Differentiability of themetric projection onto a convex set with singular boundary points

The differentiability of the metric projection P onto a closed convex set K in Rn is examined. The boundary ãK can have singular points of orders k ̈ −1Ù 0Ù 1ÜÙ n − 1. Here k ̈ −1 corresponds to the interior points of K , k ̈ 0 to regular points of the boundary (i.e., faces), k ̈ 1ÙÜ Ù n − 2 to edges and k ̈ n − 1 to vertices. It is assumed that for every k the set of all singular points forms an n − k − 1 dimensional manifold Tk+1 (possibly empty) of class p 3 2. Under a mild continuity assumption it is shown that then P is of class p − 1 on an open set W whose complement has null Lebesgue measure. The setW is the union of the interiors of inverse images of Tk+1 under PØ Moreover, a formula for the Fréchet derivative DP on each of these regions is given that relates D P to the second fundamental form of the manifoldTk+1Ø The results are illustrated (a) on the metric projection P from the space Sym of symmetric matrices onto the convex cone Sym+ of positive semidefinite symmetric matrices and (b) on the metric projection from Sym onto the unit ball under the operator norm. We prove the indefinite differentiability of these projections on explicitly determined open sets with complements of measure 0 and give explicit formulas for the derivatives. In (a) the method of proof, based on the above general result, is different from the previous treatment in [17] and applies to situations [21] where the special methods of [17] cannot be used. The case (b) is new. MSC 2010. 26B25 Primary, 26B05, 52A20.