This is an announcement for the paper "The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases" by Osman Gueler and Filiz Guertuna. Abstract: 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. Archive classification: math.OC math.FA Mathematics Subject Classification: 90C34; 46B20; 90C30; 90C46; 65K10 Remarks: 36 pages The source file(s), Ellipsoid35.bbl: 8177 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0707 or http://arXiv.org/abs/0709.0707 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0707 or in gzipped form by using subject line get 0709.0707 to: math@arXiv.org.