Banach–Mazur compactum

In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set of -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.

Definitions

If and are two finite-dimensional normed spaces with the same dimension, let denote the collection of all linear isomorphisms Denote by the operator norm of such a linear map the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between and is defined by

We have if and only if the spaces and 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

for which and

Properties

F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:

[1]

where denotes with the Euclidean norm (see the article on spaces). From this it follows that for all However, for the classical spaces, this upper bound for the diameter of is far from being approached. For example, the distance between and is (only) of order (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 for some universal

Gluskin's method introduces a class of random symmetric polytopes in and the normed spaces having 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.[2] On the other hand, is not homeomorphic to a Hilbert cube.

See also

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.
  • https://planetmath.org/BanachMazurCompactum
  • A note on the Banach-Mazur distance to the cube
  • The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold
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