View source for Banach–Mazur compactum

{{short description|Set of n-dimensional subspaces of a normed space made into a compact metric space.}}
{{distinguish|Banach–Mazur game|Banach–Mazur theorem}}
{{Cleanup bare URLs|date=September 2022}}
In the [[mathematics|mathematical]] study of [[functional analysis]], the '''Banach–Mazur distance''' is a way to define a [[Distance function|distance]] on the set <math>Q(n)</math> of <math>n</math>-dimensional [[normed space]]s. With this distance, the set of [[isometry]] classes of <math>n</math>-dimensional normed spaces becomes a [[compact metric space]], called the '''Banach–Mazur compactum'''.

== Definitions ==
If <math>X</math> and <math>Y</math> are two finite-dimensional normed spaces with the same dimension, let <math>\operatorname{GL}(X, Y)</math> denote the collection of all linear isomorphisms <math>T : X \to Y.</math>  Denote by <math>\|T\|</math> the [[operator norm]] of such a linear map &mdash; the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between <math>X</math> and <math>Y</math> is defined by
<math display="block">\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>\delta(X, Y) = 0</math> if and only if the spaces <math>X</math> and <math>Y</math> are isometrically isomorphic. Equipped with the metric ''δ'', the space of isometry classes of <math>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 display="block">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>d(X, Z) \leq d(X, Y) \, d(Y, Z)</math> and <math>d(X, X) = 1.</math>

== Properties ==
[[John ellipsoid|F. John's theorem]] on the maximal ellipsoid contained in a convex body gives the estimate:

: <math>d(X, \ell_n^2) \le \sqrt{n}, \,</math> <ref>http://users.uoa.gr/~apgiannop/cube.ps</ref>

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

A major achievement in the direction of estimating the diameter of <math>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>c\,n,</math> for some universal <math>c > 0.</math>

Gluskin's method introduces a class of random symmetric [[polytope]]s <math>P(\omega)</math> in <math>\R^n,</math> and the normed spaces <math>X(\omega)</math> having <math>P(\omega)</math> as unit ball (the vector space is <math>\R^n</math> and the norm is the [[Norm (mathematics)|gauge]] of <math>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>X(\omega).</math>

<math>Q(2)</math> is an [[absolute extensor]].<ref>[http://www.iop.org/EJ/article/0036-0279/53/1/L06/RMS_53_1_L06.pdf?request-id=e60e8254-2945-4600-b648-67deecf06a15 The Banach–Mazur compactum is not homeomorphic to the Hilbert cube]</ref> On the other hand, <math>Q(2)</math>is not homeomorphic to a [[Hilbert cube]].

== See also ==

* {{annotated link|Compact space}}
* {{annotated link|General linear group}}

== Notes ==
{{reflist}}

== References ==
* {{springer|title=Banach–Mazur compactum|id=B/b110100|first=A.A.|last=Giannopoulos}}
* {{cite journal
| last = Gluskin
| first =  Efim D.
| title = The diameter of the Minkowski compactum is roughly equal to ''n'' (in Russian)
| journal = Funktsional. Anal. I Prilozhen.
| volume = 15
| year = 1981
| issue = 1
| pages = 72&ndash;73
| doi = 10.1007/BF01082381
| mr = 0609798
| s2cid = 123649549
}}
* {{cite book
| last = Tomczak-Jaegermann
| first = Nicole | author-link = Nicole Tomczak-Jaegermann
| title = Banach-Mazur distances and finite-dimensional operator ideals
| series = Pitman Monographs and Surveys in Pure and Applied Mathematics 38
| publisher = Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York
| year = 1989
| pages = xii+395
| isbn = 0-582-01374-7
| mr = 0993774
}}
* https://planetmath.org/BanachMazurCompactum
* [http://users.uoa.gr/~apgiannop/cube.ps A note on the Banach-Mazur distance to the cube]
* [http://at.yorku.ca/p/a/a/o/26.htm The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold]

{{Banach spaces}}
{{Functional analysis}}
{{Topological vector spaces}}

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[[Category:Functional analysis]]
[[Category:Metric geometry]]
[[Category:Metric spaces]]