On the role of Convexity in Functional and Isoperimetric Inequalities

Details On the role of Convexity in Functional and Isoperimetric Inequalities On the role of Convexity in Functional and Isoperimetric Inequalities

2008-04-03

arxiv.org/abs/0804.0453v2

Mathematics

Functional Analysis

42 pages, to appear in the Proceedings of the London Mathematical Society; small revision

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

On the role of Convexity in Functional and Isoperimetric Inequalities does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the role of Convexity in Functional and Isoperimetric Inequalities, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the role of Convexity in Functional and Isoperimetric Inequalities will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-351942