Optimal Transport: Old and NewAt the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and selfcontained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.

Contents
5  
21  
29  
Basic properties  43 
Cyclical monotonicity and Kantorovich duality 51  50 
The Wasserstein distances  93 
Displacement interpolation  113 
The MongeMather shortening principle  163 
Volume control  493 
Density control and local regularity  505 
Infinitesimal displacement convexity  525 
Isoperimetrictype inequalities  545 
Concentration inequalities 567  566 
Gradient flows I  629 
Qualitative properties  693 
Functional inequalities  719 
Global approach 205  204 
Local approach  215 
The Jacobian equation  273 
Smoothness 281  280 
Qualitative picture  333 
Optimal transport and Riemannian geometry 353  354 
Ricci curvature  357 
Otto calculus  421 
Displacement convexity I  435 
Displacement convexity II 449  445 
Synthetic treatment of Ricci curvature 731  730 
Analytic and synthetic points of view  735 
Convergence of metricmeasure spaces 743  742 
Stability of optimal transport  773 
Geometric  847 
References  915 
List of short statements  957 
Some notable cost functions  971 
Other editions  View all
Common terms and phrases
References to this book
Geometric Control and Nonsmooth Analysis: In Honor of the 73rd Birthday of H ... Fabio Ancona No preview available  2008 