# 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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topPlichko, 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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- [2] H. Auerbach, Über eine Eigenschaft der Eilinien mit Mittelpunkt, Ann. Soc. Polon. Math. 9 (1930), 204.
- [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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- [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] H. Lenz, Eine Kennzeichnung des Ellipsoids, Arch. Math. (Basel) 8 (1957), 209-211. Zbl0078.35802
- [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] A. Pełczyński and S. J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in ${\mathbb{R}}^{n}$, Math. Proc. Cambridge Philos. Soc. 109 (1991), 125-148. Zbl0718.52007
- [11] A. Pietsch, Operator Ideals, Deutsch. Verlag Wiss., Berlin, 1978.
- [12] S. Rolewicz, Metric Linear Spaces, Reidel and PWN, Dordrecht-Warszawa, 1985.
- [13] A. F. Ruston, Auerbach's theorem and tensor products of Banach spaces, Proc. Cambridge Philos. Soc. 58 (1962), 476-480. Zbl0108.10902
- [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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