, 164, 165 Latala, 82 Latala-Oleszkiewicz, 204 Litvak-Milman-Schechtman, 183 Lutwak-Yang-Zhang, 175, 181 Meyer-Pajor, 161 Milman, E., 474–477, 488 Milman, V. D., 44, 52, 54, 56 Milman-Pajor, 9 Milman-Schechtman, 45 Minkowski, 12, 15 Pajor-Tomczak, 35 Author Index

Adamczak, R., 197, 334, 335, 347, 355, 547, Borell, C., 10, 31, 59, 60, 65, 66, 69, 100, 565 568 Aldaz, J. M., 134, 565 Bourgain, J., 55, 59, 61, 97, 101, 111, 117, Alesker, S., 115, 135, 183, 354, 565 118, 121, 123, 134–136, 171, 214, 215, Alexandrov, A. D., 14–16, 58, 565 220, 223, 242, 349, 354, 379, 568 Alonso-Gutiérrez, D., 134, 137, 155, 156, Brascamp, H. J., 22, 59, 569 158, 171, 377, 387, 388, 565 Brazitikos, S., 456, 569 Anderson, T. W., 171, 565 Brehm, U., 422, 569 Anttila, M., 397, 422, 423, 565 Brenier, Y., 410, 569 Aomoto, A., 171, 565 Brezis, H., 1, 569 Arias de Reyna, J., 30, 60, 171, 565 Brunn, H., 4, 58, 569 Artstein-Avidan, S., 37, 60, 273, 559, 563, Buchta, C., 384, 569 565 Burago, Y. D., 58, 569 Aubrun, G., 134, 355, 566 Busemann, H., 131, 132, 136, 398, 422, 569 Buser, P., 467, 509, 569 Bárány, I., 171, 371, 384, 385, 566 Badrikian, A., 61, 566 Campi, S., 209, 569 Bakry, D., 469, 509, 566 Carbery, A., 101, 134, 569 Ball, K. M., 18, 24, 30, 58–60, 84, 100, 101, Carl, B., 158, 171, 569 134, 136, 137, 149, 155, 166, 169, 171, Carlen,E.A.,569 172, 397, 405, 422, 423, 550, 552, 559, Cauchy, A. L., 16 562, 563, 565, 566 Chakerian, G. D., 59, 569 Banaszczyk, W., 210 Chavel, I., 509, 569 Barlow, R. E., 100, 566 Cheeger, J., 462, 466, 509, 570 Barron, A. R., 566 Chevet, S., 61, 566 Barthe, F., 22, 23, 59, 60, 101, 137, 172, Cordero-Erausquin, D., 171, 190, 210, 273, 312, 368, 423, 510, 559, 563, 565–567 567, 570 Bastero, J., 156, 332, 422, 565, 567 Costa, M., 562, 570 Batson, J., 355, 567 Cover, T., 562, 570 Bayle, V., 509, 567 Csiszár, I., 570 Beckenbach, E. F., 567 Bellman, R., 567 Dafnis, N., xx, 221, 227, 232, 233, 242, 367, Bernués, J., 156, 422, 565, 567 368, 374, 375, 381, 387, 388, 565, 570 Berwald, L., 567 Dalla, L., 135, 570 Blachman, N. M., 567 Dar, S., 117, 120, 121, 135, 155, 156, 171, Blaschke, W., 3, 11, 58, 135, 387, 567 570 Bobkov, S. G., 60, 71, 101, 141, 143, 170, Das Gupta, S., 59, 570 171, 202, 203, 210, 241, 306, 307, 312, Dembo, A., 562, 570 381, 391, 395, 402, 422, 423, 506, 509, Diaconis, P., 390, 422, 570 510, 562, 567, 568 Dudley, R. M., 39, 40, 61, 570 Bogachev, V., 1, 568 Durrett, R., 570 Bolker, E. D., 149, 171, 568 Dvoretzky, A., 20, 43, 59, 61, 570 Bonnesen, T., 58, 568 Dyer, M. E., 384, 385, 387, 570

