Countably convex $G_{\delta }$ sets

Vladimir Fonf, and Menachem Kojman. "Countably convex $G_{δ}$ sets." Fundamenta Mathematicae 168.2 (2001): 131-140. <http://eudml.org/doc/282192>.

@article{VladimirFonf2001,
abstract = {We investigate countably convex $G_\{δ\}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary. We show that for countably convex $G_\{δ\}$ subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques. Various necessary conditions on cliques and semi-cliques are obtained for countably convex $G_\{δ\}$ subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex $G_\{δ\}$ subset of ℝ³ all cliques are scattered, whereas in ℝ⁴ a countably convex $G_\{δ\}$ set may contain a dense-in-itself clique.},
author = {Vladimir Fonf, Menachem Kojman},
journal = {Fundamenta Mathematicae},
keywords = {countably convex subsets; countable union of convex sets; semi-clique; cliques},
language = {eng},
number = {2},
pages = {131-140},
title = {Countably convex $G_\{δ\}$ sets},
url = {http://eudml.org/doc/282192},
volume = {168},
year = {2001},
}

TY - JOUR
AU - Vladimir Fonf
AU - Menachem Kojman
TI - Countably convex $G_{δ}$ sets
JO - Fundamenta Mathematicae
PY - 2001
VL - 168
IS - 2
SP - 131
EP - 140
AB - We investigate countably convex $G_{δ}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary. We show that for countably convex $G_{δ}$ subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques. Various necessary conditions on cliques and semi-cliques are obtained for countably convex $G_{δ}$ subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex $G_{δ}$ subset of ℝ³ all cliques are scattered, whereas in ℝ⁴ a countably convex $G_{δ}$ set may contain a dense-in-itself clique.
LA - eng
KW - countably convex subsets; countable union of convex sets; semi-clique; cliques
UR - http://eudml.org/doc/282192
ER -