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arXiv · 2406.01773

Concurrent normals problem for convex polytopes and Euclidean distance degree

Ivan Nasonov·Gaiane Panina·Dirk Siersma
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Abstract

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.

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Ivan Nasonov, Gaiane Panina, Dirk Siersma. 2024-06-03. Concurrent normals problem for convex polytopes and Euclidean distance degree. https://arxiv.org/abs/2406.01773

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