Primal and dual problems

Abstract

In this chapter we present generalities on the transport problem from a measure μ on a space X to another measure ν on a space Y. In general, X and Y can be complete and separable metric spaces, but soon we will focus on the case where they are the same subset Ω⊂ ℝd (often compact). The cost function c: X× Y→ [ 0, + ∞] will be possibly supposed to be continuous or semi-continuous, and then we will analyze particular cases (such as c(x, y) = h(x − y) for strictly convex h). It includes a proof of the existence of an optimal transport plan, a proof of Kantorovich duality, and the Brenier Theorem stating that an optimal transport map for the quadratic cost exists, and is the gradient of a convex function. The discussion section presents connections with problems in economics and finance, and complementary topics such as the multi-marginal case, and a brief survey of the regularity of optimal maps.