An isoperimetric inequality in the plane with a log-convex density

TL;DR

This paper proves that in a plane with a log-convex density, the optimal shape for the weighted isoperimetric problem is a centered ball, extending classical isoperimetric results to weighted settings.

Contribution

It establishes that centered balls minimize the weighted perimeter for given volume under a log-convex density, providing a new isoperimetric inequality in this context.

Findings

Centered balls are minimizers for the weighted isoperimetric problem.

Uniqueness of the minimizer is established.

The result applies to densities of the form e^{h(|x|)} with convex h.

Abstract

Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $\mathbb{R}^2$ is written $P_f(E)$. We study minimisers for the weighted isoperimetric problem \[ I_f(v):=\inf\Big\{ P_f(E):E\text{ is a set of finite perimeter in }\mathbb{R}^2\text{ and }V_f(E)=v\Big\} \] for $v>0$. Suppose $f$ takes the form $f:\mathbb{R}^2\rightarrow(0,+\infty);x\mapsto e^{h(|x|)}$ where $h:[0,+\infty)\rightarrow\mathbb{R}$ is a non-decreasing convex function. Let $v>0$ and $B$ a centred ball in $\mathbb{R}^2$ with $V_f(B)=v$. We show that $B$ is a minimiser for the above variational problem and obtain a uniqueness result.