Cylindrical estimates for the Cheeger constant and applications

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 $\R^N$. This positively solves open conjectures raised by Parini (\emph{J.\ Convex Anal.}\ (2017)) and by Briani--Buttazzo--Prinari (\emph{Ann.\ Mat.\ Pura Appl.}\ (2023)).