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Working paper

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

Kolesnikov A., Lysenko N. Y.
We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to 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}, {Ji} are partitions of the line into unions of disjoint connected sets. The families of sets {Ii},{Ji} admit 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.