Quantitative Transversal Theorems in the Plane
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911296988905472 |
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| author | Axelrod-Freed, Ilani Carvalho, João Pedro Takahashi, Yuki |
| author_facet | Axelrod-Freed, Ilani Carvalho, João Pedro Takahashi, Yuki |
| contents | Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent $k$-ordering, for the existence of a hyperplane transversal for sets in $\mathbb{R}^d$. We obtain a quantitative generalization of Hadwiger's theorem in $\mathbb{R}^2$, showing that compact convex sets in $\mathbb{R}^2$ with a quantitative version of consistent ordering have a transversal satisfying quantitative requirements. Our proof generalizes the methods in Wenger's proof of Hadwiger's theorem in $\mathbb{R}^2$. We also prove colorful versions of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11024 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative Transversal Theorems in the Plane Axelrod-Freed, Ilani Carvalho, João Pedro Takahashi, Yuki Combinatorics 52C30 (Primary) 52C35 (Secondary) Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent $k$-ordering, for the existence of a hyperplane transversal for sets in $\mathbb{R}^d$. We obtain a quantitative generalization of Hadwiger's theorem in $\mathbb{R}^2$, showing that compact convex sets in $\mathbb{R}^2$ with a quantitative version of consistent ordering have a transversal satisfying quantitative requirements. Our proof generalizes the methods in Wenger's proof of Hadwiger's theorem in $\mathbb{R}^2$. We also prove colorful versions of our results. |
| title | Quantitative Transversal Theorems in the Plane |
| topic | Combinatorics 52C30 (Primary) 52C35 (Secondary) |
| url | https://arxiv.org/abs/2308.11024 |