Preface xi the text I apologize for that. Much more complete lists of references on the Monge-Kantorovich problem can be found in Gangbo and McCann [141], and especially in Rachev and R¨ uschendorf [211]. On the other hand, I did not hesitate to give references for subjects whose relation to the optimal transportation problem is not necessarily immediate, whenever I felt that this could give the reader some insights in related fields. At first I did not intend to consider the optimal mass transportation problem in a very general framework. But a graduate course that I taught in the fall of 2001 on the mean-field limit in statistical physics, made me realize the practical importance of handling mass transportation on infinite- dimensional spaces such as the Wiener space, or the space of probability measures on some phase space. Tools like the Kantorovich duality, or the metric properties induced by optimal transportation, happen to be very use- ful in such contexts — as was understood long ago by people doing research in mathematical statistics. This is why in Chapters 1 and 7 I have treated those topics under quite general assumptions, in a context of Polish spaces (which is not the most general setting that one could imagine, but which is suﬃcient for all the applications I am used to). Almost all the rest of the notes deals with finite-dimensional spaces. Let me mention that several researchers, in particular ¨ Ust¨ unel and F.-Y. Wang, are currently working to extend some of the geometrical results described below to an infinite- dimensional setting allowing for the Wiener space. A more precise overview of the contents of this book is given at the end of the Introduction, after a precise statement of the problem. I shall also summarize at the beginning the main notation used in the text to avoid devastating confusion, note carefully the definition of a “small set” in Rn, as a set of Hausdorff dimension at most n − 1. As the reader should understand, the subject is still very vivid and likely to get into new developments in the next years. Among topics which are still waiting for progress, let me only mention the numerical methods for computing optimal transportations. At the time of this writing, some noticeable progress seems to have been done on this subject by Tannenbaum and his coworkers. Even though these beautiful new schemes seem extremely promising, they need confirmation from the mathematical point of view, which is one reason why I skipped this topic (the other reason being my lack of competence). Some related results can be found in [152]. Also I wish to emphasize that optimal mass transportation, besides its own intrinsic interest, sometimes appears as a surprisingly effective tool in problems which do not a priori seem to have any relation to it. For this reason I think that getting at least superficially acquainted with it is a wise

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