3. Overview of the course 11 optimal transportation theorem. This chapter contains an overview of Bre- nier’s original motivations in fluid mechanics, but does not dig deeply into the subject, which probably would deserve a book on its own. From the point of view of partial differential equations, the Knott-Smith- Brenier theorem implies existence and uniqueness of the solution, in a very weak sense, to a certain Monge-Amp`ere equation. For this equation a diffi- cult regularity theory was developed by Caffarelli and by Urbas, using quite different tools (see Theorem 4.14). Their work allows one to study some smoothness properties of the optimal transference plan, up to now only in the quadratic case. We quickly review the theory in Chapter 4. This chapter is the only one in which the proofs of the most important results are not pro- vided. Indeed, developing the arguments in this chapter would have meant essentially writing a second book moreover, the excellent set of notes by Ur- bas [241] and the recent reference book by Guti´ errez [153] can be consulted by readers who want to know more about regularity. Next, in Chapter 5, we introduce two simple but fundamental notions due to McCann: displacement interpolation and displacement con- vexity. For many applications, displacement convexity will turn out to be the right notion of convexity. We shall spend some time explaining various ways to look at it, and some of its applications. Chapter 6 penetrates into a rather different world, which is the universe of geometric inequalities. The prototype of such inequalities is certainly the isoperimetric inequality. We shall explain in this chapter how mass transportation provides amazingly powerful tools to study some functional inequalities with geometric content. This chapter follows a series of works initiated by McCann and continued by Alesker, Barthe, Cordero-Erausquin, Dar, Milman, Schmuckenschl¨ ager, and others. In the end, we also present a new proof of the optimal Sobolev inequality, devised jointly by Cordero- Erausquin, Nazaret and the author. As the present manuscript was undergoing final revision, the author discovered (thanks to Kavian) the marvelous review paper by Gardner [142], about the Brunn-Minkowski inequality and its aftermath. There the reader can find a lot of complementary references about the contents of Chapter 6, with a somewhat different point of view. On the whole, in Chapters 2 to 6, we will only be interested in the min- imizers for the Monge-Kantorovich problem: characterize them, use their existence for various applications, etc. To go further, we shall need to be interested in Question 2 above. This is why in Chapter 7 we review the properties of the Monge-Kantorovich distances or Wasserstein dis- tances (see Theorem 7.3), which are distances induced by the value of the
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