On a Cheeger--Kohler-Jobin inequality

Abstract

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(\Omega {)}^{\frac{1}{N+2}}{h}_1(\Omega )$ among open convex bounded sets $\Omega \subset {\mathbb{R}}^N$, where $T_2(\Omega )$ denotes the torsional rigidity of a set $\Omega$ and $h_1(\Omega )$ its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existence for the problem above among several other classes of sets. As a side result we discuss the equivalence of the several definitions of Cheeger constants present in the literature and show a quite general class of sets for which those are equivalent.