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.

Not to be confused with Banach–Mazur game or Banach–Mazur theorem.

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

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

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

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

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_n² denotes Rⁿ 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_n¹ 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 Rⁿ, and the normed spaces X(ω) having P(ω) as unit ball (the vector space is Rⁿ 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

1. http://users.uoa.gr/~apgiannop/cube.ps
2. 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