Index A Aleksandrov solution (of Monge-Amp` ere equa- tion), 129, 136 Aleksandrov’s differentiability theorem, 58 Aleksandrov’s lemma, 84 Aleksandrov-Fenchel inequalities, 192 Alesker-Dar-Milman diffeomorphism, 191 approximate geodesics, 117 arithmetic-geometric inequality, 156 Arnold’s interpretation (Euler equation), 115 Aronson-B´ enilan inequalities, 303 Aronsson’s minimal Lipschitz extension prob- lem, 102 Ascoli’s theorem, 19 B Bakry-Emery theorem, 280, 290 strategy, 275, 289 with skew-symmetric perturbation, 329 Barenblatt-Pattle profile, 301 Barthe inequalities, 196 Beckner-Hirschman uncertainty principle, 286 Bellman principle, 175 Benamou-Brenier formula, 239 Biot-Savart formula, 255 Birkhoff’s theorem, 5, 15 Blachman-Stam inequality, 285 Bochner formula, 178 Boltzmann equation, 223 Borel probability measure, 18 bounded Lipschitz distance, 207 Brascamp-Lieb inequalities, 195 reverse, 196 Brenier map, 67, 85 Brenier solution (Monge-Amp` ere), 130 Brenier’s polar factorization theorem, 109 Brenier’s theorem, 66, 83, 85 Brunn-Minkowski inequality, 184 Burgers’ equation, 170 C Caffarelli’s log concave perturbation theo- rem, 291, 335 center of mass, 153 central limit theorem, 222 change of variable, 9, 125 chaos (for particle systems), 341 characteristics (for transport equations), 166 Choquet’s theorem, 5, 14 concavity, 60 c-concavity, 33, 86 semi-concavity, 91 concentration of measure, 287, 292, 342 conservation of mass, 167 convergence to equilibrium, 288 convexity, 23, 52 in Ê n , 52 semi-convexity, 59 strict convexity, 52 uniform convexity, 59 cost (transportation), 1 homogeneous, 147 optimal, 3 total, 3 Csisz´ar-Kullback-Pinsker inequality, 271, 293 Cullen’s stability principle, 325 cyclical monotonicity, 79, 80, 82, 88 c-cyclical monotonicity, 86 367

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