Metric measure geometry

TL;DR

This paper provides a detailed exposition of Gromov's metric measure geometry, focusing on the concentration of measure phenomenon and curvature bounds, with complete proofs and new insights.

Contribution

It offers comprehensive proofs and clarifies aspects of Gromov's theory that were previously omitted or not fully detailed in literature.

Findings

Detailed proofs of key results in metric measure geometry

Extension of concentration of measure with curvature bounds

Clarification of Gromov's original ideas

Abstract

In this book, we study Gromov's metric geometric theory on the space of metric measure spaces, based on the idea of concentration of measure phenomenon due to L\'evy and Milman. Although most of the details are omitted in the original article of Gromov, we present complete and detailed proofs for some main parts, in which we prove several claims that are not mentioned in any literature. We also discuss concentration with a lower bound of curvature, originally studied by Funano and the author.