Minimal cones and a problem of Euler

Abstract

The minimal cones $C=\{(x,y)\in {\mathbb{R}}^m\times {\mathbb{R}}^m;|x|\le |y|\}$ are shown to minimize the weighted perimeter $P_{\alpha }(E)=\int |x{|}^{\alpha }|D{\phi }_E|$, $\alpha \in \mathbb{R}$, $E\subset {\mathbb{R}}^m\times {\mathbb{R}}^m$, whenever $m+\alpha \ge 4$. This completes recent results of Dierkes and Huisken [Math. Ann. (2023)].