A fundamental topological property of distance functions in Hilbert spaces

Let $E$ be a closed nonempty subset of a general real Hilbert space $H$. Its singular set $\Sigma _E$ consists of those points in $H\setminus E$ at which the distance function $d_E$ fails to be Fréchet differentiable. In particular, this paper demonstrates in full generality that $\Sigma _E$ is of the same homotopy type as the open set $\mathcal G_E=\{x\in H\colon d_{\overline {\rm co}\, E}(x) \lt d_E(x)\}$ consisting of the points whose distance to the closed convex hull of $E$ is strictly smaller than to $E$ itself. Moreover, it is shown that $\mathcal G_E$ is intimately connected to the Aubry set of $E$. In the literature, the singular set $\Sigma _E$ is also known as the medial axis of $E$ when $\dim H \lt \infty $.