A weighted isoperimetric inequality in a wedge

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.