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The Mazur property for compact sets. (English) Zbl 0653.46021

The author establishes the following smoothness property. Let X be a Banach space. Every compact convex set of X is the intersection of balls containing it if and only if the cone of the extreme points of the dual X’ is dense in X’ with respect to the topology of uniform convergence on compact sets of X. This characterization is also used to derive other results and applications as stability under \(c_ 0\) and \(\ell_ p\) sums, \(1\leq p<\infty\), and renorming theorems. Some of the author’s proofs unify and simplify previous proof techniques.
Reviewer: P.Pucci

MSC:

46B20 Geometry and structure of normed linear spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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