Remarks on mass transportation minimizing expectation of a minimum of affine functions

Alexander V. Kolesnikov, Nikolay Lysenko
Theory of Stochastic Processes
Vol.21 (37), no.2, 2016, pp.22-28
We study the Monge-Kantorovich problem with one-dimensional marginals μ and ν and the cost function c = min{l1, ..., ln} that equals the minimum of a finite number n of affine functions li satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of n products Ii × Ji, where {Ii} and {Ji} are partitions of the real line into unions of disjoint connected sets. The families of sets {Ii} and {Ji} have the following properties: 1) c=li on Ii × Ji, 2) {Ii}, {Ji} is a couple of partitions solving an auxiliary n-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.