Geodesic
Remarks on mass transportation minimizing expectation of a minimum of affine functions
Teoriâ slučajnyh processov, Tome 21 (2016) no. 2, pp. 22-28 Cet article a éte moissonné depuis la source Math-Net.Ru

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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/

[1] Russian Math. Surveys, 67:5, 785–890

[2] V. I. Bogachev, Measure theory, v. 1,2, Springer, Berlin, 2007

[3] J. Delon, J. Salomon, A. Sobolevskii, “Local matching indicators for transport problems with concave costs”, SIAM J. Disc. Math., 26:2 (2012), 801–827

[4] W. Gangbo, R. J. McCann, “The geometry of optimal transportation”, Acta Math., 177 (1996), 113–161

[5] R. J. McCann, “Exact solutions to the transportation problem on the line”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 1341–1380

[6] B. Pass, “Multi-marginal optimal transport: theory and applications”, ESAIM: Math. Model. Numer. Anal., 49 (2015), 1771–1790

[7] P. Pegon, D. Piazzoli, F. Santambrogio, “Full characterization of optimal transport plans for concave costs”, Disc. Cont. Dyn. Syst. A, 35:12 (2015), 6113–6132

[8] C. Villani, Topics in optimal transportation, Amer. Math. Soc., Rhode Island, 2003