Dilation type inequalities for strongly-convex sets in weighted Riemannian manifolds

TL;DR

This paper studies dilation inequalities in weighted Riemannian manifolds, extending classical convex geometry results, and derives functional inequalities related to entropy under curvature bounds.

Contribution

It introduces the dilation profile concept and compares it with model spaces to establish new isoperimetric and functional inequalities under curvature conditions.

Findings

Dilation profiles are estimated by model spaces with Ricci curvature bounds.

New functional inequalities related to entropy are derived.

Connections between dilation inequalities and isoperimetric properties are established.

Abstract

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell's lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.