The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915032882741248 |
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| author | Lian, Yanlu Mo, Qun Xia, Yu |
| author_facet | Lian, Yanlu Mo, Qun Xia, Yu |
| contents | The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method Lian, Yanlu Mo, Qun Xia, Yu Metric Geometry 52C17, 11H31 The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain. |
| title | The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method |
| topic | Metric Geometry 52C17, 11H31 |
| url | https://arxiv.org/abs/2411.16038 |