On the Largest Convexity Number of Co-Finite Sets in the Plane
Chaya Keller, Micha A. Perles
TL;DR
This work determines the extremal convexity number for cofinite planar sets $X=\mathbb{R}^2\setminus P$, focusing on $P$ in convex or general position. The authors introduce and relate several covering and encapsulation variants ($cov$, $enc$, and their disjoint forms) and prove tight bounds in the convex-position case: $enc(P)=cov(P)=\left\lfloor\frac{n+5}{2}\right\rfloor-\delta(n)$ with $\delta(n)=1$ for $n\in\{0,1,3\}$, while $enc_{\circ}(P)=cov_{\circ}(P)=\left\lfloor\frac{2n+5}{3}\right\rfloor$; in general position, they obtain linear upper bounds $enc(P),cov(P)\leq\frac{7n}{11}+4$ and $enc_{\circ}(n)=cov_{\circ}(n)\leq\left\lfloor\frac{2n+5}{3}\right\rfloor$. These results resolve the convex- and disjoint-setting questions posed by Lawrence and Morris and provide a structural framework for how removing finite point sets controls unions of convex covers.
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].