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On \(B\)-convex and Mazur sets of Banach spaces. (English) Zbl 0836.46009

Summary: There exists an equivalent norm \(|\cdot |\) in \(\ell_2\) such that in the reflexive space \((\ell_2, |\cdot |)^*\), there is a bounded \(B\)-convex set \(K\) with the property that the closure of \(K\) is not an intersection of balls. A necessary condition is given for Banach spaces with the property that the closure of every bounded \(B\)- convex set is an intersection of balls.

MSC:

46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
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