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**Optimal transport. Old and new.**
*(English)*
Zbl 1156.53003

Grundlehren der Mathematischen Wissenschaften 338. Berlin: Springer (ISBN 978-3-540-71049-3/hbk). xxii, 973 p. (2009).

As the title suggests, the book is aimed to old and new problems of optimal transport. After the publication of his [Topics in optimal transportation. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1106.90001)], the present book offered to the author a good opportunity to approach differently the whole theory, with alternative proofs and a more probabilistic presentation, and to incorporate new results. Among these new results was John Mather’s minimal measures which has a lot to do with optimal transport, and the fact that optimal transport could provide a robust synthetic approach to Ricci curvature bounds. In comparison with the previous book, this approach is oriented more on probability, geometry, and dynamical systems, and less on analysis and physics. According to the author’s recommendation, both books can be read independently, or together, and their complementarity can have pedagogical value.

In order to give the reader a feeling for the content and applicability of the results we point out in the sequel the titles of the parts and the chapters: 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.

This meticulous work is based on very large bibliography (846 titles) that is converted into a very valuable monograph that presents many statements and theorems written specifically for this approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes. Disseminated throughout the book, several appendices contain either some domains of mathematics useful to non-experts, or proofs of important auxiliary results.

Very useful instruments such as List of short statements, List of figures, Index, and Some notable cost functions are accessible at the end of the book.

In order to give the reader a feeling for the content and applicability of the results we point out in the sequel the titles of the parts and the chapters: 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.

This meticulous work is based on very large bibliography (846 titles) that is converted into a very valuable monograph that presents many statements and theorems written specifically for this approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes. Disseminated throughout the book, several appendices contain either some domains of mathematics useful to non-experts, or proofs of important auxiliary results.

Very useful instruments such as List of short statements, List of figures, Index, and Some notable cost functions are accessible at the end of the book.

Reviewer: Mihail Voicu (Iaşi)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |