Piercing families of convex sets in the plane that avoid a certain subfamily with lines
| Autor: | McGinnis, Daniel |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 9 pages, 9 figures |
| Databáze: | arXiv |
| Externí odkaz: |
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