Submanifolds, Isoperimetric Inequalities and Optimal Transportation

TL;DR

This paper develops new isoperimetric inequalities for submanifolds in Euclidean space using optimal transportation techniques, including sharp weighted and classical inequalities, extending previous results with novel mass transportation methods.

Contribution

It introduces a new approach to isoperimetric inequalities on submanifolds via mass transportation, providing sharp and nonsharp inequalities with novel proofs.

Findings

Established a sharp weighted isoperimetric inequality.

Derived a nonsharp classical isoperimetric inequality.

Extended the Monge problem solution to submanifold-supported measures.

Abstract

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 by J. Michael and L. Simon. 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.