  Given $\Omega$ bounded open set of $\mathbb R^{n}$ and $\alpha\in \mathbb R$,
let us consider \[ \mu(\Omega,\alpha)=\min_{\substack{v\in
W_{0}^{1,2}(\Omega)\\v\not\equiv 0}} \frac{\displaystyle\int_{\Omega} |\nabla
v|^{2}dx+\alpha \left|\displaystyle\int_{\Omega}|v|v\,dx
\right|}{\displaystyle\int_{\Omega} |v|^{2}dx}. \] We study some properties of
$\mu(\Omega,\alpha)$ and of its minimizers, and, depending on $\alpha$, we
determine the set $\Omega_{\alpha}$ among those of fixed measure such that
$\mu(\Omega_{\alpha},\alpha)$ is the smallest possible.
