Weinstock inequality in higher dimensions

TL;DR

This paper extends the Weinstock inequality to higher dimensions for convex sets with fixed surface area, using a new sharp isoperimetric inequality involving surface area, volume, and boundary momentum.

Contribution

It introduces a novel sharp isoperimetric inequality that enables the extension of the Weinstock inequality to higher dimensions for convex sets.

Findings

Weinstock inequality holds in $\

n ext{ for convex sets with prescribed surface area}

New isoperimetric inequalities for the first Wentzell eigenvalue.

Abstract

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