Given a positive lower semi-continuous density f on R2 the
weighted volume Vf:=fL2 is defined on the
L2-measurable sets in R2. The f-weighted perimeter
of a set of finite perimeter E in R2 is written Pf(E). We
study minimisers for the weighted isoperimetric problem If(v):=inf{Pf(E):E is a set of finite perimeter in R2 and Vf(E)=v} for v>0. Suppose f takes the form
f:R2→(0,+∞);x↦eh(∣x∣) where
h:[0,+∞)→R is a non-decreasing convex function. Let
v>0 and B a centred ball in R2 with Vf(B)=v. We show that
B is a minimiser for the above variational problem and obtain a uniqueness
result