The topology of the Banach-Mazur compactum

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies A subset of or equal to R-n, n greater than or equal to 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let J(0)(n) be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n