  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.
