The topology of the Banach–Mazur compactum

Sergey A. Antonyan¹

¹

Fundamenta Mathematicae, Tome 166 (2000) no. 3, pp. 209-232

Résumé

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A ⊆ \Bbb R^n$, n ≥ 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 $\bold K(\Bbb Q,2)$; (4) $BM_0(2) = J_0(2)/O(2)$ is noncontractible; (5) BM(2) is a nonhomogeneous absolute retract. Other models for BM(n) are established.

Détail

DOI : 10.4064/fm-166-3-209-232

Keywords: Banach-Mazur compactum, G-ANR, orbit space, Q-manifoldhomotopy type, Eilenberg-MacLane space $\bold K(\Bbb Q, 2)$