A weighted relative isoperimetric inequality in convex cones (Q820866)

Let \(\mathcal{C}\subset\mathbb R^n\), \(n\geq2\), be an open convex cone and set \(K=B_1\cap\mathcal{C}\), where \(B_1\) is the unit ball of \(\mathbb R^n\) with center at the origin. Let \(E\subset\mathcal{C}\) be a set of finite perimeter with \(\mathcal{L}^n(E)=\mathcal{L}^n(K)\). Given \(h(x)\geq0\) with \(\inf_{\alpha\in K}\nabla h(x)\cdot\alpha\geq0\) for a.e. \(x\in\mathcal{C}\), the author proves that \[ n\int_{E}h~\mathrm{d}\mathcal{L}^n \leq \int_{\mathcal{C}\cap\hspace{0.1em}\partial^\ast E}h~\mathrm{d}\mathcal{H}^{n-1}, \] where \(\partial^\ast E\) stands for the reduced boundary of \(E\). Moreover, if \(\mathcal{C}\) is strictly contained in the half space \(\mathbb R_+^n:=\{x=(x_1,...,x_n)\in\mathbb R^n~|~x_n>0\}\), \(h=h(x_n)>0\) in \(\mathbb R_+^n\) with \(\int_Eh~\mathrm{d}\mathcal{L}^n<\infty\), and equality holds in the above inequality, then \(E=K\) up to sets of \(\mathcal{L}^n\)-measure zero. The method of proof (which is based on the Monge-Ampère equation) is further employed to improve a weighted isoperimetric inequality due to \textit{I. Cabrè} et al. [J. Eur. Math. Soc. (JEMS) 18, No. 12, 2971--2998 (2016; Zbl 1357.28007)], see Corollary 0.14. In addition, an application is given in the context of Brakke's log-convex density conjecture (resolved by \textit{G. R. Chambers} [J. Eur. Math. Soc. (JEMS) 21, No. 8, 2301--2332 (2019; Zbl 1423.49042)]), see Theorem 0.7.