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On Kantorovich Problems with a Parameter

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Abstract

In this note, we study the Kantorovich problem of optimal transportation of measures on metric spaces in the case where the cost function and marginal distributions depend on a parameter from a metric space. It is shown that the Hausdorff distance between the sets of probability measures with given marginals can be estimated by the distances between the marginals. As a corollary, it is proved that the cost of optimal transportation is continuous with respect to the parameter if the cost function and marginal distributions are continuous in this parameter.

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Funding

This research is supported by the Russian Science Foundation, grant no. 22-11-00015.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    V. I. Bogachev

  2. National Research University Higher School of Economics, Moscow, Russia

    V. I. Bogachev & S. N. Popova

  3. Saint-Tikhon’s Orthodox University, Moscow, Russia

    V. I. Bogachev

  4. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow, Russia

    S. N. Popova

Authors
  1. V. I. Bogachev
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  2. S. N. Popova
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Correspondence to V. I. Bogachev.

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Bogachev, V.I., Popova, S.N. On Kantorovich Problems with a Parameter. Dokl. Math. 106, 426–428 (2022). https://doi.org/10.1134/S1064562422700107

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  • DOI: https://doi.org/10.1134/S1064562422700107

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