### 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{l

_{1}, ..., 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.