  We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to
weights that are powers of the distance to the origin. For instance we show
that 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, we
establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg
inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear
problems.
