On closed sets with convex projections in Hilbert space

• Autorzy: Stoyu Barov , Jan J. Dijkstra
• 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.

Kategorie tematyczne

• 57N20: Topology of infinite-dimensional manifolds
• 52A07: Convex sets in topological vector spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 197
• Numer: 1
• Strony: 17-33

Daty

wydano

2007

Twórcy

autor

Stoyu Barov

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

autor

Jan J. Dijkstra

• Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Identyfikatory

DOI

10.4064/fm197-0-2