Geometric properties of optimizers for the maximum gradient of the torsion function

Consider J(Omega):= |nabla u_Omega|_∞/√|Omega| and JP(Omega):= |nabla u_Omega|_∞/P(Omega) , where Omega is a planar convex domain, u_Omega is the torsion function, P(Omega) is the perimeter of Omega and |Omega| its area. We prove that there exist planar convex domains that maximize the functionals J and JP, and any maximizer has a C1 boundary that contains a line segment on which |nabla u_Omega| attains its maximum.