Proximal Smoothness and the Lower-C² Property

A subset X of a real Hilbert space H is said to be proximally smooth provided that the function $d_X : H \to R$ (the distance to X) is continuously differentiable on an open tube U around X. It is proven that this property is equivalent to $d_X$ having a nonempty proximal subgradient at every point of U, and that the (Gâteaux = Fréchet) derivative is locally Lipschitz on U. The Lipschitz behavior of the derivative is a consequence of the fact that under proximal smoothness, the metric projection onto X is single valued and Lipschitz on U. Alternate characterizations of proximal smoothness are given as well, in terms of properties of the proximal normal cone multifunction on X and on nearby closed neighborhoods of X. In case X is weakly closed, the list of equivalences is extended to include each point of U admitting a unique closest point in X. Further specializations are given in finite dimensions. In that setting, we discuss properties of locally Lipschitz real valued functions whose epigraphs are proximally smooth in a local sense. It is demonstrated that this function class coincides with the lower–$C^2$ functions studied by Rockafellar.