Fr\'echet barycenters in the Monge-Kantorovich spaces
Abstract
We consider the space P(X) of probability measures on arbitrary Radon space X endowed with a transportation cost J(μ, ) 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 Fr\'echet barycenter, prove a version of the law of large numbers for Fr\'echet barycenters, and discuss the structure of P(X) related to the transportation cost J.
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