Some characterization of locally nonconical convex sets
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 767-771
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A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x, y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+\frac{1}{2}(y-x)\subset Q$. A set $Q$ is local cylindric (LC) if for $x,y\in Q$, $x\ne y$, $z\in (x,y)$ there exists an open neighbourhood $U$ of $z$ such that $U\cap Q$ (equivalently: $\mathrm bd(Q)\cap U$) 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 [3], where the implication ${\mathrm LNC}\Rightarrow {\mathrm LC}$ was proved in general, while the inverse implication was proved in case of Hilbert spaces.
A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x, y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+\frac{1}{2}(y-x)\subset Q$. A set $Q$ is local cylindric (LC) if for $x,y\in Q$, $x\ne y$, $z\in (x,y)$ there exists an open neighbourhood $U$ of $z$ such that $U\cap Q$ (equivalently: $\mathrm bd(Q)\cap U$) 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 [3], where the implication ${\mathrm LNC}\Rightarrow {\mathrm LC}$ was proved in general, while the inverse implication was proved in case of Hilbert spaces.
Seredyński, Witold. Some characterization of locally nonconical convex sets. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 767-771. http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a17/
@article{CMJ_2004_54_3_a17,
author = {Seredy\'nski, Witold},
title = {Some characterization of locally nonconical convex sets},
journal = {Czechoslovak Mathematical Journal},
pages = {767--771},
year = {2004},
volume = {54},
number = {3},
mrnumber = {2086732},
zbl = {1080.52500},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a17/}
}
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[3] G. C. Shell: On the geometry of locally nonconical convex sets. Geom. Dedicata 75 (1999), 187–198. | DOI | MR | Zbl