10 Introduction 3. Overview of the course In Chapter 1, we begin our study by a presentation of the Kantorovich duality, which is a very powerful tool, from both the theoretical and the numerical points of view. It was Kantorovich who formally pointed out this dual formulation (see Theorem 1.3), and, together with Rubinstein, proved it rigorously for a compact space equipped with a distance cost function, in which case it takes a simpler form (see Theorem 1.14). The Kantorovich- Rubinstein theorem was rewritten in the seventies by Dudley [116, Lect. 20], motivated by problems of mathematical statistics (see [119, Chapter 11] for a more recent exposition), and later pushed to a high degree of abstraction by several authors [211]. Our setting here will be very general (Polish space, arbitrary lower semi-continuous cost function). Then, in Chapter 2, we review the most important results about the exis- tence and characterization of the optimal transference plans, i.e. Question 1 above. As we shall see, it is usually very simple to give a proof of existence of a minimizer for the Kantorovich problem, but it is much trickier to do the same for the Monge problem. In the last fifteen years, a systematic and ele- gant theory for the Monge problem was developed by (in alphabetic order) Brenier, Evans, Gangbo, Knott and Smith, McCann, Rachev, R¨uschendorf, and others, with particular emphasis on the quadratic cost c(x, y) = |x−y|2 in Rn. The basic result for this quadratic cost (Theorem 2.12) was dis- covered at least twice, first by Knott and Smith, then by Brenier, who was motivated by considerations arising from fluid mechanics. Their results were generalized by McCann on one hand, Rachev and R¨ uschendorf on the other. For many researchers, however, it is the name of Brenier which is associated with this theorem, because he was one of the first to unveil the potential applications of mass transportation to problems of classical mechanics or mathematical physics. Apart from the quadratic setting, similar results concern the cost func- tions |x − y|p, or more generally d(x, y)p on a Riemannian manifold. In this course, we mainly focus on the quadratic cost function, then just review the basic known results for other costs. As we already said, we recommend the notes by Evans [126] and especially those by Ambrosio [11], and by Ambrosio and Pratelli [14] for a much more complete treatment of the cost function c(x, y) = |x − y|. In Chapter 3, we develop a little bit on the motivations which led Bre- nier to study the Monge-Kantorovich problem, and explain Brenier’s polar factorization theorem (Theorem 3.8), which is essentially equivalent to the

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