Each generic polytope in $\mathbb{R}^3$ has a point with ten normals to the boundary
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918138472300544 |
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| author | Nasonov, Ivan Panina, Gaiane |
| author_facet | Nasonov, Ivan Panina, Gaiane |
| contents | It is conjectured since long that each smooth convex body $\mathbf{P}\subset \mathbb{R}^n$ has a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $\mathbf{P}$. The conjecture is proven for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$ and prove that each generic polytope has a point in its interior with at least $10$ normals to the boundary. This bound is exact: there exists a tetrahedron with no more than $10$ normals emanating from a point in its interior.
The proof is based on piecewise linear analog of Morse theory, analysis of bifurcations, and some combinatorial tricks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Each generic polytope in $\mathbb{R}^3$ has a point with ten normals to the boundary Nasonov, Ivan Panina, Gaiane Metric Geometry It is conjectured since long that each smooth convex body $\mathbf{P}\subset \mathbb{R}^n$ has a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $\mathbf{P}$. The conjecture is proven for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$ and prove that each generic polytope has a point in its interior with at least $10$ normals to the boundary. This bound is exact: there exists a tetrahedron with no more than $10$ normals emanating from a point in its interior. The proof is based on piecewise linear analog of Morse theory, analysis of bifurcations, and some combinatorial tricks. |
| title | Each generic polytope in $\mathbb{R}^3$ has a point with ten normals to the boundary |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2411.12745 |