Local isoperimetric inequalities in metric measure spaces verifying measure contraction property

TL;DR

This paper establishes local isoperimetric inequalities in metric measure spaces with measure contraction properties, linking geometric conditions and volume bounds to isoperimetric constants.

Contribution

It provides new local isoperimetric inequalities in MCP spaces under specific geometric and volume conditions, extending previous global results.

Findings

Isoperimetric constants estimated in smaller geodesic balls

Volume lower bounds are crucial for inequalities

Results apply to essentially non-branching MCP spaces

Abstract

We prove that on an essentially non-branching $\mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.