Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian Manifolds

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Springer, Feb 27, 2009 - Mathematics - 160 pages
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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Contents

Introduction
1
The isoperimetric inequality on S and some consequences
9
Almost euclidean subspaces of t Spaces
19
Levy families
27
Martingales
42
Type and cotype of normed spaces and some
51
Additional applications of Levy families
60
Type and cotype of normed spaces
69
The Rademacher projection
98
Projections on random euclidean subspaces
106
Isoperimetric inequalities in Riemannian Manifolds
114
Gaussian and Rademacher averages
130
Proof of the BeurlingKato Theorem 14 4
137
Notes and Remarks
144
References
151
19
153

Krivines theorem
77
The MaureyPisier theorem
85

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