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Convexity numbers of closed sets in $\mathbb R^n$

Authors: Stefan Geschke and Menachem Kojman
Journal: Proc. Amer. Math. Soc. 130 (2002), 2871-2881
MSC (1991): Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75
Published electronically: March 25, 2002
MathSciNet review: 1908910
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Abstract: For $n>2$ let $\mathcal I_n$ be the $\sigma$-ideal in $\mathcal P(n^\omega)$ generated by all sets which do not contain $n$equidistant points in the usual metric on $n^\omega$. For each $n>2$ a set $S_n$ is constructed in $\mathbb{R} ^n$ so that the $\sigma$-ideal which is generated by the convex subsets of $S_n$ restricted to the convexity radical $K(S_n)$ is isomorphic to $\mathcal I_n$. Thus $\operatorname{cov}(\mathcal I_n)$is equal to the least number of convex subsets required to cover $S_n$ -- the convexity number of $S_n$.

For every non-increasing function $f:\omega\setminus 2\to\{\kappa\in\operatorname{card}:\operatorname{cf}(\kappa)>\aleph_0\}$ we construct a model of set theory in which $\operatorname{cov}(\mathcal I_n)=f(n)$ for each $n\in\omega\setminus 2$. When $f$ is strictly decreasing up to $n$, $n-1$uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of $\mathbb{R} ^n$. It is conjectured that $n$, but never more than $n$, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of $\mathbb{R} ^n$. This conjecture is true for $n=1$and $n=2$.

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Additional Information

Stefan Geschke
Affiliation: Freie Universität Berlin, Arnimallee 2-6, D-1495 Berlin

Menachem Kojman
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

Keywords: Convex cover, convexity number, $n$-space, forcing extension, covering number
Received by editor(s): April 19, 2001
Received by editor(s) in revised form: May 31, 2001
Published electronically: March 25, 2002
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2002 American Mathematical Society