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 $\operatorname(X, Y)$ 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
$\delta(X, Y) = \log \Bigl( \inf \left\ \Bigr).$

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
$d(X, Y) := \mathrm^ = \inf \left\,$
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:

: $d(X, \ell_n^2) \le \sqrt, \,$

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 $\ell_n^$ is (only) of order $n^$ (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.

See also

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Notes

References

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Banach-Mazur compactum

A note on the Banach-Mazur distance to the cube

{{DEFAULTSORT:Banach-Mazur compactum
Functional analysis
Metric geometry
Metric spaces