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

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 $n=2$.