  We study isoperimetric problems with respect to infinite measures on $R ^n$.
In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2} dx$, $c\geq 0$,
we prove that, among all sets with given $\mu-$measure, the ball centered at
the origin has the smallest (weighted) $\mu-$perimeter. Our results are then
applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems
and a comparison result for elliptic boundary value problems.
