Abstract

We consider the space P(X)\mathcal{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)\mathcal{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)\mathcal{P}(X) related to the transportation cost J.

Author information

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Alexey Kroshnin

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

kroshnin@phystech.edu

Keywords

  • Optimal transport
  • Wasserstein space
  • Wasserstein barycenter
  • law of large numbers

Mathematics Subject Classification

60D05, 28C99, 54E40

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Suggested citation

A. Kroshnin. “Fréchet Barycenters in the Monge-Kantorovich Spaces.” Journal of Convex Analysis 25 (2018), No. 4, 1371–1395. https://doi.org/10.68381/jca25084

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.