
Journal of Convex Analysis 25 (2018), No. 4, 13711395 Copyright Heldermann Verlag 2018 Fréchet Barycenters in the MongeKantorovich Spaces Alexey Kroshnin Moscow Institute of Physics and Technology, Institute for Information Transmission Problems, 9 Institutskiy per., DolgoprudnyMoscow Region 141701, Russia kroshnin@phystech.edu We consider the space P(X) of probability measures on arbitrary Radon space X endowed with a transportation cost J(μ, ν) generated by a nonnegative continuous cost function. For a probability distribution on P(X) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for Fréchet barycenters, and discuss the structure of P(X) related to the transportation cost J. Keywords: Optimal transport, Wasserstein space, Wasserstein barycenter, law of large numbers. MSC: 60D05, 28C99, 54E40 [ Fulltextpdf (181 KB)] for subscribers only. 