591 592 AUTHOR INDEX

Eldan, R., 415, 423, 459, 487, 510, 570 Jessen, B., 58, 571 Emery, M., 566 John, F., 18, 19, 59, 574 Erhard, A., 60, 571 Johnson, W. B., 574 Junge, M., 155, 171, 574 Füredi, Z., 171, 384, 385, 387, 566, 570 Feller, W., 1, 571 König, H., 37, 60, 150, 155, 171, 575 Fenchel, W., 15, 58, 568, 571 Kadlec, J., 408, 574 Fernique, X., 61, 571 Kahane, J.-P., 34, 60, 574 Figiel, T., 51, 60, 61, 571 Kaibel, V., 384, 571 Fleiner, T., 384, 571 Kannan, R., 101, 109, 135, 334, 354, 510, Fleury, B., 433, 571 571, 574 Folland, G. B., 571 Kanter, M., 161, 171, 574 Fradelizi, M., 67, 94, 100, 101, 137, 190, Kashin, B., 48, 50, 61, 574 210, 216, 219, 273, 567, 570, 571 Khintchine, A., 34, 60, 574 Freedman, D., 390, 422, 570 Klartag, B., 91, 100, 111, 135, 147, 171, Frieze, A., 571 193–195, 205, 207, 209, 210, 214, 215, Fukuda, K., 384, 571 220, 221, 223, 241, 242, 244, 249, 251, 256, 260, 262–264, 270, 310, 311, 332, Gardner, R. J., 58, 136, 137, 571 372, 377, 380, 388, 407, 415, 423, 429, Gatzouras, D., xx, 384–386, 571 453, 459, 485, 510, 568, 571, 574, 575 Giannopoulos, A., 17, 21, 58, 59, 100, Knothe, H., 7, 59, 575 135–137, 171, 234, 242, 254, 270, 273, Koldobsky, A., 137, 163, 172, 402, 422, 423, 279, 280, 284, 301, 310, 322, 332, 354, 567, 571, 575 359, 367, 368, 374, 375, 381, 384–388, Kozma, G., 377, 380, 388, 575 570–572 Krasnosel’skii, M. A., 100, 575 Giertz, M., 136, 572 Kubota, T., 13 Gilbarg, D., 572 Kullback, S., 575 Gluskin, E. D., 158, 171, 572 Kuperberg, G., 61, 137, 575 Goodey, P. R., 171, 572 Kuwert, E., 509, 575 Gordon, Y., 53, 61, 572 Kwapien, S., 60, 575 Gozlan, N., 546, 573 Lévy, P., 30, 60, 576 Grünbaum, B., 71, 100, 573 Larman, D. G., 135, 136, 371, 566, 570, 575 Groemer, H., 58, 135, 163, 359, 387, 573 Latala, R., 82, 101, 197, 204, 210, 510, 533, Gromov, M., 60, 100, 171, 471, 573 536, 544, 547, 565, 576 Gronchi, P., 209, 569 Ledoux, M., 60, 171, 254, 347, 348, 467, Gross, L., 458, 573 469, 509, 566, 576 Grothendieck, A., 43, 59 Lehec, J., 546, 576 Gruber, P. M., 1, 58, 573 Leindler, L., 5, 59, 576 Guédon, O., 94, 101, 210, 312, 354, 368, Lewis, D. R., 61, 149, 171, 576 381, 388, 436, 443, 567, 570, 571, 573 Li, P., 509, 510, 576 Lichnerowicz, A., 408, 576 Hörmander, L., 408, 574 Lieb, E. H., 22, 59, 563, 569, 576 Haagerup, U., 34, 60, 573 Lifshits, M. A., 199, 422, 575, 576 Hadwiger, H., 58, 136, 573 Lindenstrauss, J., 58, 60, 100, 171, 379, Hardy, G. H., 69, 573 568, 571, 574, 576 Hargé, G., 171, 573 Littlewood, J. E., 60, 69, 573, 576 Harper, L. H., 32, 60, 573 Litvak, A. E., 183, 197, 210, 332, 334, 335, Hartzoulaki, M., xx, 109, 135, 314, 332, 347, 355, 367, 369, 387, 547, 565, 576 354, 359, 367, 387, 572, 573 Loomis, L. H., 140, 170, 577 Helffer, B., 408, 574 Lovász, L., 101, 109, 135, 334, 354, 510, Hensley, D., 100, 106, 574 574, 577 Henstock, R., 58, 574 Lozanovskii, G. J., 146, 577 Hernandez Cifre, M. A., 387, 565 Lust-Piquard, F., 352, 577 Hinow, P., 422, 569 Lusternik, L., 58, 577 Hioni, L., xx Luttinger, J. M., 59, 569 Holmstedt, T., 368, 574 Lutwak, E., 136, 175, 181, 209, 241, 577 Houdré, C., 509, 568 Huet, N., 510, 574 Müller, C., 579 AUTHOR INDEX 593

