On the Largest Convexity Number of Co-Finite Sets in the Plane

TL;DR

This paper investigates the maximum convexity number of the complement of a finite point set in the plane, establishing bounds and exact values for points in convex position, thus solving a problem posed by Lawrence and Morris.

Contribution

It provides tight bounds and exact values for the convexity number of the complement of finite point sets, advancing understanding in geometric covering problems.

Findings

For all n ≥ 4, the convexity number f(n) satisfies ⌊(n+5)/2⌋ ≤ f(n) ≤ (7n+44)/11.

When points are in convex position, the convexity number is exactly ⌊(n+5)/2⌋.

The results resolve a previously open problem by Lawrence and Morris.

Abstract

The convexity number of a set $X \subset \mathbb{R}^2$ is the minimum number of convex subsets required to cover it. We study the following question: what is the largest possible convexity number $f(n)$ of $\mathbb{R}^2 \setminus S$, where $S$ is a set of $n$ points in general position in the plane? We prove that for all $n \geq 4$, $\lfloor\frac{n+5}{2}\rfloor \leq f(n) \leq \frac{7n+44}{11}$. We also show that for every $n \geq 4$, if the points of $S$ are in convex position then the convexity number of $\mathbb{R}^2 \setminus S$ is $\lfloor\frac{n+5}{2}\rfloor$. This solves a problem of Lawrence and Morris [Finite sets as complements of finite unions of convex sets, Disc. Comput. Geom. 42 (2009), 206-218].