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

Theory of Stochastic Processes. 2016. Vol. 21(37). No. 2. P. 22-28.
Kolesnikov A., Lysenko N. Y.

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and

the cost function $c = \min\{l_1, \ldots, l_n\}$

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 $I_i \times J_i$, 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 $I_i \times J_i$,

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.