Volume of the polar of random sets and shadow systems

We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities

Weighted norm inequalites and indices

We extend and simplify several classical results on weighted norm inequalities for classical operators acting on rearrangement invariant spaces using the theory of indices. As an application we obtain necessary and sufficient conditions for generalized Hardy type operators to be bounded on Λp(w), Λp,∞ (w), Γp(w) and Γp,∞(w)

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $\mu(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincar\'e inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities we had obtained in an earlier paper, and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.Comment: 32 pages, made a few minor changes and correction