Shortest closed curve to contain a sphere in its convex hull
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909149604872192 |
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| author | Ghomi, Mohammad Wenk, James |
| author_facet | Ghomi, Mohammad Wenk, James |
| contents | We show that in Euclidean 3-space any closed curve $γ$ which contains the unit sphere within its convex hull has length $L\geq4π$, and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in $n$ dimensions, we include the estimate $L\geq Cn\sqrt{n}$ by Nazarov, which is sharp up to the constant $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_05988 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shortest closed curve to contain a sphere in its convex hull Ghomi, Mohammad Wenk, James Differential Geometry Metric Geometry Primary: 53A04, 52A40, Secondary: 60G15, 58E We show that in Euclidean 3-space any closed curve $γ$ which contains the unit sphere within its convex hull has length $L\geq4π$, and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in $n$ dimensions, we include the estimate $L\geq Cn\sqrt{n}$ by Nazarov, which is sharp up to the constant $C$. |
| title | Shortest closed curve to contain a sphere in its convex hull |
| topic | Differential Geometry Metric Geometry Primary: 53A04, 52A40, Secondary: 60G15, 58E |
| url | https://arxiv.org/abs/2209.05988 |