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Osman Guler(gulermath.umbc.edu) 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 semiinfinite 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 semiinfinite 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. Keywords: John ellipsoid, Lowner ellipsoid, inscribed ellipsoid, circumscribed ellipsoid, minimum volume, maximum volume, optimality conditions, semiinfinite programming, contact points, automorphism group, symmetric convex bodies, Haar measure. Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization ) Category 2: Infinite Dimensional Optimization (Semiinfinite Programming ) Category 3: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF] Entry Submitted: 09/05/2007 Modify/Update this entry  
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