**Mathematical Surveys and Monographs**

Volume: 89;
2001;
181 pp;
Softcover

MSC: Primary 28; 46; 52; 60;
Secondary 58; 62; 82

Print ISBN: 978-0-8218-3792-4

Product Code: SURV/89.S

List Price: $72.00

Individual Member Price: $57.60

**Electronic ISBN: 978-1-4704-1316-3
Product Code: SURV/89.S.E**

List Price: $72.00

Individual Member Price: $57.60

# The Concentration of Measure Phenomenon

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*Michel Ledoux*

*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.

#### Readership

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

# Table of Contents

## The Concentration of Measure Phenomenon

- CONTENTS v6 free
- INTRODUCTION vii8 free
- 1. CONCENTRATION FUNCTIONS AND INEQUALITIES 112 free
- 2. ISOPERIMETRIC AND FUNCTIONAL EXAMPLES 2334
- 3. CONCENTRATION AND GEOMETRY 4758
- 4. CONCENTRATION IN PRODUCT SPACES 6778
- 5. ENTROPY AND CONCENTRATION 91102
- 6. TRANSPORTATION COST INEQUALITIES 117128
- 7. SHARP BOUNDS ON GAUSSIAN AND EMPIRICAL PROCESSES 133144
- 8. SELECTED APPLICATIONS 151162
- REFERENCES 171182
- INDEX 181192