We present some extensions of classical results that involve el
ements of the dual of Banach spaces, such as Bishop–Phelp's theor
em and James' compactness theor
em, 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: <
em>Let E be a Banach space such that
em>
(BE⁎,ω⁎)<
em>is convex block compact. Let A and B be bounded, closed and convex sets with distance
em>
d(A,B)>0<
em>. If every
em>
x⁎∈E⁎<
em>with
em>
<
em>attains its infimum on A and its supr
emum on B, then A and B are both weakly compact.
em> We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a
variational probl
em in dual Banach spaces.