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

Abstract

Given a positive lower semi-continuous density $f$ on ${\mathbb{R}}^2$ the weighted volume $V_f:=f{\mathcal{L}}^2$ is defined on the ${\mathcal{L}}^2$-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 $f:{\mathbb{R}}^2\to (0,+\infty );x\mapsto e^{h(|x|)}$ where $h:[0,+\infty )\to \mathbb{R}$ 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.

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