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 $V_f:=fmathscr{L}^2$ 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):=infBig{ P_f(E):Etext{ is a set of finite perimeter in }mathbb{R}^2text{ and }V_f(E)=vBig} ] for $v>0$. Suppose $f$ takes the form $f:mathbb{R}^2rightarrow(0,+infty);xmapsto e^{h(|x|)}$ where $h:[0,+infty)rightarrowmathbb{R}$ 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.