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

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Abstract

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 1n n is the unit ball of ℓ 1n , 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.

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  1. St.Petersburg State University, St.Petersburg, Russia

    F. L. Bakharev

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  1. F. L. Bakharev
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 33–42.

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Bakharev, F.L. Estimates of maximal distances between spaces whose norms are invariant under a group of operators. J Math Sci 141, 1526–1530 (2007). https://doi.org/10.1007/s10958-007-0058-9

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  • DOI: https://doi.org/10.1007/s10958-007-0058-9

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