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

TL;DR

The paper proves that in three-dimensional convex polytopes, a generic interior point has at least ten boundary normals, confirming a specific case of a long-standing conjecture about convex bodies.

Contribution

It establishes that every generic convex polytope in A3 has an interior point with at least ten boundary normals, providing the first exact bound for polytopes in this context.

Findings

Every generic polytope in A3 has an interior point with at least 10 normals.

The bound of 10 normals is sharp, as shown by a tetrahedron with no more than 10 normals from an interior point.

The proof employs piecewise linear Morse theory, bifurcation analysis, and combinatorial techniques.

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.