More on convexity numbers of closed sets in $\mathbb {R}^n$
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- by Stefan Geschke PDF
- Proc. Amer. Math. Soc. 133 (2005), 1307-1315 Request permission
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, Kubiś, 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.References
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Additional Information
- Stefan Geschke
- Affiliation: II. Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- MR Author ID: 681801
- Email: geschke@math.fu-berlin.de
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 1307-1315
- MSC (2000): Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75
- DOI: https://doi.org/10.1090/S0002-9939-04-07685-3
- MathSciNet review: 2111936