Analysis Seminar

Some isoperimetric inequality involving the boundary momentum and curvature integrals

by Dr Rossano Sannipoli (Università di Pisa)

Sala Riunioni (Dipartimento di Matematica)

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.