On exposed points and extremal points of convex sets in $\mathbb {R}^n$ and Hilbert space
Volume 232 / 2016
Abstract
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.
