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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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