A New Linear Programming Method in Sphere Packing
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866915043548856320 |
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| author | Mo, Qun Wen, Jinming Xia, Yu |
| author_facet | Mo, Qun Wen, Jinming Xia, Yu |
| contents | Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary functions $\{g_m\}_{m\in \mathbb{N}^{+}}$, where $g_m$ is a $mΛ$-periodic auxiliary function defined on $\mathbb{R}^n$, with $Λ$ being a given full-rank lattice in $\mathbb{R}^n$. This new method extends the original approach and offers a greater flexibility. Furthermore, using this new linear programming framework, we construct several effective auxiliary functions for dimensions $n=1,2,3$. We hope this approach provides valuable insights into solving sphere packing problems for $n=2,3$ and even higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Linear Programming Method in Sphere Packing Mo, Qun Wen, Jinming Xia, Yu Metric Geometry Functional Analysis Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary functions $\{g_m\}_{m\in \mathbb{N}^{+}}$, where $g_m$ is a $mΛ$-periodic auxiliary function defined on $\mathbb{R}^n$, with $Λ$ being a given full-rank lattice in $\mathbb{R}^n$. This new method extends the original approach and offers a greater flexibility. Furthermore, using this new linear programming framework, we construct several effective auxiliary functions for dimensions $n=1,2,3$. We hope this approach provides valuable insights into solving sphere packing problems for $n=2,3$ and even higher dimensions. |
| title | A New Linear Programming Method in Sphere Packing |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2410.04800 |