The topology of the Banach-Mazur compactum

Abstract

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)

(2) J(n) is an O(n)-AR

(3) J(0)(2)/SO(2) is an Eilenberg-MacLane space K(Q, 2)

(4) BM0(2) = J(0)(2)/O(2) is noncontractible

(5) BM(2) is a nonhomogeneous absolute retract. Other models for BM(n) are established.