On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

Stoyu Barov, and Jan J. Dijkstra. "On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space." Fundamenta Mathematicae 232.2 (2016): 117-129. <http://eudml.org/doc/286521>.

@article{StoyuBarov2016,
abstract = {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 $G_\{δ\}$-set.},
author = {Stoyu Barov, Jan J. Dijkstra},
journal = {Fundamenta Mathematicae},
keywords = {extremal points; exposed points},
language = {eng},
number = {2},
pages = {117-129},
title = {On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space},
url = {http://eudml.org/doc/286521},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Stoyu Barov
AU - Jan J. Dijkstra
TI - On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space
JO - Fundamenta Mathematicae
PY - 2016
VL - 232
IS - 2
SP - 117
EP - 129
AB - 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 $G_{δ}$-set.
LA - eng
KW - extremal points; exposed points
UR - http://eudml.org/doc/286521
ER -