We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop–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 4df5810ff0d1" title="Click to view the MathML source">d(A,B)>0. If every d805e3c3" title="Click to view the MathML source">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.