  Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this
paper we mainly study some properties of the second Dirichlet eigenvalue
$\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcal
Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] where
$F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We
provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of
given measure, showing the validity of a Hong-Krahn-Szego type inequality.
Furthermore, we investigate the limit problem as $p\to+\infty$.
