Strong proximinality of closed convex sets

We show that in a Banach space XX, every closed convex subset is strongly proximinal if and only if the dual norm is strongly subdifferentiable and for each norm 1 functional ff in the dual space X∗X∗, JX(f)JX(f)- the set of norm 1 elements in XX where ff attains its norm - is compact. As a consequence, it is observed that if the dual norm is strongly subdifferentiable, then every closed convex subset of XX is strongly proximinal if and only if the metric projection onto every closed convex subset of XX is upper semi-continuous.