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Characterization of reflexive Banach spaces in terms of strongly exposed points of unbounded sets. (Russian) Zbl 0629.46013

A convex cone K in a Banach space X is called normal if there is a constant \(C>0\) such that \(\| x\| \geq C\| y\|\) whenever, x,y,x-y\(\in K\). A convex set A in K is called a basis of K if for very \(x\in K\setminus \{0\}\) there is a \(t>0\) and a unique element \(y\in A\) such that \(x=ty\). An element \(a\in A\) is called strongly exposed if there is a functional \(f\in X'\) such that \(\lim_{\epsilon \to 0}diam \{x\in A|\) \(f(x)>f(a)-\epsilon \}=0.\)
In this paper it is proved that in any nonreflexive Banach space there is a closed normal cone K with an unbounded basis A, consisting only of strongly exposed points.
Reviewer: A.Torgašev

MSC:

46B20 Geometry and structure of normed linear spaces
46A55 Convex sets in topological linear spaces; Choquet theory
46A40 Ordered topological linear spaces, vector lattices
46B10 Duality and reflexivity in normed linear and Banach spaces
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