Geodesic
On the volume method in the study of Auerbach bases of finite-dimensional normed spaces
Colloquium Mathematicum, Tome 69 (1996) no. 2, pp. 267-270.

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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].
DOI : 10.4064/cm-69-2-267-270

Anatolij Plichko 1

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Anatolij Plichko. On the volume method in the study of Auerbach bases of finite-dimensional normed spaces. Colloquium Mathematicum, Tome 69 (1996) no. 2, pp. 267-270. doi : 10.4064/cm-69-2-267-270. http://geodesic.mathdoc.fr/articles/10.4064/cm-69-2-267-270/

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