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

• Autorzy: Stoyu Barov , Jan J. Dijkstra
• Języki publikacji: EN

Abstrakty

EN

Let 𝕍 be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim 𝕍, and let B be convex and closed in 𝕍. Let 𝓟 be a collection of linear k-subspaces of 𝕍. A set C ⊂ 𝕍 is called a 𝓟-imitation of B if B and C have identical orthogonal projections along every P ∈ 𝓟. An extremal point of B with respect to the projections under 𝓟 is a point that all closed subsets of B that are 𝓟-imitations of B have in common. A point x of B is called exposed by 𝓟 if there is a P ∈ 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 𝓟 is open. In addition, we discuss the question whether the exposed points form a $G_{δ}$-set.

Kategorie tematyczne

• 52A07: Convex sets in topological vector spaces
• 52A20: Convex sets in n dimensions (including convex hypersurfaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2016
• Tom: 232
• Numer: 2
• Strony: 117-129

Daty

wydano

2016

Twórcy

autor

Stoyu Barov

• Institute of Mathematics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev St., 1113 Sofia, Bulgaria

autor

Jan J. Dijkstra

• P.O. Box 1180, Crested Butte, CO 81224, U.S.A.

Identyfikatory

DOI

10.4064/fm232-2-2