Preface of the Second Edition
Preface of the First Edition
Notation
Introduction
1.Formulation of the optimal transportation problem
2.Basic questions
3.Overview of the course
Chapter 1.The Kantorovich Duality
1.1.Generalduality
1.2.Distance cost functions
1.3.Appendix: A duality argument in Cb(X × Y)
1.4.Appendix: {0,1}—valued costs and Strassen's theorem
Chapter 2.Geometry of Optimal Transportation
2.1.A duality—based proof for the quadratic cost
2.2.The real line
2.3.Alternative arguments
2.4.Generalizations to other costs
2.5.More on c—concave functions
Chapter 3.Brenier's Polar Factorization Theorem
3.1.Rearrangements and polar factorization
3.2.Historical motivations: fluid mcchanics
3.3.Proof of Brenier's polar factorization theorem
3.4.Related facts
Chapter 4.The Monge—Ampere Equation
4.1.Informal presentation
4.2.Regularity
4.3.Open problems
Chapter 5.Displacement Interpolation and Displacement Convexity
5.1.Displacement interpolation
5.2.Displacement convexity
5.3.Application: uniqueness of ground state
5.4.The Eulerian point of view
Chapter 6.Geometric and Gaussian Inequalit.ies
6.1.Brunn—Minkowski and Prekopa—Leindler inequalities
6.2.The Alesker—Dar—Milman diffeomorphism
6.3.Gaussian inequalities
6.4 Sobolev ineqiialities
Chapter 7.The Metric Side of Optimal Transportation
7.1.Monge—Kantorovich distances
7.2.Topological properties
7.3.The real line
7.4.Behavior under rescalcd convolution
7.5.An application to the Boltzmann equation
7.6.Linearization
Chapter 8.A Differential Point of View on Optimal Transportation
8.1.A differential formulation of optimal transportation
8.2.Differential calculus in (P(Rn), W2)
8.3.Monge—Kantorovich induced dynamics
8.4.Time—discretization
8.5.Differentiability of the quadratic Wasserstein distance
8.6.Non—quadratic costs
Chapter 9.Entropy Production and Transportation Inequalities
9.1.More on optimal—transportation induced dissipative equations
9.2.Logarithmic Sobolev inequalities
9.3.Talagrandinequalitics
9.4.HWI inequalities
9.5.Nonlincar generalizations: internal energy
9.6.Nonlinear generalizat.ions: interaction energy
Chapter 10.Problems
List of Problems
Bibliography
Table of Short Statements
Index