Functional analysis/Mathematical analysis
Remarks on the Monge–Kantorovich problem in the discrete setting
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 207-213.

In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9], which shows that all three quantities coincide in the discrete case.

En théorie du transport optimal, trois quantités jouent un rôle central : le coût minimal de transport, introduit par Monge, sa version relaxée, introduite par Kantorovich, et la formulation duale, due aussi à Kantorovich. L'objet de cette note est de mettre en avant une démonstration totalement élémentaire, extraite de [9], du fait que ces trois quantités coïncident dans le cas discret ; cette preuve ne requiert aucune connaissance préalable.

Accepted:
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DOI: 10.1016/j.crma.2017.12.008
Brezis, Haïm 1, 2, 3

1 Department of Mathematics, Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Departments of Mathematics and Computer Science, Technion, Israel Institute of Technology, 32000 Haifa, Israel
3 Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
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Brezis, Haïm. Remarks on the Monge–Kantorovich problem in the discrete setting. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 207-213. doi : 10.1016/j.crma.2017.12.008. http://www.numdam.org/articles/10.1016/j.crma.2017.12.008/

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