Rigidity results for variational infinity ground states

We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $Omega subset R ^n$. They state that $u$ coincides with a multiple of the distance from the boundary of $Omega$ if either $| abla u|$ is constant on $partial Omega$, or $u$ is of class $C ^ {1,1}$ outside the high ridge of $Omega$. Consequently, in both cases $Omega$ can be geometrically characterized as a ``stadium-like domain''.