On three measures of non-convexity

TL;DR

This paper investigates measures of non-convexity in sets, establishing bounds and relationships between graph-theoretic parameters and convex coverings, including resolving a conjecture for planar sets.

Contribution

It proves a conjecture linking the convexity number and chromatic number for planar sets and constructs examples demonstrating complex relationships in higher dimensions.

Findings

Bounded the convexity number by the chromatic number for planar sets.

Constructed sets with high convexity number but low chromatic number in higher dimensions.

Showed that certain geometric configurations have surprisingly small clique and chromatic numbers.

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.