On the Clarke Subdifferential of the Distance Function of a Closed Set

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, de(x) := inf I/Y.rll, be< which are valid when C is convex, remain valid when C is not convex. The assumption of subdifferential regularity for d, plays an important role. When x 4 C, the most precise results also require the nonemptiness of the set PC(x):= {XEC: 11x-ill =d,-(x)}. As an interesting side result, the equivalence between the strict differentiability and the regularity of d, at x is established when .Y 4 C, PC(x) # @, and the norm on X is smooth. ‘% 1992 Academic Press, Inc.