We study the gain in regularity of the distance to the boundary of a domain in \({\mathbb {R}}^m\) . In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be \({\mathcal {C}}^{1,1}\) regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz.
