| Title:
|
Some characterization of locally nonconical convex sets (English) |
| Author:
|
Seredyński, Witold |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
767-771 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
stable convex set |
| MSC:
|
46A55 |
| MSC:
|
46Cxx |
| MSC:
|
52A05 |
| idZBL:
|
Zbl 1080.52500 |
| idMR:
|
MR2086732 |
| . |
| Date available:
|
2009-09-24T11:17:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127927 |
| . |
| Reference:
|
[1] J. Cel: Tietze-type theorem for locally nonconical convex sets.Bull. Soc. Roy. Sci Liège 69 (2000), 13–15. Zbl 0964.46004, MR 1766658 |
| Reference:
|
[2] S. Papadopoulou: On the geometry of stable compact convex sets.Math. Ann. 229 (1977), 193–200. Zbl 0339.46001, MR 0450938, 10.1007/BF01391464 |
| Reference:
|
[3] G. C. Shell: On the geometry of locally nonconical convex sets.Geom. Dedicata 75 (1999), 187–198. Zbl 0937.52002, MR 1686757, 10.1023/A:1005080830204 |
| . |
