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 P-C: X -> C denote the (standard) metric projection operator. In this paper, we define the Gateaux directional differentiability of P-C. We investigate some properties of the Gateaux directional differentiability of P-C. 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 P-C.