Banach–Mazur compactum explained

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 - the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between X and Y is defined by$$\backslash delta(X,\; Y)\; =\; \backslash log\; \backslash Bigl(\backslash inf\; \backslash left\backslash \; \backslash Bigr).$$

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$$d(X,\; Y)\; :=\; \backslash mathrm^\; =\; \backslash inf\; \backslash left\backslash ,$$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 ^{2 )} _{n} \le \sqrt{n}, ^[1]

where

\ell ^{2} _{n} 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 ^{1} _{n} and \ell ^{inf ty} _{n} 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 cn, 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.^[2] On the other hand, Q( 2 ) is not homeomorphic to a Hilbert cube.

References

• • Gluskin. Efim D.. The diameter of the Minkowski compactum is roughly equal to n (in Russian). Funktsional. Anal. I Prilozhen.. 15. 1981. 1. 72 - 73. 10.1007/BF01082381. 0609798. 123649549.
• Book: Tomczak-Jaegermann , Nicole . Nicole Tomczak-Jaegermann. Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics 38. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. 1989. xii+395. 0-582-01374-7. 0993774.
• Banach-Mazur compactum
• A note on the Banach-Mazur distance to the cube
• The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold

Notes and References

1. http://users.uoa.gr/~apgiannop/cube.ps Cube
2. Web site: The Banach–Mazur compactum is not homeomorphic to the Hilbert cube . www.iop.org.