Banach–Mazur compactum

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

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

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

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

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

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:

$d(X,\ell _{n}^{2})\leq {\sqrt {n}},\,$

where $\ell _{n}^{2}$ denotes $\mathbb {R} ^{n}$ 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 $\ell _{n}^{\infty }$ 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 $P(\omega )$ in $\mathbb {R} ^{n},$ and the normed spaces $X(\omega )$ having $P(\omega )$ as unit ball (the vector space is $\mathbb {R} ^{n}$ 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

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

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. doi:10.1007/BF01082381. MR 0609798. S2CID 123649549.
• 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.
• 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