  Denote with $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ the first nontrivial
eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll}
-\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu
e^{h\left(|x|\right)}u & \text{in} & \Omega & & \frac{\partial u}{\partial
\nu}=0 & \text{on} & \partial \Omega , \end{array} \right. \end{equation*}
where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under
suitable assumption on $h$ we prove that the ball centered at the origin is the
unique set maximizing $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ among all
Lipschitz bounded domains $\Omega $ of $\mathbb{R}^{N}$ of prescribed
$e^{h\left(|x|\right)}dx$-measure and symmetric about the origin. Moreover, an
example in the model case $h\left(|x|\right) =|x|^{2},$ shows that, in general,
the assumption on the symmetry of the domain cannot be dropped. In the
one-dimensional case, i.e. when $\Omega$ reduces to an interval $(a,b),$ we
consider a wide class of weights (including both Gaussian and anti-Gaussian).
We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides
along the $x$-axis keeping fixed its weighted length.
