Linearly and directionally bounded weak-star closed sets and the AFPP

Linearly bounded and directionally bounded closed convex sets play a very relevant role in metric fixed point theory [12, 14, 16]. In reflexive spaces both collections of sets are identical and this fact characterizes the reflexivity of the space [14, 16]. Since closed convex sets are weakstar closed in reflexive spaces, it is natural to ask about the relationship between linearly bounded and directionally bounded sets in the case of a non-reflexive dual space if we assume, in addition, that the set is weak-star closed. We will show two divergent answers to this question depending on certain topological and isometric properties of the underlying dual Banach space.