Chapter 1 The Kantorovich Duality In this first chapter, we shall investigate a powerful duality formula due to Kantorovich. In order to emphasize the great generality of this principle, which does not require any underlying geometrical structure, we shall con- sider here abstract Polish spaces (complete, separable metric spaces). Polish spaces are commonly used in probability theory, see for instance Billings- ley [41] or Dudley [119]. In fact, an even more general setting would be possible here some of the results below are proven in [119] without assum- ing completeness, and it is shown in [211] that the topological setting can sometimes be dispended with (while measurability is a crucial assumption, see [117]). On the contrary, most of the rest of these notes, with the im- portant exception of Chapter 7, will take place in a much more restricted framework (Banach space, Euclidean space or Riemannian manifold). This chapter will also be an opportunity to recall some basic notions about probability measures and topology. 1.1. General duality 1.1.1. Definitions and preliminaries. We start by recalling the basic notions of optimal mass transportation. Let (X, μ) and (Y, ν) be proba- bility spaces, and let c be a nonnegative measurable function on X × Y . Kantorovich’s mass transportation problem consists in minimizing the lin- ear functional π −→ X×Y c(x, y) dπ(x, y) 17
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