Level sets of solutions to the stationary Hamilton-Jacobi equation are John regular

Let $u$ be the unique nonnegative viscosity solution of the Hamilton-Jacobi equation $H(x,\nabla u)=0$ in the external domain ${\mathbb R}^{ n} \setminus K$ with $u=0$ on $K$. Under general conditions on $H$, we prove that all sublevels of $u$ are John domains. Moreover, if $K$ itself is a John domain, we provide a uniform lower bound on the John constant of all sublevels. We exhibit counterexamples showing that John regularity is sharp in this setting.