Asymptotic geometric analysis. P. 1, Asymptotic geometric analysis, part I / Shiri Artstein-Avidan; Apostolos Giannopoulos; Vitali D. Milman

Subject categories: • General convexity • Normed linear spaces and Banach spaces; Banach lattices • Geometric probability and stochastic geometry • Classical measure theory • Geometry and structure of normed linear spaces • • Probabilistic methods in Banach space theory • Convex sets in n dimensions (including convex hypersurfaces) • Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) • Asymptotic theory of convex bodies • Research exposition (monographs, survey articles) pertaining to computer scienceThe 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 in 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 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.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. "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 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 $\ell $-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 Chapter 11. Elementary Convexity Chapter 12. Advanced Convexity Shiri Artstein-avidan, Apostolos Giannopoulos, Vitali D. Milman. Part 1. [without Special Title] -- Includes Bibliographical References (pages 415-437) And Indexes.