Concurrent normals problem for convex polytopes and Euclidean distance degree

Abstract

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 \(n=2,3,4\).

We treat the same problem for convex polytopes in \(\mathbb{R}^3\). 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 \(\mathbb{R}^3\) has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in \(\mathbb{R}^3\) 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.