On three measures of non-convexity

Abstract

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in $X$. We consider the following three parameters of a set $X$: the clique number $\omega(I(X))$, the chromatic number $\chi(I(X))$ and the convexity number $\gamma(X)$, which is the minimum number of convex subsets of $X$ that cover $X$. We settle a conjecture of Matou\v{s}ek and Valtr claiming that for every planar set $X$, $\gamma(X)$ can be bounded in terms of $\chi(I(X))$. As a part of the proof we show that a disc with $n$ one-point holes near its boundary has $\chi(I(X)) \ge \log\log(n)$ but $\omega(I(X))=3$. We also find sets $X$ in $\mathbb{R}^5$ with $\chi(X)=2$, but $\gamma(X)$ arbitrarily large.