  In this paper, we consider the directional differentiability of metric
projection and its properties in uniformly convex and uniformly smooth Bochner
space Lp(S; X), in which (S, A, mu) is a positive measure space and X is a
uniformly convex and uniformly smooth Banach space. Let (arbitrary) A in A with
measure of A greater than 0 and define a subspace Lp(A; X) of Lp(S; X), which
is considered as a closed and convex subset of Lp(S; X). We first study the
properties of the normalized duality mapping in Lp(S; X) and in Lp(A; X). For
any c in Lp(A; X) and r > 0, we define a closed ball BA(c; r) in Lp(A; X) and a
cylinder CA(c; r) in Lp(S; X) with base BA(c; r). Then, we investigate some
optimal properties of the corresponding metric projections P(Lp(A;X)),
P(BA(c;r)) and P(CA(c;r)) that include the inverse images, the directional
differentiability and the precise solutions of their directional derivatives.
