Banach–Mazur compactum

In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set $$Q(n)$$ 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 $$\operatorname{GL}(X, Y)$$ denote the collection of all linear isomorphisms $$T : X \to Y.$$ Denote by $$\|T\|$$ the operator norm of such a linear map &mdash; the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between $$X$$ and $$Y$$ is defined by $$\delta(X, Y) = \log \Bigl( \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\} \Bigr).$$

We have $$\delta(X, Y) = 0$$ 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 $$d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\},$$ for which $$d(X, Z) \leq d(X, Y) \, d(Y, Z)$$ and $$d(X, X) = 1.$$

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


 * $$d(X, \ell_n^2) \le \sqrt{n}, \,$$

where $$\ell_n^2$$ denotes $$\R^n$$ with the Euclidean norm (see the article on $L^p$ spaces).

From this it follows that $$d(X, Y) \leq n$$ for all $$X, Y \in Q(n).$$ However, for the classical spaces, this upper bound for the diameter of $$Q(n)$$ is far from being approached. For example, the distance between $$\ell_n^1$$ and $$\ell_n^{\infty}$$ is (only) of order $$n^{1/2}$$ (up to a multiplicative constant independent from the dimension $$n$$).

A major achievement in the direction of estimating the diameter of $$Q(n)$$ is due to E. Gluskin, who proved in 1981 that the (multiplicative) 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 $$P(\omega)$$ in $$\R^n,$$ and the normed spaces $$X(\omega)$$ having $$P(\omega)$$ as unit ball (the vector space is $$\R^n$$ and the norm is the gauge of $$P(\omega)$$). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space $$X(\omega).$$

$$Q(2)$$ is an absolute extensor. On the other hand, $$Q(2)$$is not homeomorphic to a Hilbert cube.