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A generalized localization theorem and geometric inequalities for convex bodies
Authors:M Fradelizi  O Guédon
Institution:a Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, Cedex 2, France
b Université Pierre et Marie Curie, Équipe d’Analyse Fonctionnelle, Institut de Mathématiques, bo?ˆte 186, 4 Place Jussieu, 75252 Paris Cedex 05, France
Abstract:In this article, we generalize a localization theorem of Lovász and Simonovits Random walks in a convex body and an improved volume algorithm, Random Struct. Algorithms 4-4 (1993) 359-412] which is an important tool to prove dimension-free functional inequalities for log-concave measures. In a previous paper Fradelizi and Guédon, The extreme points of subsets of s-concave probabilities and a geometric localization theorem, Discrete Comput. Geom. 31 (2004) 327-335], we proved that the localization may be deduced from a suitable application of Krein-Milman's theorem to a subset of log-concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull. Here, we generalize this result to more constraints, give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem. Finally, using this new localization theorem, we solve an open question on the comparison of the volume of sections of non-symmetric convex bodies in Rn by hyperplanes. A surprising feature of the result is that the extremal case in this geometric inequality is reached by an unusual convex set that we manage to identify.
Keywords:primary 52A40  46A55  28A25  secondary 26D15  46B22
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