More on exposed points and extremal points of convex sets in ${ℝ}^n$ and Hilbert space

Barov, Stoyu T.. "More on exposed points and extremal points of convex sets in $\mathbb {R}^n$ and Hilbert space." Commentationes Mathematicae Universitatis Carolinae 64.1 (2023): 63-72. <http://eudml.org/doc/299408>.

@article{Barov2023,
abstract = {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 =\lbrace w\rbrace $. 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.},
author = {Barov, Stoyu T.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex set; extremal point; exposed point; Hilbert space; Grassmann manifold},
language = {eng},
number = {1},
pages = {63-72},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {More on exposed points and extremal points of convex sets in $\mathbb \{R\}^n$ and Hilbert space},
url = {http://eudml.org/doc/299408},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Barov, Stoyu T.
TI - More on exposed points and extremal points of convex sets in $\mathbb {R}^n$ and Hilbert space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 1
SP - 63
EP - 72
AB - 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 =\lbrace w\rbrace $. 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.
LA - eng
KW - convex set; extremal point; exposed point; Hilbert space; Grassmann manifold
UR - http://eudml.org/doc/299408
ER -