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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 maximumvolume. We first give a unified, modern exposition of the basic theory of theseextremal ellipsoids using the semi-infinite programming approach pioneered byFritz John in his seminal 1948 paper. We then investigate the automorphismgroups of convex bodies and their extremal ellipsoids. We show that if theautomorphism group of a convex body K is large enough, then it is possible todetermine the extremal ellipsoids CE(K) and IE(K) exactly, using eithersemi-infinite programming or nonlinear programming. As examples, we compute theextremal ellipsoids when the convex body K is the part of a given ellipsoidbetween two parallel hyperplanes, and when K is a truncated second order coneor an ellipsoidal cylinder.

The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases