DMCA
Concentration Of Measure And Isoperimetric Inequalities In Product Spaces (1995)
| Citations: | 375 - 4 self |
BibTeX
@MISC{Talagrand95concentrationof,
author = {Michel Talagrand},
title = {Concentration Of Measure And Isoperimetric Inequalities In Product Spaces},
year = {1995}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product# N of probability spaces has measure at least one half, "most" of the points of# N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word "most" is made rigorous by isoperimetric-type inequalities that bound the measure of the exceptional sets. The meaning of the work "close" is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.
Keyphrases
product space isoperimetric inequality di erent inequality systematic exploration large number many case optimal result am classification number measure phenomenon main way exceptional set concrete situation optimal numerical constant geometric probability common scheme banach space talagrand table abstract tool probability space wide range isoperimetric-type inequality simple approach
