# Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds

## TL;DR

This paper proves that spaces with local Ricci curvature bounds and nearly maximal volume exhibit almost Euclidean isoperimetric inequalities in smaller regions, extending to non-smooth spaces via optimal transportation.

## Contribution

It establishes almost Euclidean isoperimetric inequalities under local Ricci bounds in both smooth and non-smooth spaces, generalizing previous results and inspired by Perelman's Pseudo Locality Theorem.

## Key findings

- Spaces with Ricci lower bounds and large volume have almost Euclidean isoperimetric properties.
- The results apply to non-smooth metric measure spaces satisfying synthetic Ricci curvature bounds.
- The inequalities are quantified in smaller regions within the manifold or space.

## Abstract

Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.