Isoperimetric Bounds on Convex Manifolds

TL;DR

This paper generalizes Cheeger-type isoperimetric bounds from convex sets in Euclidean space to Riemannian manifolds with non-negative Ricci curvature, incorporating curvature bounds and improving previous concentration-based results.

Contribution

It extends isoperimetric bounds to convex manifolds with curvature constraints and refines earlier concentration-based bounds to be sharp across all parameters.

Findings

Extended Cheeger-type bounds to Riemannian manifolds with curvature bounds.

Used comparison tools in CAT spaces for the extension.

Provided a sharp, parameter-sensitive isoperimetric bound.

Abstract

We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lov\'asz-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits us to use comparison tools in Cartan-Alexandrov-Toponogov (or CAT) spaces. Along the way, we also quantitatively improve our previous result that weak concentration assumptions imply a Cheeger-type isoperimetric bound, to a sharp bound with respect to all parameters.