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

Colloquium Mathematicae (1996)

• Volume: 69, Issue: 2, page 267-270
• ISSN: 0010-1354

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## Abstract

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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].

## How to cite

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Plichko, Anatolij. "On the volume method in the study of Auerbach bases of finite-dimensional normed spaces." Colloquium Mathematicae 69.2 (1996): 267-270. <http://eudml.org/doc/210339>.

@article{Plichko1996,
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].},
author = {Plichko, Anatolij},
journal = {Colloquium Mathematicae},
keywords = {minimal volume of -dimensional parallelepipeds; maximal volume of -dimensional crosspolytopes; orthogonality; Auerbach bases},
language = {eng},
number = {2},
pages = {267-270},
title = {On the volume method in the study of Auerbach bases of finite-dimensional normed spaces},
url = {http://eudml.org/doc/210339},
volume = {69},
year = {1996},
}

TY - JOUR
AU - Plichko, Anatolij
TI - On the volume method in the study of Auerbach bases of finite-dimensional normed spaces
JO - Colloquium Mathematicae
PY - 1996
VL - 69
IS - 2
SP - 267
EP - 270
AB - 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].
LA - eng
KW - minimal volume of -dimensional parallelepipeds; maximal volume of -dimensional crosspolytopes; orthogonality; Auerbach bases
UR - http://eudml.org/doc/210339
ER -

## References

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1. [1] H. Auerbach, O polu krzywych wypukłych o średnicach sprzężonych (On the area of convex curves with conjugate diameters), Ph.D. thesis, L'viv University, 1930 (in Polish).
2. [2] H. Auerbach, Über eine Eigenschaft der Eilinien mit Mittelpunkt, Ann. Soc. Polon. Math. 9 (1930), 204.
3. [3] H. Auerbach, Sur les groupes linéaires bornés, I-III, Studia Math. 4 (1933), 113-127, 158-166; 5 (1934), 43-49. Zbl59.1093.05
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5. [5] M. M. Day, Polygons circumscribed about closed convex curves, Trans. Amer. Math. Soc. 62 (1947), 315-319. Zbl0034.25301
6. [6] R. C. James, Inner products in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559-566. Zbl0041.43701
7. [7] R. J. Knowles and T. A. Cook, Some results on Auerbach bases for finite-dimensional normed spaces, Bull. Soc. Roy. Sci. Liège 42 (1973), 518-522. Zbl0275.46011
8. [8] H. Lenz, Eine Kennzeichnung des Ellipsoids, Arch. Math. (Basel) 8 (1957), 209-211. Zbl0078.35802
9. [9] A. Yu. Levin and Yu. I. Petunin, Some questions connected with the notion of orthogonality in a Banach space, Uspekhi Mat. Nauk 18 (3) (1963), 167-170 (in Russian). Zbl0171.33302
10. [10] A. Pełczyński and S. J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in ${ℝ}^{n}$, Math. Proc. Cambridge Philos. Soc. 109 (1991), 125-148. Zbl0718.52007
11. [11] A. Pietsch, Operator Ideals, Deutsch. Verlag Wiss., Berlin, 1978.
12. [12] S. Rolewicz, Metric Linear Spaces, Reidel and PWN, Dordrecht-Warszawa, 1985.
13. [13] A. F. Ruston, Auerbach's theorem and tensor products of Banach spaces, Proc. Cambridge Philos. Soc. 58 (1962), 476-480. Zbl0108.10902
14. [14] A. E. Taylor, A geometric theorem and its application to biorthogonal systems, Bull. Amer. Math. Soc. 53 (1947), 614-616. Zbl0031.40502

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