Article
| 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 |
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| Category: | math |
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| 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 |
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| Date available: | 2023-08-28T09:44:16Z |
| Last updated: | 2024-02-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151799 |
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