Isoperimetric inequality under Measure-Contraction property

TL;DR

This paper establishes a sharp isoperimetric inequality for metric measure spaces with Ricci curvature bounds under the Measure-Contraction property, extending classical results to a synthetic curvature setting.

Contribution

It proves a sharp isoperimetric inequality and measure theoretic rigidity for spaces satisfying the Measure-Contraction property with Ricci curvature bounds.

Findings

Sharp isoperimetric inequality proven for these spaces

Measure theoretic rigidity results obtained

Extension of classical inequalities to synthetic curvature bounds

Abstract

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality \`a la L\'evy-Gromov holds true. Measure theoretic rigidity is also obtained.