Banach–Mazur compactum

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

Definitions

If ${\displaystyle X}$ and ${\displaystyle Y}$ are two finite-dimensional normed spaces with the same dimension, let ${\displaystyle \operatorname {GL} (X,Y)}$ denote the collection of all linear isomorphisms ${\displaystyle T:X\to Y.}$ Denote by ${\displaystyle \|T\|}$ the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between ${\displaystyle X}$ and ${\displaystyle Y}$ is defined by

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

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

Many authors prefer to work with the multiplicative Banach–Mazur distance

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

${\displaystyle d(X,Z)\leq d(X,Y)\,d(Y,Z)}$

${\displaystyle d(X,X)=1.}$

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 ${\displaystyle \ell _{n}^{2}}$ denotes ${\displaystyle \mathbb {R} ^{n}}$ with the Euclidean norm (see the article on ${\displaystyle L^{p}}$ spaces).

From this it follows that ${\displaystyle d(X,Y)\leq n}$ for all ${\displaystyle X,Y\in Q(n).}$ However, for the classical spaces, this upper bound for the diameter of ${\displaystyle Q(n)}$ is far from being approached. For example, the distance between ${\displaystyle \ell _{n}^{1}}$ and ${\displaystyle \ell _{n}^{\infty }}$ is (only) of order ${\displaystyle n^{1/2}}$ (up to a multiplicative constant independent from the dimension ${\displaystyle n}$).

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

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

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

See also

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

Notes

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