On a Cheeger–Kohler-Jobin Inequality

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely (formula presented) denotes the torsional rigidity of a set Ω and h1(Ω) 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.