Helly numbers for Quantitative Helly-type results
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914955224154112 |
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| author | Ivanov, G. Naszodi, M. |
| author_facet | Ivanov, G. Naszodi, M. |
| contents | We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number \(2d\) concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number \(2d+1\) for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes) \(3d+1\); however, we have no reason to believe that this bound is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Helly numbers for Quantitative Helly-type results Ivanov, G. Naszodi, M. Combinatorics 52A27 (primary), 52A35 We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number \(2d\) concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number \(2d+1\) for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes) \(3d+1\); however, we have no reason to believe that this bound is sharp. |
| title | Helly numbers for Quantitative Helly-type results |
| topic | Combinatorics 52A27 (primary), 52A35 |
| url | https://arxiv.org/abs/2409.15048 |