Existence and Differentiability of Metric Projections in Hilbert Spaces

This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections always exist, unique and directionally differentiable at boundary points of S. These properties of metric projections are considered for possibly nonconvex sets S. In particular, existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as “nearly convex” sets.