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| मुख्य लेखकों: | , , |
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| स्वरूप: | Preprint |
| प्रकाशित: |
2024
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| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/2406.01773 |
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| _version_ | 1866909278938333184 |
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| author | Nasonov, Ivan Panina, Gaiane Siersma, Dirk |
| author_facet | Nasonov, Ivan Panina, Gaiane Siersma, Dirk |
| contents | It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for $n=2,3,4$.
We treat the same problem for convex polytopes in $\mathbb{R}^3$. It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in $\mathbb{R}^3$ has $8$ normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in $\mathbb{R}^3$ has a point in its interior with $10$ normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with $10$ normals.
Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concurrent normals problem for convex polytopes and Euclidean distance degree Nasonov, Ivan Panina, Gaiane Siersma, Dirk Metric Geometry Differential Geometry 52B10, 58K05 It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$. It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in $\mathbb{R}^3$ has $8$ normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in $\mathbb{R}^3$ has a point in its interior with $10$ normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with $10$ normals. Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed. |
| title | Concurrent normals problem for convex polytopes and Euclidean distance degree |
| topic | Metric Geometry Differential Geometry 52B10, 58K05 |
| url | https://arxiv.org/abs/2406.01773 |