The Concentration of Measure Phenomenon
About this Title
Michel Ledoux, Université Paul-Sabatier, Toulouse, France
Publication: Mathematical Surveys and Monographs
Publication Year 2001: Volume 89
ISBNs: 978-0-8218-3792-4 (print); 978-1-4704-1316-3 (online)
MathSciNet review: MR1849347
MSC: Primary 28C15; Secondary 28A35, 46B09, 60E15, 82B44
It was undoubtedly a necessary task to collect all the results on the concentration of measure during the past years in a monograph. The author did this very successfully and the book is an important contribution to the topic. It will surely influence further research in this area considerably. The book is very well written, and it was a great pleasure for the reviewer to read it.
The observation of the concentration of measure phenomenon is inspired by isoperimetric inequalities. A familiar example is the way the uniform measure on the standard sphere $S^n$ becomes concentrated around the equator as the dimension gets large. This property may be interpreted in terms of functions on the sphere with small oscillations, an idea going back to Lévy. The phenomenon also occurs in probability, as a version of the law of large numbers, due to Emile Borel. This book offers the basic techniques and examples of the concentration of measure phenomenon. The concentration of measure phenomenon was put forward in the early seventies by V. Milman in the asymptotic geometry of Banach spaces. It is of powerful interest in applications in various areas, such as geometry, functional analysis and infinite-dimensional integration, discrete mathematics and complexity theory, and probability theory. Particular emphasis is on geometric, functional, and probabilistic tools to reach and describe measure concentration in a number of settings.
The book presents concentration functions and inequalities, isoperimetric and functional examples, spectrum and topological applications, product measures, entropic and transportation methods, as well as aspects of M. Talagrand's deep investigation of concentration in product spaces and its application in discrete mathematics and probability theory, supremum of Gaussian and empirical processes, spin glass, random matrices, etc. Prerequisites are a basic background in measure theory, functional analysis, and probability theory.
Graduate students and research mathematicians interested in measure and integration, functional analysis, convex and discrete geometry, and probability theory and stochastic processes.
Table of Contents
- 1. Concentration functions and inequalities
- 2. Isoperimetric and functional examples
- 3. Concentration and geometry
- 4. Concentration in product spaces
- 5. Entropy and concentration
- 6. Transportation cost inequalities
- 7. Sharp bounds on Gaussian and empirical processes
- 8. Selected applications