Geodesic
Remarks on mass transportation minimizing expectation of a minimum of affine functions
Teoriâ slučajnyh processov, Tome 21 (2016) no. 2, pp. 22-28.

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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.
Keywords: Monge–Kantorovich problem, concave cost functions. 
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Alexander V. Kolesnikov; Nikolay Lysenko. Remarks on mass transportation minimizing expectation of a minimum of affine functions. Teoriâ slučajnyh processov, Tome 21 (2016) no. 2, pp. 22-28. http://geodesic.mathdoc.fr/item/THSP_2016_21_2_a3/

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