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

Stoyu Barov; Jan J. Dijkstra

Fundamenta Mathematicae (2016)

  • Volume: 232, Issue: 2, page 117-129
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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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 -

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