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## Fundamenta Mathematicae

2016 | 232 | 2 | 117-129
Tytuł artykułu

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

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EN
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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.
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Tom
Numer
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117-129
Opis fizyczny
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wydano
2016
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autor
• Institute of Mathematics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev St., 1113 Sofia, Bulgaria
autor
• P.O. Box 1180, Crested Butte, CO 81224, U.S.A.
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