On a Cheeger–Kohler-Jobin Inequality

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely \( \min \Big \{ T_2(\Omega ) ^{\frac{1}{N+2}}h_1(\Omega ) : \Omega \subset \mathbb R^N,\text { open, convex, bounded}\Big \}\, \) 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.