  Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with
a perturbation of the Gaussian measure of the form
$f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave
function depending only on the signed distance from the linear hyperplane
parallel to $\partial\Omega$. In this work we follow a variational approach to
show that half-spaces perpendicular to $\partial\Omega$ uniquely minimize the
weighted perimeter in $\Omega$ among sets enclosing the same weighted volume.
The main ingredient of the proof is the characterization of half-spaces
parallel or perpendicular to $\partial\Omega$ as the unique stable sets with
small singular set and null weighted capacity. Our methods also apply for
$\Omega=\mathbb{R}^{n+1}$, which produces in particular the classification of
stable sets in Gauss space and a new proof of the Gaussian isoperimetric
inequality. Finally, we use optimal transport to study the weighted minimizers
when the perturbation term $\omega$ is concave and possibly non-smooth.
