Löwner-John ellipsoids

This bachelor thesis is about Löwner-John ellipsoids and the special Steiner inellipse, as well as the geometric connection between the roots of a complex cubic polynomial and its derivative. The central result is given by Marden ́s theorem, which shows an exact position of these roots:

The roots of the derivative of the polynomial correspond to the foci of the Steiner inellipse of the triangle spanned by the roots of the polynomial. To prove this theorem, a short introduction to convex geometry is adduced and based on that Löwner-John ellipsoids are considered. In order to proof finally the theorem of Marden, the Steiner inellipse and its associated properties are discussed, to take it as a geometric configuration for that proof.