Let be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim , and let B be convex and closed in . Let be a collection of linear k-subspaces of . A set C ⊂ is called a -imitation of B if B and C have identical orthogonal projections along every P ∈ . An extremal point of B with respect to the projections under is a point that all closed subsets of B that are -imitations of B have in common. A point x of B is called exposed by if there is a P ∈ such that (x+P) ∩ B = x. In the present paper we show that all extremal points are limits of sequences of exposed points whenever is open. In addition, we discuss the question whether the exposed points form a -set.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-2,
author = {Stoyu Barov and Jan J. Dijkstra},
title = {On exposed points and extremal points of convex sets in Rn and Hilbert space},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {117-129},
zbl = {1336.52001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-2}
}
Stoyu Barov; Jan J. Dijkstra. On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space. Fundamenta Mathematicae, Tome 233 (2016) pp. 117-129. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-2/