On the Number of Vertices in a Hyperplane Section of a Polytope
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| Format: | Preprint |
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2024
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| _version_ | 1866912497639882752 |
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| author | De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. |
| author_facet | De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. |
| contents | We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Number of Vertices in a Hyperplane Section of a Polytope De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. Combinatorics 52C35, 52C07 We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes. |
| title | On the Number of Vertices in a Hyperplane Section of a Polytope |
| topic | Combinatorics 52C35, 52C07 |
| url | https://arxiv.org/abs/2412.12419 |