On exposed points and extremal points of convex sets in $\mathbb {R}^n$ and Hilbert space

Stoyu Barov ¹   ; Jan J. Dijkstra ²

¹ Institute of Mathematics Bulgarian Academy of Sciences 8 Acad. G. Bonchev St. 1113 Sofia, Bulgaria
² P.O. Box 1180 Crested Butte, CO 81224, U.S.A.

Fundamenta Mathematicae, Tome 232 (2016) no. 2, pp. 117-129

Résumé

Let $\mathbb V$ be a Euclidean space or the Hilbert space $\ell^2$, let $ 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 set $C \subset \mathbb V$ is called a $\mathcal{P}$-imitation of $B$ if $B$ and $C$ have identical orthogonal projections along every $P \in \mathcal{P}$. An extremal point of $B$ with respect to the projections under $\mathcal{P}$ is a point that all closed subsets of $B$ that are $\mathcal{P}$-imitations of $B$ have in common. A point $x$ of $B$ is called exposed by $\mathcal{P}$ if there is a $P \in \mathcal{P}$ such that $(x+P) \cap B = \{x\}$. In the present paper we show that all extremal points are limits of sequences of exposed points whenever $\mathcal{P}$ is open. In addition, we discuss the question whether the exposed points form a $G_\delta$-set.

Détail

DOI : 10.4064/fm232-2-2

Keywords: mathbb euclidean space hilbert space ell mathbb dim mathbb convex closed mathbb mathcal collection linear k subspaces mathbb set subset mathbb called mathcal imitation nbsp have identical orthogonal projections along every mathcal extremal point respect projections under mathcal point closed subsets mathcal imitations have common nbsp point called exposed mathcal there mathcal cap present paper extremal points limits sequences exposed points whenever mathcal addition discuss question whether exposed points form delta set