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The Concentration of Measure Phenomenon (Paperback)
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The Concentration of Measure Phenomenon (Paperback)
Series: Mathematical Surveys and Monographs
Expected to ship within 12 - 17 working days
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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 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. This 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.
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