Metric projections and the differentiability of distance functions

Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M ) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X . If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M . Examples are presented to show that some of our hypotheses are needed.