2024/10/30 by Nasonov, Ivan, Panina, Gaiane
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2411.12745
It is conjectured since long that each smooth convex body P⊂ ℝn has a point in its interior which belongs to at least 2n normals from different points on the boundary of P. The conjecture is proven for n=2,3,4. We treat the same problem for convex polytopes in ℝ3 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.