Limiting case of an isoperimetric inequality with radial density and applications

We prove a sharp isoperimetric inequality with radial density whose functional counterpart corresponds to a limiting case for the exponents of the Il'in (or Caffarelli-Kohn-Nirenberg) inequality in L 1. We show how the latter applies to obtain an optimal critical Sobolev weighted norm improvement to one of the L 1 weighted Hardy inequalities of [25]. Further applications include an L p version with the best constant of the functional analogue of this isoperimetric inequality and also a weighted Pólya-Szegö inequality