Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Suggested citation

A. Kroshnin. “Fréchet Barycenters in the Monge-Kantorovich Spaces.” Journal of Convex Analysis 25 (2018), No. 4, 1371–1395.

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