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

2026-01-01

AI Summary

This study investigates the minimum number $ f(n) $ of convex sets required to cover the plane after removing $ n $ points in general position. By integrating techniques from discrete geometry, combinatorial optimization, and convex covering theory, the work resolves an open problem posed by Lawrence and Morris, establishing tight bounds: for all $ n \geq 4 $, $ \lfloor (n+5)/2 \rfloor \leq f(n) \leq (7n+44)/11 $. Notably, when the removed points are in convex position, the exact value is determined as $ f(n) = \lfloor (n+5)/2 \rfloor $, yielding a complete characterization in this case.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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].

Problem

• convexity number
• co-finite sets
• convex cover
• plane
• general position

Innovation

• convexity number
• co-finite sets
• convex cover
• combinatorial geometry
• convex position