On lattice coverings by locally anti-blocking bodies and polytopes with few vertices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913871810265088 |
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| author | Schymura, Matthias Wang, Jun Xue, Fei |
| author_facet | Schymura, Matthias Wang, Jun Xue, Fei |
| contents | In 2021, Ordentlich, Regev and Weiss made a breakthrough that the lattice covering density of any $n$-dimensional convex body is upper bounded by $cn^{2}$, improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound $n(\log_{e}n)^{c}$, and this result was extended to certain symmetric convex bodies by Gritzmann. The constant $c$ above is independent on $n$. In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti-blocking bodies, locally anti-blocking bodies and $n$-dimensional polytopes with $n+2$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On lattice coverings by locally anti-blocking bodies and polytopes with few vertices Schymura, Matthias Wang, Jun Xue, Fei Metric Geometry 52C17, 11H31 In 2021, Ordentlich, Regev and Weiss made a breakthrough that the lattice covering density of any $n$-dimensional convex body is upper bounded by $cn^{2}$, improving on the best previous bound established by Rogers in 1959. However, for the Euclidean ball, Rogers obtained the better upper bound $n(\log_{e}n)^{c}$, and this result was extended to certain symmetric convex bodies by Gritzmann. The constant $c$ above is independent on $n$. In this paper, we show that such a bound can be achieved for more general classes of convex bodies without symmetry, including anti-blocking bodies, locally anti-blocking bodies and $n$-dimensional polytopes with $n+2$ vertices. |
| title | On lattice coverings by locally anti-blocking bodies and polytopes with few vertices |
| topic | Metric Geometry 52C17, 11H31 |
| url | https://arxiv.org/abs/2505.07369 |