This book investigates the geometric properties of optimal transportation maps and their connection to functional inequalities. It explores the theoretical underpinnings of optimal transport, a field that deals with finding the most efficient way to move mass from one distribution to another. The work delves into the geometric structures that arise from these optimal mappings and their implications for understanding fundamental inequalities in analysis.

This work is recognized as a significant contribution to the field of optimal transportation and its relationship with functional inequalities. It is likely to be of interest to researchers and advanced students in mathematics, particularly those specializing in analysis, geometry, and probability theory. The book's focus on theoretical aspects suggests a rigorous and in-depth treatment of the subject matter.