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 consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that 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 . Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-2,
author = {Stoyu Barov and Jan J. Dijkstra},
title = {On closed sets with convex projections in Hilbert space},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {17-33},
zbl = {1131.52001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-2}
}
Stoyu Barov; Jan J. Dijkstra. On closed sets with convex projections in Hilbert space. Fundamenta Mathematicae, Tome 193 (2007) pp. 17-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-2/