On the role of convexity in functional and isoperimetric inequalities

Abstract

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/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 (for example, 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 ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.