On the extremals of the P\'olya-Szeg\H{o} inequality

The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a translate of its symmetric decreasing rearrangement is controlled by the measure of the set of critical points.Comment: 17 pages, 3 figure

Rearrangement inequalities for functionals with monotone integrands

The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy-Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.Comment: 20 pages. Slightly re-organized, four added reference

Rate of convergence of random polarizations

After n random polarizations of Borel set on a sphere, its expected symmetric difference from a polar cap is bounded by C/n, where the constant depends on the dimension [arXiv:1104.4103]. We show here that this power law is best possible, and that the constant grows at least linearly with the dimension.Comment: 5 page