8 Introduction Remark. Identity (13) is intuitively clear: for the cost function 1x=y, the optimal transportation cost is obtained when all the mass that can stay in place (all the mass shared by μ and ν) does indeed stay in place. If one sends all the remaining mass of μ onto the remaining mass of ν, the corresponding total cost is (with sloppy notation) μ−ν≥0 d(μ − ν) = 1 2 μ−ν≥0 d(μ − ν) + μ−ν≤0 d(ν − μ) since μ−ν=0 d(μ − ν) = 0, and X d(μ − ν) = 0. In Chapter 1, we shall see that identity (13) (as well as the more general version of Strassen’s theorem) can be seen as a consequence of some basic results in the theory of optimal transportation. Although the ordering of our two basic questions seems quite natural (at least to the author of these notes!), it turns out that Question 2 was reasonably well mastered long before Question 1. Even back in the seventies and earlier, many researchers in probability theory understood the great interest that the mass transportation problem could have for them, as a tool of measuring distances between probability measures in more or less complicated settings. The history of Question 1 is much more agitated. Ironically, Monge’s original problem was extremely diﬃcult, for two reasons. First, as a general rule the Monge problem is much more tricky than the Kantorovich problem. Secondly, the particular cost function which he considered, namely the dis- tance function c(x, y) = |x − y|, is very degenerate from the point of view of convexity properties. In spite of Sudakov’s important work [228] at the end of the seventies, it is certainly fair to state that the Monge problem has begun to be reasonably well understood only in recent years. In fact, Sudakov “proved” the existence of a minimizer to the Monge problem with a distance cost function, but his argument was not completely correct, as pointed out recently by Alberti, Kirchheim and Preiss (see the explanations in Ambrosio and Pratelli [14, sect. 1] see also Section 8 in the same ref- erence). Even though Sudakov’s proof was carefully fixed in [11, 14], it is striking to realize that the first published correct (so it seems!) proof of existence for this cost function was the diﬃcult 1999 memoir by Evans and Gangbo [127], which partly relied on the theory of p-Laplace equations, and in which somewhat strong regularity assumptions were imposed on the marginals μ and ν... Even if we forget about this degeneracy of the distance cost function, the following simple remark should be suﬃcient to understand why Monge’s problem is in general so tricky. Assume that the probability measures μ and ν are defined on Rn (or a smooth manifold) and are absolutely continuous

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