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Convexity numbers of closed sets in
Authors:Stefan Geschke   Menachem Kojman
Affiliation:Freie Universität Berlin, Arnimallee 2-6, D-1495 Berlin ; Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
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 $f:omegasetminus 2to{kappainoperatorname{card}:operatorname{cf}(kappa)>aleph_0}$ we construct a model of set theory in which $operatorname{cov}(mathcal I_n)=f(n)$ for each $ninomegasetminus 2$. 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 $n=2$.
Keywords:Convex cover   convexity number   $n$-space   forcing extension   covering number
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