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040			_aIIITD
082			_a519.2 _bLED-C
100			_aLedoux, Michel
245			_aThe concentration of measure phenomenon _cby Michel Ledoux
260			_aProvidence : _bAmerican Mathematical Society, _c©2001
300			_ax, 181 p. ; _c26 cm.
490			_aMathematical surveys and monographs ; _vv. 89
504			_aIncludes bibliographical references and index.
505			_t1. Concentration functions and inequalities
505			_t2. Isoperimetric and functional examples
505			_t3. Concentration and geometry
505			_t4. Concentration in product spaces
505			_t5. Entropy and concentration
505			_t6. Transportation cost inequalities
505			_t7. Sharp bounds on Gaussian and empirical processes
505			_t8. Selected applications
520			_aThe 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 Levy. 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.
650			_aConcentration functions
650			_aMeasure theory
650			_aProbabilities
942			_cBK _2ddc
999			_c209646 _d209646