Piercing intersecting convex sets
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910821140922368 |
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| author | Bárány, Imre Dillon, Travis Pálvölgyi, Dömötör Varga, Dániel |
| author_facet | Bárány, Imre Dillon, Travis Pálvölgyi, Dömötör Varga, Dániel |
| contents | Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06472 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Piercing intersecting convex sets Bárány, Imre Dillon, Travis Pálvölgyi, Dömötör Varga, Dániel Combinatorics 52A15, 52A35, 90C05 Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal. |
| title | Piercing intersecting convex sets |
| topic | Combinatorics 52A15, 52A35, 90C05 |
| url | https://arxiv.org/abs/2409.06472 |