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Concentration Property on Probability Spaces

✍ Scribed by A.A. Giannopoulos; V.D. Milman

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 226 KB
Volume: 156
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.2000.1949

No coin nor oath required. For personal study only.

✦ Synopsis

Introduction

1.1. The starting point of this paper is the notion of concentration for metric probability spaces. Let (X, d, +) be a metric space with metric d and diameter diam(X) 1, which is also equipped with a Borel probability measure +. We then define the concentration function (or ``isoperimetric constant'') of X by

where

metric probability spaces is called a Le vy family if for every =>0 :(X n ; =) Ä 0 as n Ä . A natural example of a Le vy family is given by the family (S n&1 , \ n , _ n ), where S n&1 is the Euclidean sphere in R n , \ n is the geodesic distance, and _ n is the rotationally invariant probability measure on S n&1 . Le vy observed that the isoperimetric inequality on S n+1 implies that

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