Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 35 (1994), No. 2, 163-171.
Embedded Maximal Ellipsoids and Semi-Infinite Optimization
Friedrich Juhnke
Abstract.
The ellipsoid of maximal volume contained in a given convex body
$K \subseteq {\bf R}^n$ will be described as solution of a
nonlinear semi-infinite
optimization problem using the Minkowski support function.
The corresponding necessary optimality conditions of the
John-Kuhn/Tucker-type
turn out to be sufficient ones too. Using these conditions only we
show the uniqueness of the maximal ellipsoid. The maximal ellipsoid of $K$
also proves to be the maximal ellipsoid of the intersection of at most
$n(n+3)/2$ supporting halfspaces of $K$.
Furthermore it will be shown that the uniqueness of the
maximal ellipsoid
turns out to be a consequence of certain generalized convexity
properties of the
optimization problem investigated.