  We study a general version of the Cheeger inequality by considering the shape
functional
$\mathcal{F}_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q(\Omega)^{1/q}$.
The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of
all domains $\Omega$ of $\mathbb{R}^d$ and in the subclass of convex domains.
In the latter case the issue concerning the existence of an optimal domain for
$\mathcal{F}_{p,q}$ is discussed.
