Mass transportation and rough curvature bounds for discrete spaces (Q1019689)

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.