Topics in Optimal Transportation (Graduate Studies in Mathematics, Vol. 58) (Graduate Studies in Mathematics)

Cedric Villani's book is a lucid and very readable documentation of the tremendous recent analytic progress in “optimal mass transportation” theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics. —Lawrence C. Evans, University of California at Berkeley In 1781, Gaspard Monge defined the problem of “optimal transportation”, or the transferring of mass with the least possible amount of work, with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is at once an introduction to the field of optimal transportation and a survey of the research on the topic over the last 15 years. The book is intended for graduate students and researchers, and it covers both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. Cover......Page 1 Title page......Page 2 Contents......Page 7 Preface......Page 11 Notation......Page 15 1. Formulation of the optimal transportation problem......Page 19 2. Basic questions......Page 24 3. Overview of the course......Page 28 1.1. General duality......Page 35 1.2. Distance cost functions......Page 52 1.3. Appendix: A duality argument in Cb(XxY)......Page 57 1.4. Appendix: {0,1}-valued costs and Strassen's theorem......Page 62 Chapter 2. Geometry of Optimal Transportation......Page 65 2.1. A duality-based proof for the quadratic cost......Page 66 2.2. The real line......Page 91 2.3. Alternative arguments......Page 96 2.4. Generalizations to other costs......Page 103 2.5. More on c-concave functions......Page 121 3.1. Rearrangements and polar factorization......Page 125 3.2. Historical motivations: fluid mechanics......Page 129 3.3. Proof of Brenier' s polar factorization theorem......Page 137 3.4. Related facts......Page 140 4.1. Informal presentation......Page 143 4.2. Regularity......Page 149 4.3. Open problems......Page 159 5.1. Displacement interpolation......Page 161 5.2. Displacement convexity......Page 168 5.3. Application: uniqueness of ground state......Page 181 5.4. The Eulerian point of view......Page 183 Chapter 6. Geometric and Gaussian Inequalities......Page 201 6.1. Brunn-Minkowski and Prekopa-Leindler inequalities......Page 202 6.2. The Alesker-Dar-Milman diffeomorphism......Page 208 6.3. Gaussian inequalities......Page 210 6.4. Sobolev inequalities......Page 218 Chapter 7. The Metric Side of Optimal Transportation......Page 223 7.1. Monge-Kantorovich distances......Page 225 7.2. Topological properties......Page 230 7.3. The real line......Page 236 7.4. Behavior under resealed convolution......Page 238 7.5. An application to the Boltzmann equation......Page 241 7.6. Linearization......Page 251 Chapter 8. A Differential Point of View on Optimal Transportation......Page 255 8.1. A differential formulation of optimal transportation......Page 256 8.2. Differential calculus......Page 268 8.3. Monge-Kantorovich induced dynamics......Page 269 8.4. Time-discretization......Page 274 8.5. Differentiability of the quadratic Wasserstein distance......Page 280 8.6. Non-quadratic costs......Page 284 Chapter 9. Entropy Production and Transportation Inequalities......Page 285 9.1. More on optimal-transportation induced dissipative equations......Page 286 9.2. Logarithmic Sobolev inequalities......Page 297 9.3. Talagrand inequalities......Page 309 9.4. HWI inequalities......Page 315 9.5. Nonlinear generalizations: internal energy......Page 319 9.6. Nonlinear generalizations: interaction energy......Page 322 Chapter 10. Problems......Page 325 List of Problems......Page 326 Bibliography......Page 367 Table of Short Statements......Page 381 Index......Page 385 Ch. 1. The Kantorovich Duality -- Ch. 2. Geometry Of Optimal Transportation -- Ch. 3. Brenier's Polar Factorization Theorem -- Ch. 4. The Monge-ampere Equation -- Ch. 5. Displacement Interpolation And Displacement Convexity -- Ch. 6. Geometric And Gaussian Inequalities -- Ch. 7. The Metric Side Of Optimal Transportation -- Ch. 8. A Differential Point Of View On Optimal Transportation -- Ch. 9. Entropy Production And Transportation Inequalities -- Ch. 10. Problems -- Table Of Short Statements. Cédric Villani. Includes Bibliographical References (p. 349-362) And Index. This graduate textbook reviews results about the existence and regularity of the optimal mass transportation problem, introducing such topics as displacement interpolation, its applications to functional inequalities with geometric content, the Monge-Kantorovich problem, and a differential formulation of the transportation problem inspired by fluid mechanics. Annotation (c)2003 Book News, Inc., Portland, OR Presents a comprehensive introduction to the theory of mass transportation with its many - and sometimes unexpected - applications. This book defines the problem of 'optimal transportation' (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. Assume that we are given a pile of sand (say), and a hole that we have to completely fill up with the sand.