Cylindrical estimates for the Cheeger constant and applications

We prove a lower bound for the Cheeger constant of a cylinder $\mathrm{\Omega }\times (0,L)$, where $\mathrm{\Omega }$ 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)).