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 GL(X,Y) 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

\( \delta(X, Y) = \log \Bigl( \inf \{ \|T\| \|T^{-1}\| : T \in \operatorname{GL}(X, Y) \} \Bigr). \)

We have δ(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

\( d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \{ \|T\| \|T^{-1}\| : T \in \operatorname{GL}(X, Y) \}, \)

for which d(X, Z) ≤ 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:

\( {\displaystyle d(X,\ell _{n}^{2})\leq {\sqrt {n}},\,} [1] \)

where ℓ_{n}^{2} denotes **R**^{n} with the Euclidean norm (see the article on *L*^{p} spaces). From this it follows that *d*(*X*, *Y*) ≤ *n* for all *X*, *Y* ∈ *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 ℓ_{n}^{1} and ℓ_{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 *c* *n*, for some universal *c* > 0.

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

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

Notes

http://users.uoa.gr/~apgiannop/cube.ps

The Banach–Mazur compactum is not homeomorphic to the Hilbert cube

References

Giannopoulos, A.A. (2001) [1994], "Banach–Mazur compactum", Encyclopedia of Mathematics, EMS Press

Gluskin, Efim D. (1981). "The diameter of the Minkowski compactum is roughly equal to n (in Russian)". Funktsional. Anal. i Prilozhen. 15 (1): 72–73. MR 0609798.

Tomczak-Jaegermann, Nicole (1989). 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. pp. xii+395. ISBN 0-582-01374-7. MR 0993774.

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

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