One-sided James' compactness theorem

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⁎)(BE⁎,ω⁎)is convex block compact. Let A and B be bounded, closed and convex sets with distance  d(A,B)>0d(A,B)>0. If every  x⁎∈E⁎x⁎∈E⁎withsup⁡(x⁎,B)