More on convexity numbers of closed sets in $\mathbb{R} ^n$

The convexity number of a set $S\subseteq\mathbb R^n$ is the least size of a family $\mathcal F$ of convex sets with $\bigcup\mathcal F=S$. $S$ is countably convex if its convexity number is countable. Otherwise $S$ is uncountably convex.

Uncountably convex closed sets in $\mathbb R^n$ have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all $n\geq 2$, it is consistent that there is an uncountably convex closed set $S\subseteq\mathbb R^{n+1}$ whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of $\mathbb R^n$.

Moreover, we construct a closed set $S\subseteq\mathbb R^3$ whose convexity number is $2^{\aleph_0}$ and that has no uncountable $k$-clique for any $k>1$. Here $C\subseteq S$ is a $k$-clique if the convex hull of no $k$-element subset of $C$ is included in $S$. Our example shows that the main result of the above-named authors, a closed set $S\subseteq\mathbb R^2$ either has a perfect $3$-clique or the convexity number of $S$ is $<2^{\aleph_0}$ in some forcing extension of the universe, cannot be extended to higher dimensions.