More on convexity numbers of closed sets in ℝⁿ

Abstract

The convexity number of a set S ⊆ R n S\subseteq \mathbb R^n is the least size of a family F \mathcal F of convex sets with ⋃ F = S \bigcup \mathcal F=S . S S is countably convex if its convexity number is countable. Otherwise S S is uncountably convex. Uncountably convex closed sets in R n \mathbb R^n have been studied recently by Geschke, Kubiś, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all n ≥ 2 n\geq 2 , it is consistent that there is an uncountably convex closed set S ⊆ R n + 1 S\subseteq \mathbb R^{n+1} whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of R n \mathbb R^n . Moreover, we construct a closed set S ⊆ R 3 S\subseteq \mathbb R^3 whose convexity number is 2 ℵ 0 2^{\aleph _0} and that has no uncountable k k -clique for any k > 1 k>1 . Here C ⊆ S C\subseteq S is a k k -clique if the convex hull of no k k -element subset of C C is included in S S . Our example shows that the main result of the above-named authors, a closed set S ⊆ R 2 S\subseteq \mathbb R^2 either has a perfect 3 3 -clique or the convexity number of S S is > 2 ℵ 0 >2^{\aleph _0} in some forcing extension of the universe, cannot be extended to higher dimensions.