The geometry of optimal transportation

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal measures . . . . . . . . . . . . . . . . . . . 126 Part I. Strictly convex costs . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. Existence and uniqueness of optimal maps . . . . . . . . . . . . . 133 4. Characterization of the optimal map . . . . . . . . . . . . . . . . . 137 Part II. Costs which are strictly concave as a function of d i s t a n c e . . . 141 5. The role of optimal maps . . . . . . . . . . . . . . . . . . . . . . . . 141 6. Uniqueness of optimal solutions . . . . . . . . . . . . . . . . . . . . 144 Part III. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A. Legendre transforms and conjugate costs . . . . . . . . . . . . . . 148 B. Examples of c-concave potentials . . . . . . . . . . . . . . . . . . . 152 C. Regularity of c-concave potentials . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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