We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop&
ndash;Phelp's theorem and James' compactness theorem, but restricting ourselves to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following:
Let E be a Banach space such that (BE⁎,ω⁎)is convex block compact. Let A and B be bounded, closed and convex sets with distance d(A,B)>0. If every x⁎∈E⁎withattains its infimum on A and its supremum on B, then A and B are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.