Geometry of isotropic convex bodies

618 Pages·2014·3.745 MB·English

by  Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, Beatrice-helen Vritsiou

Mathematical
Surveys
and
Monographs
Volume 196
Geometry of Isotropic
Convex Bodies
Silouanos Brazitikos
Apostolos Giannopoulos
Petros Valettas
Beatrice-Helen Vritsiou
American Mathematical Society
Geometry of Isotropic
Convex Bodies

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 Michael I. Weinstein
Michael A. Singer
2010 Mathematics Subject Classification. Primary 52Axx, 46Bxx, 60Dxx, 28Axx.
For additional informationand 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-HelenVritsiou.
pagescm. –(Mathematicalsurveysandmonographs;volume196)
Includesbibliographicalreferencesandindexes.
ISBN978-1-4704-1456-6(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
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Contents
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 (cid:2)-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. ψ -estimates 78
α
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 ψ -case 123
2
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. L -centroid bodies and concentration of mass 173
q
5.1. L -centroid bodies 174
q
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 I (K,Z◦(K)) 234
1 q
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 ψ -body 284
2
8.3. Distribution of the ψ -norm 292
2
8.4. Super-Gaussian directions 298
8.5. ψ -estimates for marginals of isotropic log-concave measures 301
α
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 Gu´edon and E. Milman 436
13.3. Notes and references 458
Chapter 14. Kannan-Lova´sz-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. Poincar´e 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 concen-
tratedinsomecanonicalwayandthemainquestioniswhether,undersomenatural
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 L >0 such that
K
(cid:2)
(cid:2)x,θ(cid:3)2dx=L2
K
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(orslicingproblem),whichisoneofthemainproblemsintheasymptotic
theory of convex bodies, and asks if there exists an absolute constant c > 0 such
that maxθ∈Sn−1|K ∩θ⊥| (cid:2) c for every convex body K of volume 1 in Rn that
has barycenter at the origin. This question was posed by Bourgain [99], who was
interestedinfindingL -boundsformaximaloperatorsdefinedintermsofarbitrary
p
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
L :=max{L :K is isotropic in Rn}(cid:3)C.
n K
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 conjec-
turecanbereformulatedinthelanguageoflogarithmically-concave(orlog-concave
ix