Cylindrical estimates for the Cheeger constant and applications

We prove a lower bound for the Cheeger constant of a cylinder Ω× (0,L), where Ω 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 RN. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani-Buttazzo-Prinari (Ann. Mat. Pura Appl. (2023)).