  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.
