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Diametrically complete sets and normal structure. (English) Zbl 1320.46014

E. Meissner [Zürich. Naturf.-Ges. 56, 42–50 (1911; JFM 42.0091.01)] introduced the notion of diametrically complete sets as a generalization of sets of constant width. A set \(C\) in a metric space \(X\) is diametrically complete if the inclusion of an additional point to the set \(C\) increases its diameter: i.e., if \(\text{diam} (C\cup \{x\}) > \text{diam}\,C\) for all \(x\) in \(X\setminus C\). H. G. Eggleston [Convexity. Cambridge: At the University Press (1958; Zbl 0086.15302)] introduced the notion of a set of constant radius from its boundary (sets \(C\) for which \(\sup\{\| y-x\| :y\in C\}= \mathrm{diam}\,C\) for all \(x\in \partial C\)) as a generalization of diametrically complete sets and proved that, in finite-dimensional spaces, a set is diametrically complete if and only if the set has constant radius from its boundary. Although a diametrically complete set is a set of constant radius from its boundary, the converse fails to hold in infinite-dimensional normed linear spaces [J. P. Moreno et al., J. Convex Anal. 13, No. 3–4, 823–837 (2006; Zbl 1142.52002)].
In the article under review, the authors prove that, in Banach spaces with normal structure, the class of diametrically complete sets and the class of sets of constant radius from its boundary coincide. In spaces without normal structure, there exist nontrivial diametral sets, i.e., closed, bounded, convex sets of positive diameter consisting only of diametral points. The authors show that, in a certain class of reflexive Banach spaces without normal structure, there exist diametrically complete diametral sets. This answers a question posed by J. P. Moreno et al. [Can. J. Math. 58, No. 4, 820–842 (2006; Zbl 1110.52003)] and supplements the results there by proving the existence of diametrically complete sets with empty interior in certain reflexive Banach spaces.
The article contains several interesting open questions.

MSC:

46B20 Geometry and structure of normed linear spaces
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
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