Submanifolds, Isoperimetric Inequalities and Optimal Transportation
نویسنده
چکیده
The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp “weighted isoperimetric inequality” and a nonsharp classical inequality similar to the one obtained in [Mi-Si]. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension. Résumé : Le but de cet article est de démonter des inégalités isopérimétriques sur les sous-variétés de l’espace euclidien en utilisant des méthodes de transport optimal de mesures. On obtient ainsi une “inégalité isopérimétrique à poids” avec constante optimale et une inégalité classique similaire à celle obtenue dans [Mi-Si]. La preuve repose sur la description d’une solution du problème de Monge entre une mesure initiale supportée par une sous-variété et une mesure finale supportée par un sous-espace de même dimension. Mathematics Subject Classifications (2000) : 53C42, 51M16.
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