  We study the isoperimetric problem for Euclidean space endowed with a
continuous density. In dimension one, we characterize isoperimetric regions for
a unimodal density. In higher dimensions, we prove existence results and we
derive stability conditions, which lead to the conjecture that for a radial
log-convex density, balls about the origin are isoperimetric regions. Finally,
we prove this conjecture and the uniqueness of minimizers for the density $\exp
(|x|^2)$ by using symmetrization techniques.
