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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 $$begin{aligned} I_f(v):=inf Big  P_f(E):Etext { is a set of finite perimeter in }mathbb {R}^2text { and }V_f(E)=vBig } end{aligned}$$for (vu003e0). Suppose f takes the form (f:mathbb {R}^2rightarrow (0,+infty );xmapsto e^{h(|x|)}) where (h:[0,+infty )rightarrow mathbb {R}) is a non-decreasing convex function. Let (vu003e0) 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.

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