Banach–Mazur compactum

Short description: Set of n-dimensional subspaces of a normed space made into a compact metric space.

In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set [math]\displaystyle{ Q(n) }[/math] of [math]\displaystyle{ n }[/math]-dimensional normed spaces. With this distance, the set of isometry classes of [math]\displaystyle{ n }[/math]-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.

Definitions

If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are two finite-dimensional normed spaces with the same dimension, let [math]\displaystyle{ \operatorname{GL}(X, Y) }[/math] denote the collection of all linear isomorphisms [math]\displaystyle{ T : X \to Y. }[/math] Denote by [math]\displaystyle{ \|T\| }[/math] the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] is defined by [math]\displaystyle{ \delta(X, Y) = \log \Bigl( \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\} \Bigr). }[/math]

We have [math]\displaystyle{ \delta(X, Y) = 0 }[/math] if and only if the spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are isometrically isomorphic. Equipped with the metric δ, the space of isometry classes of [math]\displaystyle{ n }[/math]-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.

Many authors prefer to work with the multiplicative Banach–Mazur distance [math]\displaystyle{ d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\}, }[/math] for which [math]\displaystyle{ d(X, Z) \leq d(X, Y) \, d(Y, Z) }[/math] and [math]\displaystyle{ d(X, X) = 1. }[/math]

Properties

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

[math]\displaystyle{ d(X, \ell_n^2) \le \sqrt{n}, \, }[/math]

where [math]\displaystyle{ \ell_n^2 }[/math] denotes [math]\displaystyle{ \R^n }[/math] with the Euclidean norm (see the article on [math]\displaystyle{ L^p }[/math] spaces). From this it follows that [math]\displaystyle{ d(X, Y) \leq n }[/math] for all [math]\displaystyle{ X, Y \in Q(n). }[/math] However, for the classical spaces, this upper bound for the diameter of [math]\displaystyle{ Q(n) }[/math] is far from being approached. For example, the distance between [math]\displaystyle{ \ell_n^1 }[/math] and [math]\displaystyle{ \ell_n^{\infty} }[/math] is (only) of order [math]\displaystyle{ n^{1/2} }[/math] (up to a multiplicative constant independent from the dimension [math]\displaystyle{ n }[/math]).

A major achievement in the direction of estimating the diameter of [math]\displaystyle{ Q(n) }[/math] is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by [math]\displaystyle{ c\,n, }[/math] for some universal [math]\displaystyle{ c \gt 0. }[/math]

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

[math]\displaystyle{ Q(2) }[/math] is an absolute extensor. On the other hand, [math]\displaystyle{ Q(2) }[/math]is not homeomorphic to a Hilbert cube.

See also

• Compact space – Type of mathematical space
• General linear group – Group of n×n invertible matrices

References

• Hazewinkel, Michiel, ed. (2001), "Banach–Mazur compactum", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b110100 
• 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. doi:10.1007/BF01082381. 
• Tomczak-Jaegermann (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. 
• 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