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 BanachâMazur distance between and is defined by
We have if and only if the spaces and are isometrically isomorphic. Equipped with the metric ÎŽ, the space of isometry classes of -dimensional normed spaces becomes a
compact metric space, called the BanachâMazur compactum.
Many authors prefer to work with the multiplicative BanachâMazur distance
for which and
Properties
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:
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 BanachâMazur 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
Gluskin, Efim D. (1981). "The diameter of the Minkowski compactum is roughly equal to n (in Russian)". Funktsional. Anal. I Prilozhen. 15 (1): 72â73.
doi:
10.1007/BF01082381.
MR0609798.
S2CID123649549.
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.
ISBN0-582-01374-7.
MR0993774.