A weighted isoperimetric inequality in a wedge

Abstract

Let $c,k_1,...,k_N$ be non-negative numbers, and define a measure $\mu$ in the wedge $W:=\{x\in {\mathbb{R}}^N:x_i>0,i=1,...,N\}$ by $d\mu =e^{c|x{|}^2}{x}_1^{k_1}...x_N^{k_N}dx$. It is shown that among all measurable subsets of $W$ with fixed $\mu$ -measure, the intersection of $W$ with a ball centered at the origin renders the weighted perimeter relative to $W$ a minimum.