论文标题
在飞机上刺穿凸的家族，该家族避免了一定的亚家族
Piercing families of convex sets in the plane that avoid a certain subfamily with lines
论文作者
论文摘要
我们将$ c（k）$定义为$ k $ sets $ f_1，\ dots，f_k $的家庭，使得$ \ textrm {cons}（f_i \ cup f_i \ cup f_ {i+1}）\ cap \ cap \ cap \ textrm {cons} $ \ {i，i+1 \} \ cap \ {j，j+1 \} = \ emptyset $（索引索引$ k $）。我们表明，如果$ \ Mathcal {f} $是一个紧凑的家族，则不包含$ c（k）$的凸套，则有$ k-2 $ lines purece $ \ mathcal {f} $。此外，我们举例说明了一个紧凑型家族的示例，其中包含$ c（k）$，并且不能被$ \ left \ lceil \ frac {k} {2} {2} \ right \ rceil -1 $ lines刺穿。
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.
