A short proof of the ${\mathcal{C}}^{1,1}$ regularity for the eikonal equation

We give a short and self-contained proof of the interior ${\mathcal{C}}^{1,1}$ regularity of solutions $\phi :\mathrm{\Omega }\to \mathbb{R}$ to the eikonal equation $\mathrm{\mid }\mathrm{\nabla }\phi \mathrm{\mid }=1$ in an open set $\mathrm{\Omega }\subset {\mathbb{R}}^N$ in dimension $N\ge 1$ under the assumption that $\phi$ is pointwise differentiable in $\mathrm{\Omega }$.