Ageev, S. M.; Bogatyi, S. A. The Banach-Mazur compactum is not homeomorphic to the Hilbert cube. (English. Russian original) Zbl 0923.46015 Russ. Math. Surv. 53, No. 1, 205-207 (1998); translation from Usp. Mat. Nauk 53, No. 1, 209-210 (1998). The Banach-Mazur compactum \(Q(n)\) is the set of isometric equivalence classes of \(n\)-dimensional Banach spaces equipped with the Banach-Mazur metric \[ d(E,F):= \log\inf\{\| T\| \cdot\| T^{-1}\| : T \text{ an isomorphism from \(E\) to }F\}. \] The precise nature (both topologically and metrically) of this compactum is an old and interesting question. The authors show that \(Q(2)\) is not homeomorphic to the Hilbert cube and thus answer, for \(n=2\), Problem 899 in [Open problems in topology (1990; Zbl 0718.54001)]. The exact result is that whereas the complement of a single point in the Hilbert cube is homotopically trivial, the complement of \(\{\mathcal E\}\) (the equivalence class of ellipses) in \(Q(2)\) is not contractible to a point; indeed it has non-trivial four-dimensional homotopy. Reviewer: A.C.Thompson (Halifax) Cited in 1 Document MSC: 46B20 Geometry and structure of normed linear spaces 46B04 Isometric theory of Banach spaces 52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) Keywords:Banach-Mazur metric; Hilbert cube; homeomorphism; contractible; homotopy Citations:Zbl 0718.54001 PDFBibTeX XMLCite \textit{S. M. Ageev} and \textit{S. A. Bogatyi}, Russ. Math. Surv. 53, No. 1, 205--207 (1998; Zbl 0923.46015); translation from Usp. Mat. Nauk 53, No. 1, 209--210 (1998) Full Text: DOI