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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
DOI: https://doi.org/10.1090/S0002-9939-02-06437-7
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
MR Author ID: 681801
Email: geschke@math.fu-berlin.de

Menachem Kojman
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
Email: kojman@math.bgu.ac.il

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