Asymptotic geometric analysis /

Asymptotic geometric analysis /

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Bibliographic Details
Author / Creator:Artstein-Avidan, Shiri, 1978- author.
Imprint:Providence, Rhode Island : American Mathematical Society, [2015]-
Description:volumes : illustrations ; 26 cm.
Language:English
Series:Mathematical surveys and monographs ; volume 202, 261
Mathematical surveys and monographs ; no. 202, 261.
Subject:
Geometric analysis.
Functional analysis.
Functional analysis.
Geometric analysis.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/10763264
Hidden Bibliographic Details
Other authors / contributors:Giannopoulos, Apostolos, 1963- author.
Milman, Vitali D., 1939- author.
ISBN:9781470421939
1470421933
9781470463601
1470463601
Notes:Includes bibliographical references and indexes.
Summary:"The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here ini a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more"--Back cover.
Table of Contents:
  • Part 1. [Without special title]
  • Chapter 1. Convex bodies: Classical geometric inequalities Chapter 2. Classical positions of convex bodies Chapter 3. Isomorphic isoperimetric inequalities and concentration of measure Chapter 4. Metric entropy and covering numbers estimates Chapter 5. Almost Euclidean subspaces of finite dimensional normed spaces Chapter 6. The ℓ-position and the Rademacher projection Chapter 7. Proportional theory Chapter 8. M-position and the reverse Brunn-Minkowski inequality Chapter 9. Gaussian approach Chapter 10. Volume distribution in convex bodies Appendix A. Elementary convexity Appendix B. Advanced convexity
  • Part 2. [Without special title]
  • Chapter 1. Functional inequalities and concentration of measure
  • Chapter 2. Isoperimetric constants of log-concave measures and related problems
  • Chapter 3. Inequalities for Gaussian measures
  • Chapter 4. Volume inequalities
  • Chapter 5. Local theory of finite dimensional normed spaces: type and cotype
  • Chapter 6. Geometry of the Banach-Mazur compactum
  • Chapter 7. Asymptotic convex geometry and classical symmetrizations
  • Chapter 8. Restricted invertibility and the Kadison-Singer problem
  • Chapter 9. Functionalization of geometry

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