Journal of Mathematical Sciences

, Volume 141, Issue 5, pp 1526–1530 | Cite as

Estimates of maximal distances between spaces whose norms are invariant under a group of operators

  • F. L. Bakharev


We consider the class Aг of n-dimensional normed spaces with unit balls of the form: \(B_U = conv\mathop \cup \limits_{\gamma \in \Gamma } \gamma (B_n^1 \cup U(B_n^1 ))\), where B n 1 n is the unit ball of ℓ n 1 , Γ is a finite group of orthogonal operators acting in ℝn, and U is a “random” orthogonal transformation. It is proved that this class contains spaces with a large Banach-Mazur distance between them. If the cardinality of Γ is of order nc, it is shown that, in the power scale, the diameter of Aг in the modified Banach-Mazur distance behaves as the classical diameter and is of order n. Bibliography: 8 titles.


Random Vector Unit Ball Classical Diameter Orthogonal Transformation Polar Decomposition 
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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • F. L. Bakharev
    • 1
  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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