A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold
type: Item
instance of: scholarly article
title: A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold
zbMATH Open document ID: 0711.53037
DOI: 10.1007/BF02764780
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The concentration function \(\alpha\) (X;\(\epsilon\)), \(\epsilon >0\), of a metric space (X,d) with respect to a probability measure has been introduced by \textit{M. Gromov} and \textit{V. D. Milman} [Am. J. Math. 105, 843-854 (1983; Zbl 0522.53039)]. A geometric invariant of the Laplace operator on a compact Riemannian manifold is the hypercontractive constant \(\rho_ 0(X)\); see \textit{O. S. Rothaus} [J. Funct. Anal. 42, 102-109 (1981; Zbl 0471.58027)]. Using the method of M. Gromov and V. D. Milman, the author obtains the following estimation of \(\alpha\) (X;\(\epsilon\)) in terms of \(\rho_ 0(X)\) \[ \alpha (X;\epsilon)\leq C \exp (-c\epsilon \sqrt{\rho_ 0(X)}\log (1+\epsilon \sqrt{\rho_ 0(X)}) \] for all \(\epsilon >0\), where \(C,c>0\) are numerical constants. Some examples are given.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 4169600
zbMATH Keywords: concentration function
zbMATH Keywords: metric space
zbMATH Keywords: probability measure
zbMATH Keywords: Laplace operator
zbMATH Keywords: hypercontractive constant
Wikidata QID: Q115391669
author: Michel Ledoux
reviewed by: Ion Mihai
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4169600