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].

翻译：集合 $X \subset \mathbb{R}^2$ 的凸性数是指覆盖该集合所需凸子集的最小数目。我们研究以下问题：对于平面上处于一般位置的 $n$ 个点构成的集合 $S$，其补集 $\mathbb{R}^2 \setminus S$ 可能的最大凸性数 $f(n)$ 是多少？我们证明对于所有 $n \geq 4$，有 $\lfloor\frac{n+5}{2}\rfloor \leq f(n) \leq \frac{7n+44}{11}$。我们还证明，对于每个 $n \geq 4$，若 $S$ 的点处于凸位置，则 $\mathbb{R}^2 \setminus S$ 的凸性数为 $\lfloor\frac{n+5}{2}\rfloor$。这解决了 Lawrence 和 Morris 提出的一个问题 [Finite sets as complements of finite unions of convex sets, Disc. Comput. Geom. 42 (2009), 206-218]。