On Nieuwland Numbers and Polar Duality

Abstract

A convex 3D-polytope is said to have Rupert's property if it can pass through a copy of itself. The Nieuwland number of a convex polytope $P$ is the largest $\nu \in \mathbb{R}^+$ such that $\nu P$ can pass through $P$. We reduce showing $P$ passes through $Q$ to a feasibility problem over a quadratic constraint set. Using this, we prove that the Nieuwland number of the octahedron is $\frac{3\sqrt2}{4}$ and that the computation of the Nieuwland number of a convex polytope can be reduced to polynomially many semialgebraic optimization problems in a fixed number of variables.