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, ..., l_n} that equals the minimum of a finite number n of affine functions l_i 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 I_i × J_i, where {I_i} and {J_i} are partitions of the real line into unions of disjoint connected sets. The families of sets {I_i} and {J_i} have the following properties: 1) c=l_i on I_i × J_i, 2) {I_i}, {J_i} 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.