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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.12268 |
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| _version_ | 1866916253524819968 |
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| author | Frankl, Nóra Jung, Attila Tomon, István |
| author_facet | Frankl, Nóra Jung, Attila Tomon, István |
| contents | Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $\mathcal{F}$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $α\binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ have an intersection of volume at least 1, then one can select $Ω_{d,α}(n)$ members of $\mathcal{F}$ whose intersection has volume at least $Ω_d(1)$.
Furthermore, with the help of this theorem, we establish a quantitative version of the $(p,q)$ theorem of Alon and Kleitman. Let $p\geq q\geq d+1$ and let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ such that among any $p$ elements of $\mathcal{F}$, there are $q$ that have an intersection of volume at least $1$. Then, we prove that there exists a family $T$ of $O_{p,q}(1)$ ellipsoids of volume $Ω_d(1)$ such that every member of $\mathcal{F}$ contains at least one element of $T$.
Finally, we present extensions about the diameter version of the Quantitative Helly theoerm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Quantitative Fractional Helly theorem Frankl, Nóra Jung, Attila Tomon, István Combinatorics Computational Geometry Metric Geometry Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $\mathcal{F}$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $α\binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ have an intersection of volume at least 1, then one can select $Ω_{d,α}(n)$ members of $\mathcal{F}$ whose intersection has volume at least $Ω_d(1)$. Furthermore, with the help of this theorem, we establish a quantitative version of the $(p,q)$ theorem of Alon and Kleitman. Let $p\geq q\geq d+1$ and let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ such that among any $p$ elements of $\mathcal{F}$, there are $q$ that have an intersection of volume at least $1$. Then, we prove that there exists a family $T$ of $O_{p,q}(1)$ ellipsoids of volume $Ω_d(1)$ such that every member of $\mathcal{F}$ contains at least one element of $T$. Finally, we present extensions about the diameter version of the Quantitative Helly theoerm. |
| title | The Quantitative Fractional Helly theorem |
| topic | Combinatorics Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2402.12268 |