Vitali D. Milman's Books

Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

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Asymptotic Geometric Analysis, Part II

By: Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

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Page Count

645

Publisher

American Mathematical Society

Published Date

ISBN 10

1470463601

ISBN 13

9781470463601

Asymptotic Geometric Analysis, Part I

By: Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

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Asymptotic Geometric Analysis, Part I

By: Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

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.

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Page Count

473

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470421933

ISBN 13

9781470421939

Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

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Functional Analysis

An Introduction

By: Yuli Eidelman,Vitali D. Milman,Antonis Tsolomitis

Introduces the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators and spectral theory of self-adjoint operators. This work presents the theorems and methods of abstract functional analysis and applications of these methods to Banach algebras and theory of unbounded self-adjoint operators.

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Page Count

344

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821836463

ISBN 13

9780821836460

Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

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Asymptotic Theory of Finite Dimensional Normed Spaces

Isoperimetric Inequalities in Riemannian Manifolds

By: Vitali D. Milman,Gideon Schechtman

This book deals with the geometrical structure of finite dimensional normed spaces, as the dimension grows to infinity. This is a part of what came to be known as the Local Theory of Banach Spaces (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie].

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Page Count

166

Publisher

Springer

Published Date

ISBN 10

3540388222

ISBN 13

9783540388227

Book's by Vitali D. Milman

Asymptotic Geometric Analysis, Part II

Asymptotic Geometric Analysis, Part II

Asymptotic Geometric Analysis, Part I

Asymptotic Geometric Analysis, Part I

Functional Analysis

Functional Analysis

Asymptotic Theory of Finite Dimensional Normed Spaces

Asymptotic Theory of Finite Dimensional Normed Spaces

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