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stable convex set
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.
[1] J. Cel: Tietze-type theorem for locally nonconical convex sets. Bull. Soc. Roy. Sci Liège 69 (2000), 13–15. MR 1766658 | Zbl 0964.46004
[2] S. Papadopoulou: On the geometry of stable compact convex sets. Math. Ann. 229 (1977), 193–200. DOI 10.1007/BF01391464 | MR 0450938 | Zbl 0339.46001
[3] G. C. Shell: On the geometry of locally nonconical convex sets. Geom. Dedicata 75 (1999), 187–198. DOI 10.1023/A:1005080830204 | MR 1686757 | Zbl 0937.52002
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