Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set formula_1 of formula_2-dimensional normed spaces. With this distance, the set of isometry classes of formula_2-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Definitions. If formula_4 and formula_5 are two finite-dimensional normed spaces with the same dimension, let formula_6 denote the collection of all linear isomorphisms formula_7 Denote by formula_8 the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur  distance between formula_4 and formula_5 is defined by We have formula_12 if and only if the spaces formula_4 and formula_5 are isometrically isomorphic. Equipped with the metric "δ", the space of isometry classes of formula_2-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Many authors prefer to work with the multiplicative Banach–Mazur distance for which formula_17 and formula_18 Properties. F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate: where formula_20 denotes formula_21 with the Euclidean norm (see the article on formula_22 spaces). From this it follows that  formula_23 for all formula_24 However, for the classical spaces, this upper bound for the diameter of formula_1 is far from being approached. For example, the distance between formula_26 and formula_27 is (only) of order formula_28 (up to a multiplicative constant independent from the dimension formula_2). A major achievement in the direction of estimating the diameter of formula_1 is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by formula_31 for some universal formula_32 Gluskin's method introduces a class of random symmetric polytopes formula_33 in formula_34 and the normed  spaces formula_35 having formula_33 as unit ball (the vector space is formula_21 and the norm is the gauge of formula_33). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space formula_39 formula_40 is an absolute extensor. On the other hand, formula_40is not homeomorphic to a Hilbert cube.