Asymptotic geometric analysis
Author(s)
Bibliographic Information
Asymptotic geometric analysis
(Mathematical surveys and monographs, v. 202)
American Mathematical Society, c2015
- pt. 1
Available at / 30 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
pt. 1S||MSM||202200032395290
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Note
Includes bibliographical references (p. 415-437) and indexes
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"