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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.

Remarks on the Monge–Kantorovich problem in the discrete setting