On the isoperimetric problem for radial log-convex densities

Given a smooth, radial, uniformly log-convex density e V on ℝn , n ≥ 2, we characterize isoperimetric sets E with respect to the corresponding weighted perimeter and weighted volume as balls centered at the origin, provided m∈[0,m0) for some (potentially computable) m 0>0; this affirmatively answers conjecture (Rosales et al. Calc Var Part Differ Equat 31(1):27–46, 2008, Conjecture 3.12) for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when V belongs to a C^2-neighborhood of c|x|^2 (c> 0).