On a Convex Set with Nondifferentiable Metric Projection
  A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property. 
Shyan S. Akmal, Nguyen Mau Nam, J. J. P. Veerman
math.OC