Concentration of Measure and Isoperimetric Inequalities in Product\n Spaces

The concentration of measure prenomenon roughly states that, if a set $A$ in\na product $\\Omega^N$ of probability spaces has measure at least one half,\n``most'' of the points of $\\Omega^N$ are ``close'' to $A$. We proceed to a\nsystematic exploration of this phenomenon. The meaning of the word ``most'' is\nmade rigorous by isoperimetric-type inequalities that bound the measure of the\nexceptional sets. The meaning of the work ``close'' is defined in three main\nways, each of them giving rise to related, but different inequalities. The\ninequalities are all proved through a common scheme of proof. Remarkably, this\nsimple approach not only yields qualitatively optimal results, but, in many\ncases, captures near optimal numerical constants. A large number of\napplications are given, in particular in Percolation, Geometric Probability,\nProbability in Banach Spaces, to demonstrate in concrete situations the\nextremely wide range of application of the abstract tools.\n