Stoyu T. Barov
More on exposed points and extremal points of convex sets in \mathbb{R}^n and Hilbert space

Comment.Math.Univ.Carolin. 64,1 (2023) 63-72.

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 =\{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.

Keywords: convex set; extremal point; exposed point; Hilbert space; Grassmann manifold

DOI: DOI 10.14712/1213-7243.2023.018
AMS Subject Classification: 52A20 52A07