Müller, D., 134, 579 310, 311, 322, 332, 355, 359, 371, 372, Müller, J., 384, 569 387, 388, 422, 565, 570–573, 577, 579 Macbeath, A. M., 58, 359, 387, 574, 577 Papadimitrakis, M., 59, 136, 171, 572, 579 Madiman, M., 562, 568 Payne, L. E., 510, 580 Mahler, K., 59, 61, 577 Pelczynski, A., 172, 579 Markessinis, E., xx, 59, 577 Perissinaki, E., xx, 397, 405, 422, 423, 565, Markoulakis, N., xx, 384–386, 572 566 Marshall, A. W., 100, 566, 577 Petty, C. M., 22, 59, 132, 136, 569, 580 Maurey, B., 32, 60, 137, 190, 210, 273, 516, Pichorides, S. K., 136, 580 546, 567, 570, 577 Pilipczuk, M., 423, 580 Maz’ya, V. G., 462, 466, 509, 577 Pinsker, M. S., 580 McDiarmid, C., 384, 385, 387, 570 Pisier, G., 1, 52, 57–59, 61, 227, 352, 577, Meckes, E., 422, 577 580 Meckes, M. W., 422, 577 Pitt, L., 159, 171, 580 Mehta, M. L., 171, 577 Pivovarov, P., 311, 315, 332, 364, 387, 388, Mendelson, S., 312, 355, 368, 567, 577 580 Meyer, M., 11, 59, 61, 150, 155, 161, 162, Podkorytov, A. N., 167, 172, 579 171, 569, 573, 575, 577 Prékopa, A., 5, 59, 580 Milman, E., 135, 147, 155, 171, 201, 241, Prochno, J., 387, 565 256, 260, 262–264, 270, 325, 330, 372, Proschan, F., 100, 566, 577 422, 436, 443, 453, 474, 475, 477, 488, Rao, M. M., 100, 580 573, 575, 578 Reisner, S., 61, 573, 580 Milman, V. D., 1, 9, 17, 21, 37, 43–45, 52, Ren, Z. D., 100, 580 54–56, 58–61, 100, 101, 105, 109, 111, Rinott, Y., 171, 580 134, 135, 170, 171, 183, 185, 209, 210, Rockafellar, R. T., 1, 580 214, 215, 220, 221, 223, 228, 240, 242, Rogers, C. A., 8, 20, 43, 59, 136, 570, 575, 273, 281, 325, 332, 354, 371, 379, 471, 580 566, 568, 569, 571–573, 576, 578 Romik, D., 422, 579 Milman,E., 476 Rosales, C., 509, 567 Minkowski, H., 4, 12, 13, 15, 58, 578 Rote, G., 384, 571 Montgomery-Smith, S. J., 579 Rotem, L., 100, 580 Rothaus, O. S., 462, 509 Naor, A., 134, 172, 312, 368, 422, 559, 563, Rudelson, M., 332, 333, 352, 354, 367, 369, 565–567, 579 387, 573, 577, 580 Nazarov, F., 61, 101, 141, 143, 167, 170, Rutickii, J. B., 100, 575 172, 202, 203, 210, 241, 306, 307, 312, Ryabogin, D., 172, 575 381, 568, 579 Nguyen, V. H., 423, 552, 563, 566 Saint-Raymond, J., 61, 151, 152, 171, 581 Santaló, L. A., 11, 59, 581 Ohmann, D., 58, 573 Saroglou, C., 59, 135, 577, 581 Oksendal, B., 579 Schechtman, G., 1, 45, 58, 60, 61, 101, 171, Oleszkiewicz, K., 172, 197, 204, 210, 565, 183, 185, 210, 576, 578, 581 576, 579 Schlumprecht, T., 171, 571, 581 Olkin, I., 100, 577 Schmidt, E., 60, 581 Schmuckenschläger, M., 170, 171, 210, 581 Pólya, G., 69, 573 Schneider, R., 1, 58, 250, 581 Pór, A., 384, 385, 566 Shannon, C. E., 581 Pajor, A., 9, 11, 35, 53, 59–61, 100, 101, Shephard, G. C., 8, 59, 580 105, 109, 134, 135, 150, 155, 158, 161, Sherman, S., 171, 581 162, 170, 171, 197, 209, 228, 240, 280, Sidák, Z., 158, 171, 581 310, 332, 334, 335, 347, 355, 367, 369, Simonovits, M., 101, 135, 334, 354, 510, 371, 387, 547, 565, 566, 569, 572, 574, 577 575–577, 579 Sjöstrand, J., 408, 574 Paley, R. E. A. C., 274, 579 Slepian, D., 40, 61, 581 Paouris, G., xx, 59, 101, 113, 175, 177, 179, Sodin, M., 101, 579 181–183, 187, 188, 190, 195–197, 201, Sodin, S., 422, 459, 510, 578, 581 209, 210, 221, 227, 232–234, 242, 254, Soffer, A., 569 270, 273, 279, 280, 284, 298, 300, 301, Spielman, D., 355, 567 594 AUTHOR INDEX

