The fractional Helly number for separable convexity spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909498941112320 |
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| author | Holmsen, Andreas F. Patáková, Zuzana |
| author_facet | Holmsen, Andreas F. Patáková, Zuzana |
| contents | A convex lattice set in $\mathbb{Z}^d$ is the intersection of a convex set in $\mathbb{R}^d$ and the integer lattice $\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, while a remarkable theorem of Bárány and Matoušek states that the fractional Helly number is only $d+1$. In this paper we generalize their result to abstract convexity spaces that are equipped with a suitable separation property. We also disprove a conjecture of Bárány and Kalai about an existence of fractional Helly property for a family of solutions of bounded-degree polynomial inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The fractional Helly number for separable convexity spaces Holmsen, Andreas F. Patáková, Zuzana Combinatorics A convex lattice set in $\mathbb{Z}^d$ is the intersection of a convex set in $\mathbb{R}^d$ and the integer lattice $\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, while a remarkable theorem of Bárány and Matoušek states that the fractional Helly number is only $d+1$. In this paper we generalize their result to abstract convexity spaces that are equipped with a suitable separation property. We also disprove a conjecture of Bárány and Kalai about an existence of fractional Helly property for a family of solutions of bounded-degree polynomial inequalities. |
| title | The fractional Helly number for separable convexity spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.01445 |