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

Consider $J(\mathrm{\Omega }):=\mathrm{\parallel }\mathrm{\nabla }{u}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}\sqrt{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}$ and $J_P(\mathrm{\Omega }):=\mathrm{\parallel }\mathrm{\nabla }{u}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}P(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a planar convex domain, $u_{\mathrm{\Omega }}$ is the torsion function, $P(\mathrm{\Omega })$ is the perimeter of $\mathrm{\Omega }$ and $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $\mathrm{\mid }\mathrm{\nabla }{u}_{\mathrm{\Omega }}\mathrm{\mid }$ attains its maximum.