Norms that generate the same Wijsman topology on convex sets.(English)Zbl 0857.46004

Summary: 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: If $$X$$ is reflexive and the duality map is $$n$$-to-$$n$$-usco, then the Wijsman topology coincides with the Mosco topology on $$C(X)$$.

MSC:

 46A50 Compactness in topological linear spaces; angelic spaces, etc. 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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