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

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Plichko A. M.

Colloquium Mathematicum

1996

tom 69

nr 2

267-270

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].

Ograniczanie wyników

1 Colloquium Mathematicum

1 Plichko A. M.

1 1996

Wyniki wyszukiwania

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