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We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect toweights that are powers of the distance to the origin. For instance we showthat if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \,\mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin.Our results also imply a weighted Polya-Sz\"ego principle. In turn, weestablish radiality of optimizers in some Caffarelli-Kohn-Nirenberginequalities, and we obtain sharp bounds for eigenvalues of some nonlinearproblems.

Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $