  We provide isoperimetric Szeg\"{o}-Weinberger type inequalities for the first
nontrivial Neumann eigenvalue $\mu_{1}(\Omega)$ in Gauss space, where $\Omega$
is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in
showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin,
having prescribed Gaussian measure, $\mu_{1}(\Omega)$ is maximum if and only if
$\Omega$ is the euclidean ball centered at the origin.
