On the Clarke subdifferential of the distance function of a closed set

Abstract

Let C be a nonempty closed subset of the real normed linear space X. In this paper we determine the extent to which formulas for the Clarke subdifferential of the distance for C, d c(x) := inf y∈c ‖x − y‖ , which are valid when C is convex, remain valid when C is not convex. The assumption of subdifferential regularity for d c plays an important role. When x ∉ C, the most precise results also require the nonemptiness of the set P C(x) := { x ̄ ϵ C : ∥x − x ̄ ∥ = d C(x)} . As an interesting side result, the equivalence between the strict differentiability and the regularity of d C at x is established when x ∉ C, P C ( x) ≠ Ø, and the norm on X is smooth.