**Graduate Studies in Mathematics**

Volume: 58;
2003;
370 pp;
Softcover

MSC: Primary 49;
Secondary 35; 60

**Print ISBN: 978-1-4704-6726-5
Product Code: GSM/58.S**

List Price: $74.00

AMS Member Price: $59.20

MAA Member Price: $66.60

**Electronic ISBN: 978-1-4704-1804-5
Product Code: GSM/58.E**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

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# Topics in Optimal Transportation

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*Cédric Villani*

This is the first comprehensive introduction to the theory of mass
transportation with its many—and sometimes unexpected—applications.
In a novel approach to the subject, the book both surveys the topic and
includes a chapter of problems, making it a particularly useful graduate
textbook.

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 intended for
graduate students and researchers, covering both theory and applications.
Readers are only assumed to be familiar with the basics of measure theory and
functional analysis.

#### Readership

Graduate students and research mathematicians interested in probability theory, functional analysis, isoperimetry, partial differential equations, and meteorology.

#### Reviews & Endorsements

Villani writes with enthusiasm, and his approachable style is aided by pleasant typography. The exposition is far from rigid. ... As an introduction to an active and rapidly growing area of research, this book is greatly to be welcomed. Much of it is accessible to the novice research student possessing a solid background in real analysis, yet even experienced researchers will find it a stimulating source of novel applications, and a guide to the latest literature.

-- Geoffrey Burton, Bulletin of the LMS

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

The book is clearly written and well organized and can be warmly recommended as an introductory text to this multidisciplinary area of research, both pure and applied - the mass transportation problem.

-- Studia Universitatis Babes-BolyaiMathematica

This is a very interesting book: it is the first comprehensive introduction to the theory of mass transportation with its many - and sometimes unexpected - applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook.

-- Olaf Ninnemann for Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Topics in Optimal Transportation

- Front Cover 11 free
- Contents 88
- Preface of the Second Edition 1212
- Preface of the First Edition 1616
- Notation 2020
- Introduction 2424
- Chapter 1: The Kantorovich Duality 4040
- Chapter 2: Geometry of Optimal Transportation 7272
- Chapter 3: Brenier’s Polar Factorization Theorem 132132
- Chapter 4: The Monge-Ampère Equation 150150
- Chapter 5: Displacement Interpolation and Displacement Convexity 168168
- Chapter 6: Geometric and Gaussian Inequalities 210210
- Chapter 7: The Metric Side of Optimal Transportation 232232
- Chapter 8: A Differential Point of View on Optimal Transportation 264264
- Chapter 9: Entropy Production and Transportation Inequalities 294294
- Chapter 10: Problems 334334
- Bibliography 376376
- Table of Short Statements 392392
- Index 398398
- Back Cover 402402