  We prove that, if $\Omega$ is an open bounded domain with smooth and
connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet
eigenvalue of the normalized $p$-Laplacian is simple in the sense that two
positive eigenfunctions are necessarily multiple of each other. We also give a
(non-optimal) lower bound for the eigenvalue in terms of the measure of
$\Omega$, and we address the open problem of proving a Faber-Krahn type
inequality with balls as optimal domains.
