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Some characterization of locally nonconical convex sets. (English) Zbl 1080.52500

Summary: A closed convex set \(Q\) in a local convex topological Hausdorff space \(X\) is called locally nonconical (LNC) if for every \(x, y\in Q\) there exists an open neighbourhood \(U\) of \(x\) such that \((U\cap Q)+\frac 12(y-x)\subset Q\). A set \(Q\) is local cylindric (LC) if for \(x,y\in Q\), \(x\neq y\), \(z\in (x,y)\) there exists an open neighbourhood \(U\) of \(z\) such that \(U\cap Q\) (equivalently: \(\text{bd} (Q)\cap U\)) is a union of open segments parallel to \([x,y]\). In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in G.C.Shell [Geom.Dedicata 75, 187-198 (1999; Zbl 0937.52002)], where the implication \(\text{LNC}\Rightarrow \text{LC}\) was proved in general, while the inverse implication was proved in case of Hilbert spaces.

MSC:

52A05 Convex sets without dimension restrictions (aspects of convex geometry)
46A55 Convex sets in topological linear spaces; Choquet theory

Citations:

Zbl 0937.52002
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References:

[1] J. Cel: Tietze-type theorem for locally nonconical convex sets. Bull. Soc. Roy. Sci Liège 69 (2000), 13–15. · Zbl 0964.46004
[2] S. Papadopoulou: On the geometry of stable compact convex sets. Math. Ann. 229 (1977), 193–200. · Zbl 0349.46001 · doi:10.1007/BF01391464
[3] G. C. Shell: On the geometry of locally nonconical convex sets. Geom. Dedicata 75 (1999), 187–198. · Zbl 0937.52002 · doi:10.1023/A:1005080830204
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