One side 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 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 $(B_{E^\ast}, \omega^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with $$\sup(x^\ast,B) < \inf(x^\ast,A)$$ attains 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.