Each generic polytope in $\mathbb{R}^3$ has a point with ten normals to the boundary

Abstract

It is conjectured since long that each smooth convex body $\mathbf{P}\subset \mathbb{R}^n$ has a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $\mathbf{P}$. The conjecture is proven for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$ 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.