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Geometry of Isotropic Convex Bodies [Hardback]

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  • Formāts: Hardback, 594 pages, height x width: 254x178 mm, weight: 1225 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-May-2014
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470414562
  • ISBN-13: 9781470414566
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Geometry of Isotropic Convex BodiesPalielināt
  • Formāts: Hardback, 594 pages, height x width: 254x178 mm, weight: 1225 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-May-2014
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470414562
  • ISBN-13: 9781470414566
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470414566.html
Keywords:
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 normalisation, 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-Lovasz-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.
Preface ix
Chapter 1 Background from asymptotic convex geometry
1(62)
1.1 Convex bodies
1(3)
1.2 Brunn--Minkowski inequality
4(4)
1.3 Applications of the Brunn-Minkowski inequality
8(4)
1.4 Mixed volumes
12(4)
1.5 Classical positions of convex bodies
16(6)
1.6 Brascamp-Lieb inequality and its reverse form
22(3)
1.7 Concentration of measure
25(9)
1.8 Entropy estimates
34(4)
1.9 Gaussian and sub-Gaussian processes
38(5)
1.10 Dvoretzky type theorems
43(7)
1.11 The l-position and Pisier's inequality
50(2)
1.12 Milman's low M*-estimate and the quotient of subspace theorem
52(3)
1.13 Bourgain-Milman inequality and the M-position
55(3)
1.14 Notes and references
58(5)
Chapter 2 Isotropic log-concave measures
63(40)
2.1 Log-concave probability measures
63(3)
2.2 Inequalities for log-concave functions
66(6)
2.3 Isotropic log-concave measures
72(6)
2.4 ψα-estimates
78(6)
2.5 Convex bodies associated with log-concave functions
84(10)
2.6 Further reading
94(6)
2.7 Notes and references
100(3)
Chapter 3 Hyperplane conjecture and Bourgain's upper bound
103(36)
3.1 Hyperplane conjecture
104(4)
3.2 Geometry of isotropic convex bodies
108(8)
3.3 Bourgain's upper bound for the isotropic constant
116(7)
3.4 The ψ2-case
123(5)
3.5 Further reading
128(6)
3.6 Notes and references
134(5)
Chapter 4 Partial answers
139(34)
4.1 Unconditional convex bodies
139(5)
4.2 Classes with uniformly bounded isotropic constant
144(6)
4.3 The isotropic constant of Schatten classes
150(5)
4.4 Bodies with few vertices or few facets
155(6)
4.5 Further reading
161(9)
4.6 Notes and references
170(3)
Chapter 5 Lq-centroid bodies and concentration of mass
173(40)
5.1 Lq-centroid bodies
174(8)
5.2 Paouris' inequality
182(8)
5.3 Small ball probability estimates
190(7)
5.4 A short proof of Paouris' deviation inequality
197(5)
5.5 Further reading
202(7)
5.6 Notes and references
209(4)
Chapter 6 Bodies with maximal isotropic constant
213(30)
6.1 Symmetrization of isotropic convex bodies
214(9)
6.2 Reduction to bounded volume ratio
223(4)
6.3 Regular isotropic convex bodies
227(4)
6.4 Reduction to negative moments
231(3)
6.5 Reduction to I1(K, Zoq(K))
234(5)
6.6 Further reading
239(3)
6.7 Notes and references
242(1)
Chapter 7 Logarithmic Laplace transform and the isomorphic slicing problem
243(28)
7.1 Klartag's first approach to the isomorphic slicing problem
244(5)
7.2 Logarithmic Laplace transform and convex perturbations
249(2)
7.3 Klartag's solution to the isomorphic slicing problem
251(3)
7.4 Isotropic position and the reverse Santalo inequality
254(2)
7.5 Volume radius of the centroid bodies
256(14)
7.6 Notes and references
270(1)
Chapter 8 Tail estimates for linear functionals
271(42)
8.1 Covering numbers of the centroid bodies
273(11)
8.2 Volume radius of the ψ2-body
284(8)
8.3 Distribution of the ψ2-norm
292(6)
8.4 Super-Gaussian directions
298(3)
8.5 ψα-estimates for marginals of isotropic log-concave measures
301(3)
8.6 Further reading
304(6)
8.7 Notes and references
310(3)
Chapter 9 M and M*-estimates
313(20)
9.1 Mean width in the isotropic case
313(9)
9.2 Estimates for M(K) in the isotropic case
322(8)
9.3 Further reading
330(2)
9.4 Notes and references
332(1)
Chapter 10 Approximating the covariance matrix
333(24)
10.1 Optimal estimate
334(15)
10.2 Further reading
349(5)
10.3 Notes and references
354(3)
Chapter 11 Random polytopes in isotropic convex bodies
357(32)
11.1 Lower bound for the expected volume radius
358(5)
11.2 Linear number of points
363(4)
11.3 Asymptotic shape
367(10)
11.4 Isotropic constant
377(4)
11.5 Further reading
381(6)
11.6 Notes and references
387(2)
Chapter 12 Central limit problem and the thin shell conjecture
389(36)
12.1 From the thin shell estimate to Gaussian marginals
391(6)
12.2 The log-concave case
397(5)
12.3 The thin shell conjecture
402(5)
12.4 The thin shell conjecture in the unconditional case
407(8)
12.5 Thin shell conjecture and the hyperplane conjecture
415(7)
12.6 Notes and references
422(3)
Chapter 13 The thin shell estimate
425(36)
13.1 The method of proof and Fleury's estimate
427(9)
13.2 The thin shell estimate of Guedon and E. Milman
436(22)
13.3 Notes and references
458(3)
Chapter 14 Kannan-Lovasz-Simonovits conjecture
461(50)
14.1 Isoperimetric constants for log-concave probability measures
462(11)
14.2 Equivalence of the isoperimetric constants
473(4)
14.3 Stability of the Cheeger constant
477(3)
14.4 The conjecture and the first lower bounds
480(5)
14.5 Poincare constant in the unconditional case
485(1)
14.6 KLS-conjecture and the thin shell conjecture
486(19)
14.7 Further reading
505(4)
14.8 Notes and references
509(2)
Chapter 15 Infimum convolution inequalities and concentration
511(38)
15.1 Property (τ)
512(11)
15.2 Infimum convolution conjecture
523(4)
15.3 Concentration inequalities
527(6)
15.4 Comparison of weak and strong moments
533(2)
15.5 Further reading
535(11)
15.6 Notes and references
546(3)
Chapter 16 Information theory and the hyperplane conjecture
549(16)
16.1 Entropy gap and the isotropic constant
550(2)
16.2 Entropy jumps for log-concave random vectors with spectral gap
552(7)
16.3 Further reading
559(3)
16.4 Notes and references
562(3)
Bibliography 565(20)
Subject Index 585(6)
Author Index 591
Silouanos Brazitikos, University of Athens, Greece.

Apostolos Giannopoulos, University of Athens, Greece.

Petros Valettas, Texas A & M University, College Station, TX.

Beatrice-Helen Vritsiou, University of Athens, Greece.