Mathematical Surveys and Monographs 2001; 181 pp; softcover Volume: 89 Reprint/Revision History: reprinted 2005 ISBN10: 0821837923 ISBN13: 9780821837924 List Price: US$68 Member Price: US$54.40 Order Code: SURV/89.S
 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. Mathematical Reviews 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 infinitedimensional 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  Concentration functions and inequalities
 Isoperimetric and functional examples
 Concentration and geometry
 Concentration in product spaces
 Entropy and concentration
 Transportation cost inequalities
 Sharp bounds of Gaussian and empirical processes
 Selected applications
 References
 Index
