Contents Preface ix Notation xiii Introduction 1 1. Formulation of the optimal transportation problem 1 2. Basic questions 6 3. Overview of the course 10 Chapter 1. The Kantorovich Duality 17 1.1. General duality 17 1.2. Distance cost functions 34 1.3. Appendix: A duality argument in Cb(X × Y ) 39 1.4. Appendix: {0, 1}-valued costs and Strassen’s theorem 44 Chapter 2. Geometry of Optimal Transportation 47 2.1. A duality-based proof for the quadratic cost 48 2.2. The real line 73 2.3. Alternative arguments 78 2.4. Generalizations to other costs 85 2.5. More on c-concave functions 103 Chapter 3. Brenier’s Polar Factorization Theorem 107 3.1. Rearrangements and polar factorization 107 3.2. Historical motivations: fluid mechanics 111 3.3. Proof of Brenier’s polar factorization theorem 119 v

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