| Title:
|
More on exposed points and extremal points of convex sets in $\mathbb{R}^n$ and Hilbert space (English) |
| Author:
|
Barov, Stoyu T. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2023 |
| Pages:
|
63-72 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let ${\mathbb{V}}$ be a separable real Hilbert space, $k \in {\mathbb{N}}$ with $k < \dim {\mathbb{V}}$, and let $B$ be convex and closed in ${\mathbb{V}}$. Let ${\mathcal{P}}$ be a collection of linear $k$-subspaces of ${\mathbb{V}}$. A point $w \in B$ is called exposed by ${\mathcal{P}}$ if there is a $P \in {\mathcal{P}}$ so that $(w + P) \cap B =\{w\}$. We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by ${\mathcal{P}}$ points whenever ${\mathcal{P}}$ is dense and $G_{\delta}$. In addition, we discuss the question when the set of exposed by some ${\mathcal{P}}$ points forms a $G_{\delta}$-set. (English) |
| Keyword:
|
convex set |
| Keyword:
|
extremal point |
| Keyword:
|
exposed point |
| Keyword:
|
Hilbert space |
| Keyword:
|
Grassmann manifold |
| MSC:
|
52A07 |
| MSC:
|
52A20 |
| idZBL:
|
Zbl 07790582 |
| idMR:
|
MR4631790 |
| DOI:
|
10.14712/1213-7243.2023.018 |
| . |
| Date available:
|
2023-08-28T09:44:16Z |
| Last updated:
|
2025-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151799 |
| . |
| Reference:
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| Reference:
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| Reference:
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