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JOURNALS // Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia // Archive

Dokl. RAN. Math. Inf. Proc. Upr., 2022 Volume 507, Pages 26–28 (Mi danma313)

This article is cited in 2 papers

MATHEMATICS

On Kantorovich problems with a parameter

V. I. Bogachevabc, S. N. Popovabd

a Lomonosov Moscow State University, Moscow, Russia
b National Research University Higher School of Economics, Moscow, Russia
c Saint-Tikhon’s Orthodox University, Moscow, Russia
d Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow, Russia

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.

Keywords: Kantorovich problem, Kantorovich metric, optimal plan, Hausdorff distance, continuity with respect to a parameter.

UDC: 517.987

Presented: V. V. Kozlov
Received: 01.06.2022
Revised: 30.10.2022
Accepted: 17.11.2022

DOI: 10.31857/S2686954322600380


 English version:
Doklady Mathematics, 2022, 106:3, 426–428

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© Steklov Math. Inst. of RAS, 2024