An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space
type: Item
instance of: scholarly article
title: An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space
zbMATH Open document ID: 0883.60031
published in: The Annals of Probability
publication date: 1998-02-23T00:00:00Z
review text: The author finds an elementary method of proving isoperimetric inequalities for the canonical Gaussian measure \(\gamma _n\) on \(\mathbb{R}^n\). In particular, the method permits to prove that for any Borel set \(A\subset\mathbb{R}^n\), \(\gamma_n^+(A)\geq \varphi (\Phi^{-1}(\gamma_n(A))\), where \(\gamma_n^+\) is the Minkowski surface measure with respect to \(\gamma_n\), \(\Phi \) is the distribution function of a standard Gaussian random variable, while \(\varphi \) is its density function. As a corollary the following well-known original form of isoperimetric inequality is derived: For any Borel \(A\subset\mathbb{R}^n \) and any \(h>0\), \(\gamma_n(A^h) \geq \Phi (\Phi^{-1}(\gamma_n(A)) + h)\), where \(A^h\) stands for the open \(h\)-neighborhood of \(A\).
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 1018387
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: Gaussian measure
zbMATH Keywords: discrete cube
author: S. G. Bobkov
reviewed by: S. A. Chobanjan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1024404285
OpenAlex ID: W2049220294
cites work: Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator
cites work: A functional form of the isoperimetric inequality for the Gaussian measure
cites work: Symétrisation dans l'espace de Gauss.
cites work: Q4284289
cites work: Q3124805
cites work: Extremal properties of half-spaces for spherically invariant measures
cites work: Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem
DOI: 10.1214/AOP/1024404285
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description: scientific article; zbMATH DE number 1018387