Optimal Concavity of the Torsion Function

It is well known that the torsion function of a convex domain has a square root which is concave. The power one half is optimal in the sense that no greater power ensures concavity for every convex set. In this paper, we investigate concavity, not of a power of the torsion function itself, but of the complement to its maximum. Requiring that the torsion function enjoys such a property for the power one half leads to an unconventional overdetermined problem. Our main result is to show that solutions of this problem exist, if and only if they are quadratic polynomials, finding, in fact, a new characterization of ellipsoids.