Some characterization of locally nonconical convex sets

Seredyński, Witold

Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 767-771.

Résumé

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.

Détail

Classification : 46A55, 46Cxx, 52A05
Keywords: stable convex set