On the role of Convexity in Functional and Isoperimetric Inequalities

TL;DR

This paper explores how convexity assumptions enable reversing the usual implications between isoperimetric and functional inequalities, unifying various inequalities and extending known results to broader geometric contexts.

Contribution

It develops a unified framework linking isoperimetric, Orlicz-Sobolev, and capacity inequalities under convexity, and extends stability and equivalence results to Riemannian manifolds with density.

Findings

Reversal of isoperimetric-functional inequality implications under convexity.

Extension of stability of isoperimetric profiles under tensorization.

Equivalence of q-log-Sobolev and isoperimetric inequalities on convex spaces.

Abstract

This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, $q$-log-Sobolev inequalities ($q \in [1,2]$) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry--Ledoux and Bobkov--Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\infty)$ curvature-dimension condition of Bakry--\'Emery.