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].
@article{bwmeta1.element.bwnjournal-article-cmv69i2p267bwm,
author = {Anatolij Plichko},
title = {On the volume method in the study of Auerbach bases of finite-dimensional normed spaces},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {267-270},
zbl = {0884.46012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv69i2p267bwm}
}
Plichko, Anatolij. On the volume method in the study of Auerbach bases of finite-dimensional normed spaces. Colloquium Mathematicae, Tome 70 (1996) pp. 267-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv69i2p267bwm/
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