Piercing families of convex sets in the plane that avoid a certain subfamily with lines

TL;DR

This paper investigates conditions under which a family of convex sets in the plane can be pierced by a limited number of lines, introducing a new geometric property called $C(k)$ and establishing bounds related to it.

Contribution

The paper introduces the concept of $C(k)$ families of convex sets and proves that families avoiding $C(k)$ can be pierced by $k-2$ lines, providing new insights into geometric piercing problems.

Findings

Families without a $C(k)$ can be pierced by $k-2$ lines.

Existence of convex set families avoiding $C(k)$ that require at least rac{k}{2}-1 lines to pierce.

Abstract

We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). We show that if $\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\mathcal{F}$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\left\lceil \frac{k}{2} \right\rceil -1$ lines.