In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set
-dimensional normed spaces.
With this distance, the set of isometry classes of
-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.
are two finite-dimensional normed spaces with the same dimension, let
{\displaystyle \operatorname {GL} (X,Y)}
denote the collection of all linear isomorphisms
the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors.
δ (
) = log
are isometrically isomorphic.
Equipped with the metric δ, the space of isometry classes of
-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.
Many authors prefer to work with the multiplicative Banach–Mazur distance
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate: where
ℓ
with the Euclidean norm (see the article on
However, for the classical spaces, this upper bound for the diameter of
ℓ
ℓ
(up to a multiplicative constant independent from the dimension
A major achievement in the direction of estimating the diameter of
is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by
Gluskin's method introduces a class of random symmetric polytopes
and the normed spaces
as unit ball (the vector space is
and the norm is the gauge of
The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space
is an absolute extensor.
is not homeomorphic to a Hilbert cube.