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

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.

Categories: math.CO math.MG