Some isoperimetric inequality involving the boundary momentum and curvature integrals

by Dr Rossano Sannipoli (Università di Pisa)

Thursday 11 Jan 2024, 17:00 → 18:00 Europe/Rome

Sala Riunioni (Dipartimento di Matematica)

The aim of this talk is twofold. In the first part we deal with a shape optimization problem of 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. 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 bidimensional case. For this functional we prove that the ball is a stable maximizer in the class of nearly spherical sets in any dimension.  In the second 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.