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ArticleOriginal scientific text

Title

On the volume method in the study of Auerbach bases of finite-dimensional normed spaces

Authors 1

Affiliations

  1. Institute of Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, Naukova 3b, 290053 L'viv, Ukraine

Abstract

In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach bases and (iii) the finite-dimensional normed space X is an inner product space provided any two Auerbach bases are isometrically equivalent. Property (i) generalizes a result of Lenz [8], and (iii) answers a question of R. J. Knowles and T. A. Cook [7].

Bibliography

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Colloquium Mathematicum
1996/69/2
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Pages:
267-270
Main language of publication
English
Received
1994-04-15
Accepted
1995-02-15
Published
1996
Exact and natural sciences