December 1995, Volume 81, Issue 1, pp 73-205
Concentration of measure and isoperimetric inequalities in product spaces
- Michel Talagrand
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The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous by isoperimetrictype 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 to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.
Dedicated to Vitali Milman
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- Concentration of measure and isoperimetric inequalities in product spaces
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
Volume 81, Issue 1 , pp 73-205
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- Primary 60E15, 28A35, 60G99
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- Michel Talagrand (1)
- Author Affiliations
- 1. Equipe d’Analyse - Tour 48 UA au CNRS no 754, Université Paris VI, 4 Pl. Jussieu, 75230, Paris Cedex 05