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Fundamenta Mathematicae

2007 | 197 | 1 | 17-33
Tytuł artykułu

On closed sets with convex projections in Hilbert space

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Języki publikacji
EN
Abstrakty
EN
Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set $𝓔^{k}(B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $𝓔^{k}(B)$ is precisely the intersection of all k-imitations C of B, i.e., closed sets C that have the same projections as B onto all k-hyperplanes. For every closed convex set B in ℓ² with nonempty interior we construct "minimal" k-imitations C, in the sense that $dim(C∖𝓔^{k}(B)) ≤ 0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.
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Rocznik
Tom
Numer
Strony
17-33
Opis fizyczny
Daty
wydano
2007
Twórcy
autor
• Institute of Mathematics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev St., 1113 Sofia, Bulgaria
autor
• Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
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