On the singularities of distance functions in Hilbert spaces (Q6536633)

Let \(E\) be a closed nonempty subset of a real Hilbert space \(H\). The singular set \(\Sigma_E\) is the set of all \(x \in H \setminus E\) where the distance function \(d_E(x) = \inf_{y \in E} \|x-y\|\) is not Fréchet differentiable. In previous work [\textit{T. Strömberg}, ``A fundamental topological property of distance functions in Hilbert spaces'', Stud. Math. (to appear; \url{doi:10.4064/sm230920-27-8})], the author has shown that \(\Sigma_E\) is a weak deformation retract of the open set \(\mathcal G_E := \{x \in H : d_{\overline{\operatorname{co}}\, E}(x) < d_E(x) \}\), where \(\overline{\operatorname{co}}\, E\) is the closed convex hull of \(E\). The paper under review clarifies the relationship between the connected components of the three sets \(\Sigma_E \subseteq \mathcal G_E \subseteq H\setminus E\). It generalizes results obtained in [\textit{P. Cannarsa} and \textit{R. Peirone}, Trans. Am. Math. Soc. 353, No. 11, 4567--4581 (2001; Zbl 1021.49013)] in finite dimensions to the general Hilbert space setting. The proof builds on the author's work in [\textit{T. Strömberg}, Math. Ann. 388, No. 2, 1119--1161 (2024; Zbl 1532.49023)] and [\textit{T. Strömberg}, ``A fundamental topological property of distance functions in Hilbert spaces'', Stud. Math. (to appear; \url{doi:10.4064/sm230920-27-8})].