Spingarn, J., 228, 581 Yau, S. T., 509, 510, 576 Srivastava, N., 355, 567, 581 Stam, A., 581 Zalgaller, V. A., 58, 569 Stavrakakis, P., xx, 322, 332, 456, 569, 572, Zhang, G., 136, 175, 181, 209, 241, 577, 583 581 Ziegler, G., 384, 583 Stein, E., 134, 581 Zinn, J., 171, 210, 581 Steiner, J., 13, 58, 582 Zong, C., 583 Sternberg, P., 509, 582 Zumbrun, K., 509, 582 Str¨omberg, J. O., 134, 582 Zvavitch, A., 172, 575 Stroock, D. W., 60, 582 Zygmund, A., 274, 579 Sudakov, V. N., 31, 35, 39, 40, 60, 61, 390, Zymonopoulou, M., 172, 575 422, 582 Szarek, S. J., 34, 37, 48, 60, 61, 273, 281, 383, 566, 578, 582

Talagrand, M., 32, 35, 42, 43, 60, 61, 117, 135, 254, 347, 348, 516, 544, 576, 582 Tao, T., 134, 579 Thomas, J., 562, 570 Tichy, R. F., 384, 569 Tomczak-Jaegermann, N., 1, 35, 36, 48, 51, 53, 58, 60, 61, 197, 254, 332, 334, 335, 347, 355, 367, 369, 387, 547, 565, 571, 576, 579, 582 Trudinger, N. S., 572 Tsirelson, B. S., 31, 60, 582 Tsolomitis, A., xx, 135, 322, 332, 354, 359, 367, 368, 374, 375, 387, 570, 572 Tzafriri, L., 100, 171, 576

