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.


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


Zbl 0937.52002
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[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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