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In the [[mathematics|mathematical]] study of [[functional analysis]], the '''Banach–Mazur distance''' is a way to define a [[distance function|distance]] on the set ''Q''(''n'') of ''n''-dimensional [[normed space]]s. If ''X'' and ''Y'' are two finite-dimensional normed spaces with the same dimension, let GL(''X'',''Y'') denote the collection of all linear isomorphisms ''T'' : ''X'' → ''Y''. The Banach–Mazur distance between ''X'' and ''Y'' is defined by
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: <math>\delta(X, Y) = \log \Bigl( \inf \{ \|T\| \|T^{-1}\| : T \in \operatorname{GL}(X, Y) \} \Bigr).</math>
 
Equipped with the metric ''δ'', the space ''Q''(''n'') is a [[compact metric space]], called the '''Banach–Mazur compactum'''.
 
Many authors prefer to work with the ''multiplicative Banach–Mazur distance''
:<math>d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \{ \|T\| \|T^{-1}\| : T \in \operatorname{GL}(X, Y) \},</math>
for which ''d''(''X'', ''Z'')&nbsp;≤ ''d''(''X'', ''Y'') ''d''(''Y'', ''Z'') and ''d''(''X'', ''X'')&nbsp;= 1.
[[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 ℓ<sub>''n''</sub><sup>2</sup> denotes '''R'''<sup>''n''</sup> with the Euclidean norm (see the article on [[Lp space|''L''<sup>''p''</sup> spaces]]).
From this it follows that ''d''(''X'', ''Y'')&nbsp;≤ ''n'' for every couple (''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 ℓ<sub>''n''</sub><sup>1</sup> and ℓ<sub>''n''</sub><sup>∞</sup> is (only) of order ''n''<sup>1/2</sup> (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''&nbsp;''n'', for some universal ''c''&nbsp;> 0.
 
Gluskin's method introduces a class of random symmetric polytopes ''P''(''ω'') in '''R'''<sup>''n''</sup>, and the normed spaces ''X''(''ω'') having ''P''(''ω'') as unit ball (the vector space is '''R'''<sup>''n''</sup> and the norm is the [[Norm (mathematics)|gauge]] of ''P''(''ω'')). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space ''X''(''ω'').
 
''Q''(2) 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, ''Q''(2) is not homeomorphic to a [[Hilbert cube]].
 
==Notes==
<references/>
 
== 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'' (Russian)
| journal = Funktsional. Anal. i Prilozhen.
| volume = 15
| year = 1981
| issue = 1
| pages = 72&ndash;73
| mr = 0609798
}}
* {{cite book
| last = Tomczak-Jaegermann
| first = Nicole | authorlink = 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
}}
*http://planetmath.org/encyclopedia/BanachMazurCompactum.html
*[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]
 
{{DEFAULTSORT:Banach-Mazur compactum}}
[[Category:Functional analysis]]
[[Category:Metric geometry]]

Latest revision as of 20:47, 18 August 2014

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