Some characterization of locally nonconical convex sets

Seredyński, Witold. "Some characterization of locally nonconical convex sets." Czechoslovak Mathematical Journal 54.3 (2004): 767-771. <http://eudml.org/doc/30898>.

@article{Seredyński2004,
abstract = {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: $\mathrm \{b\}d(Q)\cap U$) 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 $\{\mathrm \{L\}NC\}\Rightarrow \{\mathrm \{L\}C\}$ was proved in general, while the inverse implication was proved in case of Hilbert spaces.},
author = {Seredyński, Witold},
journal = {Czechoslovak Mathematical Journal},
keywords = {stable convex set},
language = {eng},
number = {3},
pages = {767-771},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some characterization of locally nonconical convex sets},
url = {http://eudml.org/doc/30898},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Seredyński, Witold
TI - Some characterization of locally nonconical convex sets
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 767
EP - 771
AB - 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: $\mathrm {b}d(Q)\cap U$) 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 ${\mathrm {L}NC}\Rightarrow {\mathrm {L}C}$ was proved in general, while the inverse implication was proved in case of Hilbert spaces.
LA - eng
KW - stable convex set
UR - http://eudml.org/doc/30898
ER -