Concurrent normals problem for convex polytopes

This is a joint work with I. Nasonov.
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 $n=2,3$ (E. Heil, 1985) and $n=4$ (J. Pardon, 2012).
We treat the same problem for convex polytopes and prove that each simple polytope in $\mathbb{R}^3$ has a point in its interior with $10$ normals to the boundary. This is an exact bound: there exists a tetrahedron with at most $10$ normals from a point in its interior. The proof is based on Morse–Cerf theory adjusted for polytopes.