Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces.

Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let PC: X → C denote the (standard) metric projection operator. In this paper, we define the G a ^ teaux directional differentiability of PC. We investigate some properties of the G a ^ teaux directional differentiability of PC. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of PC. [ABSTRACT FROM AUTHOR]