A weighted relative isoperimetric inequality in convex cones

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