Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set-dimensional normed spaces.With this distance, the set of isometry classes of-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.are two finite-dimensional normed spaces with the same dimension, let{\displaystyle \operatorname {GL} (X,Y)}denote the collection of all linear isomorphismsthe operator norm of such a linear map — the maximum factor by which it "lengthens" vectors.{\displaystyle \delta (X,Y)=\log {\Bigl (}\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\}{\Bigr )}.}are isometrically isomorphic.Equipped with the metric δ, the space of isometry classes of-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.Many authors prefer to work with the multiplicative Banach–Mazur distance{\displaystyle d(X,Y):=\mathrm {e} ^{\delta (X,Y)}=\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\},}F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate: wherewith the Euclidean norm (see the article onHowever, for the classical spaces, this upper bound for the diameter ofFor example, the distance between(up to a multiplicative constant independent from the dimensionA major achievement in the direction of estimating the diameter ofis due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below byGluskin's method introduces a class of random symmetric polytopesand the normed spacesas unit ball (the vector space isand the norm is the gauge ofThe proof consists in showing that the required estimate is true with large probability for two independent copies of the normed spaceis an absolute extensor.is not homeomorphic to a Hilbert cube.