Weinstock inequality in higher dimensions

Abstract

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n \geq 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.