Banach–Mazur compactum


In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q of n-dimensional normed spaces. With this distance, the set of isometry classes of n-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.

Definitions

If X and Y are two finite-dimensional normed spaces with the same dimension, let GL denote the collection of all linear isomorphisms T : X → Y. With ||T|| we denote the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between X and Y is defined by
We have δ = 0 if and only if the spaces X and Y are isometrically isomorphic. Equipped with the metric δ, the space of isometry classes of n-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 d ≤ d d and d = 1.

Properties

on the maximal ellipsoid contained in a convex body gives the estimate:
where ℓn2 denotes Rn with the Euclidean norm.
From this it follows that d ≤ n for all X, Y ∈ Q. However, for the classical spaces, this upper bound for the diameter of Q is far from being approached. For example, the distance between ℓn1 and ℓn∞ is of order n1/2.
A major achievement in the direction of estimating the diameter of Q is due to E. Gluskin, who proved in 1981 that the diameter of the Banach–Mazur compactum is bounded below by c n, for some universal c > 0.
Gluskin's method introduces a class of random symmetric polytopes P in Rn, and the normed spaces X having P as unit ball. The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space X.
Q is an absolute extensor. On the other hand, Q is not homeomorphic to a Hilbert cube.