  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.
