Norms that Generate the Same Wijsman Topology on Convex Sets

In a Banach space X, we introduce a criterion for comparing the Wijsman topologies that are induced by two equivalent norms of X on the hyperspace of closed convex sets C(X). Thereafter, we study the duality map associated with the unit ball of a given norm of X in relation to its composition with the polarity map. This more geometrical description of the norm allows us to give a direct proof of a known theorem (see [3] and [1]): If X is reflexive and the duality map is n-to-n-usco, then the Wijsman topology coincides with the Mosco topology on C(X).