Geodesic
Fréchet Barycenters in the Monge-Kantorovich Spaces
Journal of convex analysis, Tome 25 (2018) no. 4, pp. 1371-1395
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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.
Classification : 60D05, 28C99, 54E40
Mots-clés : Optimal transport, Wasserstein space, Wasserstein barycenter, law of large numbers 
@article{JCA_2018_25_4_JCA_2018_25_4_a16,
     author = {A. Kroshnin},
     title = {Fr\'echet {Barycenters} in the {Monge-Kantorovich} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {1371--1395},
     year = {2018},
     volume = {25},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2018_25_4_JCA_2018_25_4_a16/}
}
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A. Kroshnin. Fréchet Barycenters in the Monge-Kantorovich Spaces. Journal of convex analysis, Tome 25 (2018) no. 4, pp. 1371-1395. http://geodesic.mathdoc.fr/item/JCA_2018_25_4_JCA_2018_25_4_a16/