The Concentration of Measure Phenomenon  
Michel Ledoux
Amer Mathematical Society
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  L?vy. The phenomenon also occurs in probability, as a  version of the law of large numbers, due to Emil 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.