  A convex body K has associated with it a unique circumscribed ellipsoid CE(K)
with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum
volume. We first give a unified, modern exposition of the basic theory of these
extremal ellipsoids using the semi-infinite programming approach pioneered by
Fritz John in his seminal 1948 paper. We then investigate the automorphism
groups of convex bodies and their extremal ellipsoids. We show that if the
automorphism group of a convex body K is large enough, then it is possible to
determine the extremal ellipsoids CE(K) and IE(K) exactly, using either
semi-infinite programming or nonlinear programming. As examples, we compute the
extremal ellipsoids when the convex body K is the part of a given ellipsoid
between two parallel hyperplanes, and when K is a truncated second order cone
or an ellipsoidal cylinder.
