From: greg@math.ucdavis.edu (Greg Kuperberg) Subject: Re: Compact convex sets in R^n Date: 14 Jan 2001 13:50:49 -0800 Newsgroups: sci.math.research Summary: Convex sets with a transitive linear automorphism group In article <93soqc$m8j$1@nnrp1.deja.com>, wrote: >Balls centered at the origin of R^n have the property that there is a >group of linear transformations that acts transitively on their >boundary (the orthogonal group of course), >and this is the case also for ellipsoids, in this case the group is a >conjugate of the orthogonal group. > >I asked a topologist if ellipsoids are the only compact convex sets >with this property An elementary lemma shows that there is a unique minimum-volume ellipsoid containing any compact set in Euclidean R^n. Either this or the dual ellipsoid (available for convex sets) is called the "Fritz John ellipsoid". After a linear transformation we can take this ellipsoid to be the unit sphere. Since your transitive group fixes this sphere it must be a subgroup of O(n). But now since your set S is convex, it is the graph of a polar function. (The center of the bounding ellipsoid must be in S.) Transitivity implies that the polar function is constant. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print *