  This paper addresses the study and applications of polyhedral duality of
locally convex topological vector (LCTV) spaces. We first revisit the classical
Rockafellar's proper separation theorem for two convex sets one which is
polyhedral and then present its LCTV extension with replacing the relative
interior by its quasi-relative interior counterpart. Then we apply this result
to derive enhanced calculus rules for normals to convex sets, coderivatives of
convex set-valued mappings, and subgradients of extended-real-valued functions
under certain polyhedrality requirements in LCTV spaces by developing a
geometric approach. We also establish in this way new results on conjugate
calculus and duality in convex optimization with relaxed qualification
conditions in polyhedral settings. Our developments contain significant
improvements to a number of existing results obtained by Ng and Song in [31].
