A Cheeger--Kohler-Jobin inequality.

In this paper we prove the following Kohler-Jobin type inequality: for any open, bounded set $\Omega \subset \mathbb R^N$ and any ball $\subset \mathbb R^N$ we have $T (\Omega)^{ 1/ N +2} h\_1 (\Omega)\geq T (B)^{1/ N +2} h\_1 (B)$, where $T$ denotes the torsional rigidity and $h\_1$ the Cheeger constant. Moreover, equality holds if and only if $$\Omega$$ is a ball. We then exploit such inequality to provide a new proof of the sharp quantitative Cheeger inequality. Eventually, we extend these results to the nonlocal framework.