Urysohn, P. S., 12, 59, 61, 582

Vaaler, J. D., 161, 171, 582 Valettas, P., 59, 273, 279, 284, 301, 310, 322, 332, 572, 581 Vempala, S., 577 Vershynin, R., 193–195, 205, 210, 332, 333, 355, 575, 581, 582 Villa, R., 30, 60, 171, 565 Villani, C., 410, 582 Vogt, H., 90, 422, 569, 582 Voigt, J. A., 90, 422, 569, 582 Volberg, A., 101, 579 von Weizs¨aker, H., 390, 582 Vritsiou, B.-H., 234, 242, 254, 266, 270, 322, 332, 572, 582

Weaver, W., 581 Weil, W., 171, 572 Weinberger, H. F., 510, 580 Werner, E., 371, 579 Whitney, H., 140, 170, 577 Wojtaszczyk, J., 422, 423, 510, 533, 547, 576, 582 Wolﬀ, P., 156, 565 Wright, J., 101, 569

Yang, D., 175, 181, 209, 241, 577 Yaskin, V., 172, 575 Published Titles in This Series

196 Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of Isotropic Convex Bodies, 2014 195 Ching-Li Chai, Brian Conrad, and Frans Oort, Complex Multiplication and Lifting Problems, 2013 194 Samuel Herrmann, Peter Imkeller, Ilya Pavlyukevich, and Dierk Peithmann, Stochastic Resonance, 2014 193 Robert Rumely, Capacity Theory with Local Rationality, 2013 192 Messoud Efendiev, Attractors for Degenerate Parabolic Type Equations, 2013 191 Grégory Berhuy and Frédérique Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication, 2013 190 Aleksandr Pukhlikov, Birationally Rigid Varieties, 2013 189 Alberto Elduque and Mikhail Kochetov, Gradings on Simple Lie Algebras, 2013 188 David Lannes, The Water Waves Problem, 2013 187 Nassif Ghoussoub and Amir Moradifam, Functional Inequalities: New Perspectives and New Applications, 2013 186 Gregory Berkolaiko and Peter Kuchment, Introduction to Quantum Graphs, 2013 185 Patrick Iglesias-Zemmour, Diffeology, 2013 184 Frederick W. Gehring and Kari Hag, The Ubiquitous Quasidisk, 2012 183 Gershon Kresin and Vladimir Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, 2012 182 Neil A. Watson, Introduction to Heat Potential Theory, 2012 181 Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay Representations, 2012 180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, 2012 179 Stephen D. Smith, Subgroup complexes, 2011 178 Helmut Brass and Knut Petras, Quadrature Theory, 2011 177 Alexei Myasnikov, Vladimir Shpilrain, and Alexander Ushakov, Non-commutative Cryptography and Complexity of Group-theoretic Problems, 2011 176 Peter E. Kloeden and Martin Rasmussen, Nonautonomous Dynamical Systems, 2011 175 Warwick de Launey and Dane Flannery, Algebraic Design Theory, 2011 174 Lawrence S. Levy and J. Chris Robson, Hereditary Noetherian Prime Rings and Idealizers, 2011 173 Sariel Har-Peled, Geometric Approximation Algorithms, 2011 172 Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon, The Classification of Finite Simple Groups, 2011 171 Leonid Pastur and Mariya Shcherbina, Eigenvalue Distribution of Large Random Matrices, 2011 170 Kevin Costello, Renormalization and Effective Field Theory, 2011 169 Robert R. Bruner and J. P. C. Greenlees, Connective Real K-Theory of Finite Groups, 2010 168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, Rings and Modules, 2010 167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster Algebras and Poisson Geometry, 2010 166 Kyung Bai Lee and Frank Raymond, Seifert Fiberings, 2010 165 Fuensanta Andreu-Vaillo, JoséM.Mazón, Julio D. Rossi, and J. Julián Toledo-Melero, Nonlocal Diffusion Problems, 2010 164 Vladimir I. Bogachev, Differentiable Measures and the Malliavin Calculus, 2010 163 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects, 2010

The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan- Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

For additional information and updates on this book, visit AMS on the Web www.ams.org/bookpages/surv-196 www.ams.org

SURV/196