  We prove a lower bound for the Cheeger constant of a cylinder $\Omega\times
(0,L)$, where $\Omega$ is an open and bounded set. As a consequence, we obtain
existence of minimizers for the shape functional defined as the ratio between
the first Dirichlet eigenvalue of the $p$-Laplacian and the $p$-th power of the
Cheeger constant, within the class of bounded convex sets in any
$\mathbb{R}^N$. This positively solves open conjectures raised by Parini (J.
Convex Anal. (2017)) and by Briani-Buttazzo-Prinari (Ann. Mat. Pura Appl.
(2023)).
