Concurrent normals problem for convex polytopes and Euclidean distance degree

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 polyt...

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Veröffentlicht in:arXiv.org 2024-08
Hauptverfasser: Nasonov, Ivan, Panina, Gaiane, Siersma, Dirk
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:2331-8422