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On a weak* drop property

Rolewicz said that the norm of a Banach space (X, ║·║) has the drop property if for every closed set C disjoint from the closed unit ball B(X), there exists an x ∈ C such that co(x, B(X)) ∩ C = {x}. He then showed that if the norm of a Banach space (X, ║·║) has the drop property then it is reflexive. Later, in 1990 Giles, Sims and Yorke showed that the norm of a Banach space (X, ║·║) has the drop property if and only if the duality mapping f → D(f) from the dual sphere S(X*) into subsets of the second dual sphere S(X**) is norm upper semi-continuous and norm compact-valued on S(X*).