  A weighted relative isoperimetric inequality in convex cones is obtained via
the Monge-Ampere equation. The method improves several inequalities in the
literature, e.g. constants in a theorem of Cabre--Ros--Oton--Serra.
Applications are given in the context of a generalization of the log-convex
density conjecture due to Brakke and resolved by Chambers: in the case of
$\alpha-$homogeneous ($\alpha>0$), concave densities, (mod translations) balls
centered at the origin and intersected with the cone are proved to uniquely
minimize the weighted perimeter with a weighted mass constraint. In particular,
if the cone is taken to be $\{x_n>0\}$, reflecting the density, balls
intersected with $\{x_n>0\}$ remain (mod translations) unique minimizers in the
$\mathbb{R}^n$ analog in the case when the density vanishes on $\{x_n=0\}$.
