A weighted relative isoperimetric inequality in convex cones

A weighted relative isoperimetric inequality in convex cones is obtained via the Monge–Ampére 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 the $\operatorname{log}$‑convex density conjecture due to Brakke and resolved by Chambers: in the case of $\alpha$‑homogeneous $(\alpha \gt 0)$, concave densities, ($\operatorname{mod}$ 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 $\lbrace x_n \gt 0 \rbrace$, reflecting the density, balls intersected with $\lbrace x_n \gt 0 \rbrace$ remain ($\operatorname{mod}$ translations) unique minimizers in the $\mathbb{R}^n$ analog in the case when the density vanishes on $\lbrace x_n = 0\rbrace$.

Keywords

Monge–Ampére, isoperimetric inequality, $\operatorname{log}$-convex density conjecture, convex cones, relative isoperimetric inequalities, partial differential equations

2010 Mathematics Subject Classification

35-xx, 49-xx, 51-xx, 52-xx, 53-xx

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Received 15 June 2020

Accepted 15 December 2020

Published 1 December 2021