On the kissing number of the cross-polytope
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915139383459840 |
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| author | Miller, Niklas |
| author_facet | Miller, Niklas |
| contents | A new upper bound $κ_T(K_n)\leq 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound $κ_T(K_n)\leq 3^n-1$ from 1957. Furthermore, it is shown that there exist kissing configurations satisfying $κ_T(K_n)\geq 1.1637^{(1-o(1))n}$, which improves on the previous best lower bound $ κ_T(K_n)\geq 1.1348^{(1-o(1))n}$ by Talata. It is also shown that the lattice kissing number satisfies $κ_L(K_n)< 12(2^n-1)$ for all $n\geq 1$, and that the lattice $D_4^+$ is the unique lattice, up to signed permutations of coordinates, attaining the maximum lattice kissing number $κ_L(K_4)=40$ in four dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the kissing number of the cross-polytope Miller, Niklas Metric Geometry 52C17, 11H31 A new upper bound $κ_T(K_n)\leq 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound $κ_T(K_n)\leq 3^n-1$ from 1957. Furthermore, it is shown that there exist kissing configurations satisfying $κ_T(K_n)\geq 1.1637^{(1-o(1))n}$, which improves on the previous best lower bound $ κ_T(K_n)\geq 1.1348^{(1-o(1))n}$ by Talata. It is also shown that the lattice kissing number satisfies $κ_L(K_n)< 12(2^n-1)$ for all $n\geq 1$, and that the lattice $D_4^+$ is the unique lattice, up to signed permutations of coordinates, attaining the maximum lattice kissing number $κ_L(K_4)=40$ in four dimensions. |
| title | On the kissing number of the cross-polytope |
| topic | Metric Geometry 52C17, 11H31 |
| url | https://arxiv.org/abs/2501.09245 |