Banach–Mazur compactum

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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. 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

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

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

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

for which d(X, Z) ≤ d(X, Y) d(Y, Z) and d(X, X) = 1. F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:

d(X, \ell^n_2) \le \sqrt{n}, \, [1]

where ℓn2 denotes Rn with the Euclidean norm (see the article on Lp spaces). From this it follows that d(X, Y) ≤ 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 ℓn1 and ℓn is (only) of order n1/2 (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(ω) in Rn, and the normed spaces X(ω) having P(ω) as unit ball (the vector space is Rn and the norm is the 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.[2] On the other hand, Q(2) is not homeomorphic to a Hilbert cube.

Notes[edit]

  1. ^ http://users.uoa.gr/~apgiannop/cube.ps
  2. ^ The Banach–Mazur compactum is not homeomorphic to the Hilbert cube

References[edit]