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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2504.17184 |
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| _version_ | 1866913965536182272 |
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| author | Bannai, Eiichi Kurihara, Hirotake Nozaki, Hiroshi |
| author_facet | Bannai, Eiichi Kurihara, Hirotake Nozaki, Hiroshi |
| contents | This paper investigates the existence of $m$-stiff configurations in the unit sphere $S^{d-1}$, which are spherical $(2m-1)$-designs that lie on $m$ parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $d$; (2) for each fixed integer $d > 10$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $m$. Furthermore, we provide a complete classification of the dimensions where $m$-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of $m$-stiff configurations in $S^{d-1}$ for small $d$ ($3 \leq d \leq 120$) with arbitrary $m$, and also for small $m$ ($6 \leq m \leq 10$) with arbitrary $d$. Finally, we conjecture that there is no $m$-stiff configuration in $S^{d-1}$ for $(d,m)$ with $d\geq 3$ and $m\geq 6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence and non-existence of spherical $m$-stiff configurations Bannai, Eiichi Kurihara, Hirotake Nozaki, Hiroshi Combinatorics 05B30, 33C45 This paper investigates the existence of $m$-stiff configurations in the unit sphere $S^{d-1}$, which are spherical $(2m-1)$-designs that lie on $m$ parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $d$; (2) for each fixed integer $d > 10$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $m$. Furthermore, we provide a complete classification of the dimensions where $m$-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of $m$-stiff configurations in $S^{d-1}$ for small $d$ ($3 \leq d \leq 120$) with arbitrary $m$, and also for small $m$ ($6 \leq m \leq 10$) with arbitrary $d$. Finally, we conjecture that there is no $m$-stiff configuration in $S^{d-1}$ for $(d,m)$ with $d\geq 3$ and $m\geq 6$. |
| title | On the existence and non-existence of spherical $m$-stiff configurations |
| topic | Combinatorics 05B30, 33C45 |
| url | https://arxiv.org/abs/2504.17184 |