Some isoperimetric inequalities involving the boundary momentum

Abstract

The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a stronger form of the classical Aleksandrov-Fenchel inequality involving the $(n-1)$ and $(n-2)$-quermassintegrals, and consequently a stronger form of the classical isoperimetric inequality in the planar case. Moreover, quantitative estimates are proved. In the second part we deal with a shape optimization problem for a functional involving the boundary momentum. It is known that in dimension two the ball is a maximizer among simply connected sets when the perimeter and centroid is fixed. We show that the result still holds in the class of undecomposable sets. In higher dimensions the same result does not hold and we consider a new scaling invariant functional that might be a good candidate to generalize the planar case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.