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MARC 21

Asymptotic geometric analysis. Part 1
Tag	Description
020	$a9781470421939
082	$a515.1 (DDC 23.)
099	$a517.972 ART pt. 1
100	$aArtstein-Avidan, Shiri
245	$aAsymptotic geometric analysis. Part 1$hM$cShiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman
260	$aProvidence, Rhode Island$bAmerican Mathematical Society$c[2015]
300	$axix, 451 pages$billustrations$c27 cm
440	$aMathematical surveys and monographs$v202
505	$aConvex 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
520	$aThe 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.
700	$aGiannopoulos, Apostolos$d1963-
700	$aMilman, Vitali D.$d1939-