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

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 $$I_f(v):=\inf \{P_f(E):E\text{ is a set of finite perimeter in }{\mathbb{R}}^2\text{ and }{V}_f(E)=v\}$$ for $v>0$. Suppose $f$ takes the form $f:{\mathbb{R}}^2\to (0,+\mathrm{\infty });x\mapsto e^{h(\mathrm{\mid }x\mathrm{\mid })}$ where $h:[0,+\mathrm{\infty })\to \mathbb{R}$ 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.