Countably convex $G_{δ}$ sets

• Autorzy: Vladimir Fonf , Menachem Kojman
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 52A37: Other problems of combinatorial convexity
• 46B20: Geometry and structure of normed linear spaces
• 52A15: Convex sets in 3 dimensions (including convex surfaces)
• 51M20: Polyhedra and polytopes; regular figures, division of spaces
• 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
• 52A07: Convex sets in topological vector spaces
• 51M05: Euclidean geometries (general) and generalizations
• 52B10: Three-dimensional polytopes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 168
• Numer: 2
• Strony: 131-140

Daty

wydano

2001

Twórcy

autor

Vladimir Fonf

• Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

autor

Menachem Kojman

• Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

Identyfikatory

DOI

10.4064/fm168-2-4