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 \ E at which the distance function d E fails to be Frechet differentiable. In particular, this paper demonstrates in full generality that Sigma(E) is of the same homotopy type as the open set g(E) = { x E H : d(co<overline> E) (x) < dE(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 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 < oo.