On closed sets with convex projections in Hilbert space

Stoyu Barov, and Jan J. Dijkstra. "On closed sets with convex projections in Hilbert space." Fundamenta Mathematicae 197.1 (2007): 17-33. <http://eudml.org/doc/282994>.

@article{StoyuBarov2007,
abstract = {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 $^\{k\}(B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $^\{k\}(B)$ 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 $dim(C∖^\{k\}(B)) ≤ 0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.},
author = {Stoyu Barov, Jan J. Dijkstra},
journal = {Fundamenta Mathematicae},
keywords = {Hilbert space; shadow; convex projection; hyperplane; -imitation},
language = {eng},
number = {1},
pages = {17-33},
title = {On closed sets with convex projections in Hilbert space},
url = {http://eudml.org/doc/282994},
volume = {197},
year = {2007},
}

TY - JOUR
AU - Stoyu Barov
AU - Jan J. Dijkstra
TI - On closed sets with convex projections in Hilbert space
JO - Fundamenta Mathematicae
PY - 2007
VL - 197
IS - 1
SP - 17
EP - 33
AB - 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 $^{k}(B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $^{k}(B)$ 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 $dim(C∖^{k}(B)) ≤ 0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.
LA - eng
KW - Hilbert space; shadow; convex projection; hyperplane; -imitation
UR - http://eudml.org/doc/282994
ER -