On the Dieudonné theorem in reflexive Banach spaces
The authors first characterize local compactness of closed convex sets containing no lines in a reflexive Banach space provided with its weak topology by a geometrical condition, which they call well-positionedness. For the proof of this characterization they utilize technical results of a forthcoming article of their own, which has not yet appeared. Next, they investigate the Dieudonné theorem about the closedness of the algebraic difference of two closed convex sets and prove the following Theorem: A closed convex subset \(C_1\) containing no lines in a reflexive Banach space \(X\) is well-positioned if and only if, for every closed convex subset \(C_2\) in \(X\) such that \(C^\infty_1\cap C^\infty_2= \{0\}\), the algebraic difference \(C_1- C_2\) is closed (where \(C^\infty:= \{d\in X:\{c+\mu d: c\in C,\,\mu> 0\}\subset C\}\)). From these two results, they obtain that Dieudonné's sufficient condition becomes also a necessary one -- at least in reflexive Banach spaces with their weak topology.
scientific article; zbMATH DE number 1978188