2024/06/03 by Ivan Nasonov, Gaiane Panina, Nasonov, Ivan +3 · 1 citation
Mathematics · #52B10 #58K05 #Differential Geometry (math.DG) #FOS: Mathematics #Graph theory and applications #Mathematical Inequalities and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2406.01773
openalex publication_date 2024/06/03 · openalex created_date 2024/06/08 · openalex updated_date 2026/07/28
It is conjectured since long that for any convex body P⊂ ℝn 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 ℝ3. 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 ℝ3 has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in ℝ3 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.