  In this paper we study some relationships between polyhedral convex sets
(PCS) and generalized polyhedral convex sets (GPCS). In particular, we clarify
by a counterexample that the necessary and sufficient conditions for the
separation of a convex set and a PCS obtained by Kung Fu Ng and Wen Song in
[Fenchel duality in finite-dimensional setting and its applications, Nonlinear
Anal. 55(2003), 845--858; Theorem~3.1] are no longer valid when considering
GPCS instead of PCS. We also introduce and study the notions of generalized
polyhedral set-valued mappings and optimal value functions generated by
generalized polyhedral convex set-valued mappings along with their generalized
differentiation calculus rules.
