Isoperimetric constants for product probability measures
type: Item
instance of: scholarly article
title: Isoperimetric constants for product probability measures
zbMATH Open document ID: 0878.60013
published in: The Annals of Probability
publication date: 1997-12-14T00:00:00Z
review text: Suppose that \((X,d)\) is a metric space equipped with a separable Borel probability measure \(\mu\), then the authors study \(\text{Is}(\mu)=\inf[\mu^+(A)/\min\{\mu(A),1- \mu(A)\}]\), the infimum is taken over all Borel sets \(A\subset X\) of measure \(0<\mu(A)<1\) and \(\mu^+(A)\) denotes the surface measure of \(A\). The main result is the inequality (1) \(\text{Is}(\mu^n)\geq (1/2\sqrt 6)\text{Is}(\mu)\), for any \(n\geq 1\), \(\mu^n\) being equal to \(\mu\otimes\cdots\otimes\mu\). An equivalent functional form (with a dimension free constant \(K(\mu)\)) is an \(L^1\)-Poincaré type inequality: (2) \(K{\mathbf E}|f|\leq{\mathbf E}|\nabla f|\), \(f\) being a function on \(\mathbb{R}^n\) with mean \({\mathbf E}f\) equal to 0, and modulus of gradient \(|\nabla f(x)|=\limsup_{y\to x} |f(y)- f(x)|/d(x,y)\). The connection between (1) and the double exponential distribution \(\nu\) on \(\mathbb{R}\) (i.e. \(\nu(dx)= 2^{-1}e^{-|x|}dx\)) is given by the following equivalent properties: (i) the probability \(\mu^n\) verifies (2) with some constant \(K\) independent of the dimension, (ii) \(\mu\) verifies (2), (iii) there exists a function \(U:\mathbb{R}\to\mathbb{R}\) with finite Lipschitz constant which transforms \(\nu\) into \(\mu\), (iv) \(\text{Is}(\mu)>0\). In addition (2) holds with \(K=\text{Is}(\mu)/(2\sqrt 6)\).
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 28A35
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 1018386
zbMATH Keywords: \(L^ 1\)-Poincaré type inequality
zbMATH Keywords: double exponential distribution
author: S. G. Bobkov
author: Christian Houdré
reviewed by: Pierre Vallois
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1024404284
OpenAlex ID: W2062939536
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DOI: 10.1214/AOP/1024404284
description: scientific article; zbMATH DE number 1018386