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

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 $n=2$.