RUS  ENG
Full version
JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2024 Volume 215, Number 1, Pages 33–58 (Mi sm9920)

This article is cited in 3 papers

Hausdorff distances between couplings and optimal transportation

V. I. Bogachevab, S. N. Popovacb

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b National Research University Higher School of Economics, Moscow, Russia
c Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia

Abstract: We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of probability measures with prescribed marginals is estimated in terms of the distances between the marginals themselves. This estimate is used to prove the continuity of the cost of optimal transportation with respect to the parameter in the case of the continuous dependence of the cost function and marginal distributions on this parameter. Existence of approximate optimal plans continuous with respect to the parameter is established. It is shown that the optimal plan is continuous with respect to the parameter in the case of uniqueness. However, examples are constructed when there is no continuous selection of optimal plans. Another application of the estimate for the Hausdorff distance concerns discrete approximations of the transportation problem. Finally, a general result on the convergence of Monge optimal mappings is proved.
Bibliography: 46 titles.

Keywords: Kantorovich problem, Monge problem, Hausdorff distance, coupling, weak convergence, continuity with respect to a parameter.

MSC: Primary 49Q22; Secondary 60A10

Received: 05.04.2023 and 30.09.2023

DOI: 10.4213/sm9920


 English version:
Sbornik: Mathematics, 2024, 215:1, 28–51

Bibliographic databases:


© Steklov Math. Inst. of RAS, 2024