Submanifolds, isoperimetric inequalities and optimal transportation

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 [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...

Which Ambient Spaces Admit Isoperimetric Inequalities for Submanifolds?

We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold.

A Mass Transportation Approach to Quantitative Isoperimetric Inequalities

A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for...

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...