  Kippenhahn discovered that the numerical range of a complex square matrix is
the convex hull of a plane real algebraic curve. Here, we present an example of
a convex set, which has a similar algebraic description as the numerical range,
whereas the analogue of Kippenhahn's construction fails regarding isolated,
singular points of the curve. This example prompted us to carefully review
Kippenhahn's assertion and to highlight aspects of a complete proof that was
achieved with methods of convex geometry and real algebraic geometry.
