A weighted relative isoperimetric inequality in convex cones

TL;DR

This paper establishes a new weighted relative isoperimetric inequality in convex cones using Monge-Ampere equations, improving existing inequalities and applying them to a generalized log-convex density conjecture with unique minimizers.

Contribution

It introduces a novel approach via Monge-Ampere equations to improve isoperimetric inequalities in convex cones and applies these results to a generalized density minimization problem.

Findings

Improved constants in isoperimetric inequalities.

Identification of unique minimizers in weighted density problems.

Extension of results to cones with vanishing densities on boundaries.

Abstract

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 $\{x_n=0\}$.