Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature

TL;DR

This paper establishes a sharp, dimension-free isoperimetric inequality in metric measure spaces with non-negative Ricci curvature using optimal transport, revealing conditions for equality and rigidity in both smooth and non-smooth settings.

Contribution

It introduces a new sharp isoperimetric inequality involving volume entropy and characterizes equality cases via foliation and splitting properties.

Findings

The inequality is sharp and dimension-free.

Equality cases correspond to specific foliation or splitting conditions.

Provides new insights into logarithmically concave measures.

Abstract

By using optimal transport theory, we prove a sharp dimension-free isoperimetric inequality involving the volume entropy, in metric measure spaces with non-negative Ricci curvature in the sense of Lott--Sturm--Villani. We show that this isoperimetric inequality is attained by a non-trivial open set, if and only if the space satisfies a certain foliation property. For metric measure spaces with non-negative Riemannian Ricci curvature, we show that the sharp Cheeger constant is achieved by a non-trivial measurable set, if and only if a one-dimensional space is split off. Our isoperimetric inequality and the rigidity theorems are proved in non-smooth framework, but new even in the smooth setting. In particular, our results provide some new understanding of logarithmically concave measures.