## Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian ManifoldsThis 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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Asymptotic Theory of Finite Dimensional Normed Spaces Vitali D. Milman,Gideon Schechtman No preview available - 2014 |

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absolute constant applications assume average ball Banach space block boundary bounded called Chapter choose clearly close compact complete conclude Consequently consider constant contains continuous convex corollary cotype curvature define definition denote depending dimension distance distribution easily elements equal estimate euclidean example exists extends fact finite dimensional finitely representable follows function geodesic given gives Haar measure holds implies independent inequality integer invariant isometric Lemma length Levy family linear manifold mean measure metric n-dimensional natural normal normed space Note obtain operator particular positive probability projection proof Proposition prove random variables REMARK resp respect result Riemannian satisfies scalars sequence similar sphere structure subset subspace symmetric Take Theorem theory unit vector vector basis volume