On the singularities of distance functions in Hilbert spaces

For a given closed nonempty subset E of a Hilbert space H, the singular set \(\Sigma _E\) ${\mathrm{\Sigma }}_E$ consists of the points in \(H\setminus E\) $H\backslash E$ where the distance function \(d_E\) $d_E$ is not Fréchet differentiable. It is known that \(\Sigma _E\) ${\mathrm{\Sigma }}_E$ is a weak deformation retract of the open set \(\mathcal {G}_E=\{x\in H: d_{\overline{{\text {co}}}\,E}(x)< d_E(x)\}\) ${\mathcal{G}}_E=\{x\in H:d_{\overline{\text{co}}E}(x)<d_E(x)\}$ . This short paper sheds light on the relationship between the connected components of the three sets \(\Sigma _E\subset \mathcal {G}_E\subseteq H{\setminus } E\) ${\mathrm{\Sigma }}_E\subset {\mathcal{G}}_E\subseteq H\backslash E$ .