  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.
