# Countably convex ${G}_{\delta}$ sets

Vladimir Fonf; Menachem Kojman

Fundamenta Mathematicae (2001)

- Volume: 168, Issue: 2, page 131-140
- ISSN: 0016-2736

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topVladimir 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 -

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