| Dewey Class |
515.1 (DDC 23.) |
| Title |
Asymptotic geometric analysis. Part 1 (M) / Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman |
| Author |
Artstein-Avidan, Shiri |
| Added Personal Name |
Giannopoulos, Apostolos , 1963- |
| Milman, Vitali D., , 1939- |
| Publication |
Providence, Rhode Island : American Mathematical Society , [2015] |
| Physical Details |
xix, 451 pages : illustrations ; 27 cm |
| Series |
Mathematical surveys and monographs ; 202 |
| ISBN |
9781470421939 |
| Summary Note |
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 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 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.: |
| Contents note |
Convex bodies: Classical geometric inequalities Classical positions of convex bodies Isomorphic isoperimetric inequalities and concentration of measure Metric entropy and covering numbers estimates Almost Euclidean subspaces of finite dimensional normed spaces The $\ell$-position and the Rademacher projection Proportional theory $M$-position and the reverse Brunn-Minkowski inequality Gaussian approach Volume distribution in convex bodies Elementary convexity Advanced convexity Bibliography Subject index Author index |
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