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Concentration of Measure and Isoperimetric Inequalities in Product Spaces
The concentration of measure prenomenon roughly states that, if a set in
a product of probability spaces has measure at least one half,
``most'' of the points of are ``close'' to . We proceed to a
systematic exploration of this phenomenon. The meaning of the word ``most'' is
made rigorous by isoperimetric-type inequalities that bound the measure of the
exceptional sets. The meaning of the work ``close'' is defined in three main
ways, each of them giving rise to related, but different inequalities. The
inequalities are all proved through a common scheme of proof. Remarkably, this
simple approach not only yields qualitatively optimal results, but, in many
cases, captures near optimal numerical constants. A large number of
applications are given, in particular in Percolation, Geometric Probability,
Probability in Banach Spaces, to demonstrate in concrete situations the
extremely wide range of application of the abstract tools