Concentration of Measure and Isoperimetric Inequalities in Product Spaces

TL;DR

This paper systematically explores the concentration of measure phenomenon in product spaces, establishing isoperimetric inequalities that quantify how most points are close to large sets, with broad applications across probability and geometry.

Contribution

It introduces a unified proof scheme for isoperimetric inequalities in product spaces, achieving near optimal constants and broad applicability.

Findings

Derived qualitatively optimal inequalities with near optimal constants

Unified proof scheme applicable to various notions of 'closeness'

Demonstrated wide-ranging applications in probability and geometry

Abstract

The concentration of measure prenomenon roughly states that, if a set $A$ in a product $\Omega^N$ of probability spaces has measure at least one half, ``most'' of the points of $\Omega^N$ 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 in Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.