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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


More on convexity numbers of closed sets in $\mathbb{R} ^n$

Author: Stefan Geschke
Journal: Proc. Amer. Math. Soc. 133 (2005), 1307-1315
MSC (2000): Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75
Published electronically: November 1, 2004
MathSciNet review: 2111936
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Abstract: The convexity number of a set $S\subseteq\mathbb R^n$ is the least size of a family $\mathcal F$ of convex sets with $\bigcup\mathcal F=S$. $S$ is countably convex if its convexity number is countable. Otherwise $S$ is uncountably convex.

Uncountably convex closed sets in $\mathbb R^n$ have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all $n\geq 2$, it is consistent that there is an uncountably convex closed set $S\subseteq\mathbb R^{n+1}$ whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of $\mathbb R^n$.

Moreover, we construct a closed set $S\subseteq\mathbb R^3$ whose convexity number is $2^{\aleph_0}$ and that has no uncountable $k$-clique for any $k>1$. Here $C\subseteq S$ is a $k$-clique if the convex hull of no $k$-element subset of $C$ is included in $S$. Our example shows that the main result of the above-named authors, a closed set $S\subseteq\mathbb R^2$ either has a perfect $3$-clique or the convexity number of $S$ is $<2^{\aleph_0}$ in some forcing extension of the universe, cannot be extended to higher dimensions.

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Additional Information

Stefan Geschke
Affiliation: II. Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

PII: S 0002-9939(04)07685-3
Keywords: Convex cover, convexity number, clique, $n$-space, forcing extension, covering number
Received by editor(s): August 18, 2003
Received by editor(s) in revised form: December 1, 2003, and January 16, 2004
Published electronically: November 1, 2004
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2004 American Mathematical Society