On a theorem by Kippenhahn

Kippenhahn discovered a real algebraic plane curve whose convex hull is the numerical range of a matrix. The correctness of this theorem was called into question when Chien and Nakazato found an example where the spatial analogue fails. They showed that the mentioned plane curve indeed lies inside the numerical range. We prove the easier converse direction of the theorem. Finding higher-dimensional generalizations of Kippenhahn's theorem is a challenge in real algebraic geometry.