Mass transportation and rough curvature bounds for discrete spaces
type: Item
instance of: scholarly article
title: Mass transportation and rough curvature bounds for discrete spaces
zbMATH Open document ID: 1184.28015
author: Karl-Theodor Sturm
author: Anca Iuliana Bonciocat
published in: Journal of Functional Analysis
publication date: 2009-06-04T00:00:00Z
review text: The article is devoted to a discretization of metric measure spaces. A metric measure space is a triple \((M,d,m)\), where \(d\) is a metric on the set \(M\), while \(m\) is a non-negative measure on the Borel \(\sigma \)-algebra of the metric space \((M,d)\). In case \(m(M)=1\), a metric measure space \((M,d,m)\) is called normalized. For two metric measure spaces an \(L_2\)-transportation distance is used. A family of absolutely continuous relative to the measure \(m\) non-negative measures \(\nu = \rho m\) is considered, where \(\rho \) is a function. Then a relative entropy of \(\nu \) with respect to \(m\) is defined by the equality \(Ent(\nu | m) = \lim_{0<\varepsilon \to 0} \int_{\rho >\varepsilon} \rho \ln \rho \,dm\). By definition a metric measure space \((M,d,m)\) has a curvature not less than \(K\) if the relative entropy is weakly \(K\)-convex on the subspace of measures of finite entropy. The authors have demonstrated that an arbitrary metric measure space of positive curvature can be approximated by a sequence of discrete measure spaces. Examples of such spaces are also described. The discretization of a hyperbolic space may produce the precise value of \(K\). Moreover, a perturbed transportation cost inequality is studied.
Mathematics Subject Classification ID: 28C99
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5561661
zbMATH Keywords: optimal transport
zbMATH Keywords: Ricci curvature
zbMATH Keywords: GH-limits
zbMATH Keywords: graphs
zbMATH Keywords: concentration of measure
reviewed by: Sergey V. Ludkovsky
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.01.029
OpenAlex ID: W1972708599
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DOI: 10.1016/J.JFA.2009.01.029
description: scientific article; zbMATH DE number 5561661