On Lattice Diameter Segments and A Discrete Borsuk Partition Problem
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909756883468288 |
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| author | Brose, Anouk E. De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. |
| author_facet | Brose, Anouk E. De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. |
| contents | The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple invariant yields interesting applications and challenges. We describe a polynomial-time algorithm that computes lattice diameter segments of lattice polygons and show that computing lattice diameters of semi-algebraic sets in dimensions three and higher is NP-hard. We prove that the function that counts lattice diameter segments in dilations of a lattice polygon is eventually a quasi-polynomial in the dilation factor. We also study the number of directions that lattice diameter segments can have. Finally, we prove a Borsuk-type theorem on the number of parts needed to partition a set of lattice points such that each part has strictly smaller lattice diameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Lattice Diameter Segments and A Discrete Borsuk Partition Problem Brose, Anouk E. De Loera, Jesús A. Lopez-Campos, Gyivan Torres, Antonio J. Combinatorics 52A38, 52B20, 52B55, 52C07, 52C17 The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple invariant yields interesting applications and challenges. We describe a polynomial-time algorithm that computes lattice diameter segments of lattice polygons and show that computing lattice diameters of semi-algebraic sets in dimensions three and higher is NP-hard. We prove that the function that counts lattice diameter segments in dilations of a lattice polygon is eventually a quasi-polynomial in the dilation factor. We also study the number of directions that lattice diameter segments can have. Finally, we prove a Borsuk-type theorem on the number of parts needed to partition a set of lattice points such that each part has strictly smaller lattice diameter. |
| title | On Lattice Diameter Segments and A Discrete Borsuk Partition Problem |
| topic | Combinatorics 52A38, 52B20, 52B55, 52C07, 52C17 |
| url | https://arxiv.org/abs/2508.20009 |