  The isoperimetric problem with a density or weighting seeks to enclose
prescribed weighted volume with minimum weighted perimeter. According to
Chambers' recent proof of the log-convex density conjecture, for many densities
on $\mathbb{R}^n$ the answer is a sphere about the origin. We seek to
generalize his results to some other spaces of revolution or to two different
densities for volume and perimeter. We provide general results on existence and
boundedness and a new approach to proving circles about the origin
isoperimetric.
