# Some characterization of locally nonconical convex sets

Czechoslovak Mathematical Journal (2004)

- Volume: 54, Issue: 3, page 767-771
- ISSN: 0011-4642

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topSeredyński, 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 -

## References

top- Tietze-type theorem for locally nonconical convex sets, Bull. Soc. Roy. Sci Liège 69 (2000), 13–15. (2000) Zbl0964.46004MR1766658
- 10.1007/BF01391464, Math. Ann. 229 (1977), 193–200. (1977) Zbl0339.46001MR0450938DOI10.1007/BF01391464
- 10.1023/A:1005080830204, Geom. Dedicata 75 (1999), 187–198. (1999) Zbl0937.52002MR1686757DOI10.1023/A:1005080830204

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