**Mathematical Surveys and Monographs**

Volume: 196;
2014;
594 pp;
Hardcover

MSC: Primary 52; 46; 60; 28;

Print ISBN: 978-1-4704-1456-6

Product Code: SURV/196

List Price: $134.00

Individual Member Price: $107.20

**Electronic ISBN: 978-1-4704-1526-6
Product Code: SURV/196.E**

List Price: $134.00

Individual Member Price: $107.20

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#### Supplemental Materials

# Geometry of Isotropic Convex Bodies

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*Silouanos Brazitikos; Apostolos Giannopoulos; Petros Valettas; Beatrice-Helen Vritsiou*

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.

#### Table of Contents

# Table of Contents

## Geometry of Isotropic Convex Bodies

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Chapter 1. Background from asymptotic convex geometry 122 free
- Chapter 2. Isotropic log-concave measures 6384
- Chapter 3. Hyperplane conjecture and Bourgain’s upper bound 103124
- Chapter 4. Partial answers 139160
- Chapter 5. 𝐿_{𝑞}-centroid bodies and concentration of mass 173194
- Chapter 6. Bodies with maximal isotropic constant Back Cover33234
- Chapter 7. Logarithmic Laplace transform and the isomorphic slicing problem Back Cover63264
- Chapter 8. Tail estimates for linear functionals Back Cover91292
- Chapter 9. 𝑀 and 𝑀*-estimates Back Cover133334
- Chapter 10. Approximating the covariance matrix Back Cover153354
- Chapter 11. Random polytopes in isotropic convex bodies Back Cover177378
- Chapter 12. Central limit problem and the thin shell conjecture Back Cover209410
- Chapter 13. The thin shell estimate Back Cover245446
- Chapter 14. Kannan-Lovász-Simonovits conjecture Back Cover281482
- Chapter 15. Infimum convolution inequalities and concentration Back Cover331532
- Chapter 16. Information theory and the hyperplane conjecture Back Cover369570
- Bibliography Back Cover385586
- Subject index Back Cover405606
- Author index Back Cover411612
- Back Cover Back Cover417618

#### Readership

Graduate students and research mathematicians interested in geometric and analytic study of convex bodies.