  This paper deals with weighted isoperimetric inequalities relative to cones
of $\mathbb{R}^{N}$. We study the structure of measures that admit as
isoperimetric sets the intersection of a cone with balls centered at the vertex
of the cone. For instance, in case that the cone is the half-space
$\mathbb{R}_{+}^{N}={x \in \mathbb{R}^{N} : x_{N}>0}$ and the measure is
factorized, we prove that this phenomenon occurs if and only if the measure has
the form $d\mu=ax_{N}^{k}\exp(c|x|^{2})dx $, for some $a>0$, $k,c\geq 0$. Our
results are then used to obtain isoperimetric estimates for Neumann eigenvalues
of a weighted Laplace-Beltrami operator on the sphere, sharp Hardy-type
inequalities for functions defined in a quarter space and, finally, via
symmetrization arguments, a comparison result for a class of degenerate PDE's.
