Submanifolds, isoperimetric inequalities and optimal transportation

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 Michael and Simon (1973) [19]. 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. © 2010 Elsevier Inc. All rights reserved