Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set  of -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. Definitions 
If  and  are two finite-dimensional normed spaces with the same dimension, let  denote the collection of all linear isomorphisms   Denote by  the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between  and  is defined by

We have  if and only if the spaces  and  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

for which  and

Properties 
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:

  

where  denotes  with the Euclidean norm (see the article on  spaces). From this it follows that  for all  However, for the classical spaces, this upper bound for the diameter of  is far from being approached. For example, the distance between  and  is (only) of order  (up to a multiplicative constant independent from the dimension ). A major achievement in the direction of estimating the diameter of  is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by  for some universal 

Gluskin's method introduces a class of random symmetric polytopes  in  and the normed spaces  having  as unit ball (the vector space is  and the norm is the gauge of ). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space 

 is an absolute extensor. On the other hand, is not homeomorphic to a Hilbert cube. See also

Notes

References 
 
 
 
 https://planetmath.org/BanachMazurCompactum
 A note on the Banach-Mazur distance to the cube
 The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold

Functional analysis
Metric geometry
Metric spaces