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

Volume 232 / 2016

Stoyu Barov, Jan J. Dijkstra Fundamenta Mathematicae 232 (2016), 117-129 MSC: Primary 52A07, 52A20. DOI: 10.4064/fm232-2-2


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.


  • Stoyu BarovInstitute of Mathematics
    Bulgarian Academy of Sciences
    8 Acad. G. Bonchev St.
    1113 Sofia, Bulgaria
  • Jan J. DijkstraP.O. Box 1180
    Crested Butte, CO 81224, U.S.A.

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