 | Geometry of isotropic convex bodies / S.Brazitikos, A.Giannopoulos, P.Valettas, B.-H.Vritsiou. - Providence: American mathematical society, 2014. - xx, 594 p. - (Mathematical surveys and monographs; vol.196). - Bibliogr.: p.565-583. - Sub.-, auth. ind.: p.585-594. - ISBN 978-1-4704-1456-6
Шифр: (И/В18-G37) 02
|
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 ℓ-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 Va-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 ^2-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
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, Z°q(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 Santaló 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 tpa-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
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 Poincare 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
|
|
