维拉尼所著《最优输运（第1分册）（英文版）》是全面讲述最优输运——无论新老问题的专著。本书讲学严谨，基于大量的文献扩充改变而成，使得这本书成为一本相当有价值的宝典类书籍，证明完整自成体系，扩充了文献注解。适于最优输运方面的每个科研人员和研究生，博士及以上的人员不需要预备知识可以完全读懂该书。

Preface

Conventions

Introduction

 1 Couplings and changes of variables

 2 Three examples of coupling techniques

 3 The founding fathers of optimal transport

Part Ⅰ Qualitative description of optimal transport

 4 Basic properties

 5 Cyclical monotonicity and Kantorovich duality

 6 The Wasserstein distances

 7 Displacement interpolation

 8 The Monge-Mather shortening principle

 9 Solution of the Monge problem I: Global approach

 10 Solution of the Monge problem II: Local approach

 11 The Jacobian equation

 12 Smoothness

 13 Qualitative picture

Part Ⅱ Optimal transport and Riemannian geometry

 14 Ricci curvature

 15 Otto calculus

 16 Displacement convexity I

 17 Displacement convexity II

 18 Volume control

 19 Density control and local regularity

 20 Infinitesimal displacement convexity

 21 Isoperimetric-type inequalities

 22 Concentration inequalities

 23 Gradient flows I

 24 Gradient flows II: Qualitative properties

 25 Gradient flows III: Functional inequalities

Part Ⅲ Synthetic treatment of Ricci curvature

 26 Analytic and synthetic points of view

 27 Convergence of metric-measure spaces

 28 Stability of optimal transport

 29 Weak Ricci curvature bounds I: Definition and Stability

 30 Weak Ricci curvature bounds II: Geometric and analytic properties

Conclusions and open problems

 References

 List of short statements

 List of figures

 Index

 Some notable cost functions