Weinstock inequality in higher dimensions

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in ${\mathbb{R}}^n$, for $n\ge 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultanously 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