vi Contents 3.4. Related facts 122 Chapter 4. The Monge-Amp` ere Equation 125 4.1. Informal presentation 125 4.2. Regularity 131 4.3. Open problems 141 Chapter 5. Displacement Interpolation and Displacement Convexity 143 5.1. Displacement interpolation 143 5.2. Displacement convexity 150 5.3. Application: uniqueness of ground state 163 5.4. The Eulerian point of view 165 Chapter 6. Geometric and Gaussian Inequalities 183 6.1. Brunn-Minkowski and Pr´ ekopa-Leindler inequalities 184 6.2. The Alesker-Dar-Milman diffeomorphism 190 6.3. Gaussian inequalities 192 6.4. Sobolev inequalities 200 Chapter 7. The Metric Side of Optimal Transportation 205 7.1. Monge-Kantorovich distances 207 7.2. Topological properties 212 7.3. The real line 218 7.4. Behavior under rescaled convolution 220 7.5. An application to the Boltzmann equation 223 7.6. Linearization 233 Chapter 8. A Differential Point of View on Optimal Transportation 237 8.1. A differential formulation of optimal transportation 238 8.2. Differential calculus 250 8.3. Monge-Kantorovich induced dynamics 251 8.4. Time-discretization 256 8.5. Differentiability of the quadratic Wasserstein distance 262 8.6. Non-quadratic costs 266 Chapter 9. Entropy Production and Transportation Inequalities 267 9.1. More on optimal-transportation induced dissipative equations 268
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