A Blaschke-Lebesgue theorem for the Cheeger constant

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p-Laplacian for any p between 1 and plus infinity (this paper covers the case p=1 whereas the case p=+∞ was already known).