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Journal of Convex Analysis 25 (2018), No. 4, 1371--1395
Copyright Heldermann Verlag 2018

Fréchet Barycenters in the Monge-Kantorovich Spaces

Alexey Kroshnin
Moscow Institute of Physics and Technology, Institute for Information Transmission Problems, 9 Institutskiy per., Dolgoprudny--Moscow Region 141701, Russia

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

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