Topics in Optimal Transportation (Graduate Studies in Mathematics)

About this book

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 Monges 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.