  This paper deals with various questions related to the isoperimetic problem
for smooth positive measure $d\mu = \varphi(x)dx$, with $x \in \Omega \subset
\mathbb{R}^N$. Firstly we find some necessary conditions on the density of the
measure $ \varphi(x)$ that render the intersection of half spaces with $\Omega$
a minimum in the isoperimetric problem. We then identify the unique
isoperimetric set for a wide class of factorized finite measures. These results
are finally used in order to get sharp inequalities in weighted Sobolev spaces
and a comparison result for solutions to boundary value problems for degenerate
elliptic equations.
