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REVISITING THE RECTANGULAR CONSTANT IN BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 April 2021

M. BARONTI Open the ORCID record for M. BARONTI [Opens in a new window] ,
E. CASINI Open the ORCID record for E. CASINI [Opens in a new window]  and
P. L. PAPINI Open the ORCID record for P. L. PAPINI [Opens in a new window]
M. BARONTI*
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16100 Genova, Italy
E. CASINI
Affiliation:
Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy e-mail: emanuele.casini@uninsubria.it
P. L. PAPINI
Affiliation:
Via Martucci 19, 40136 Bologna, Italy e-mail: pierluigi.papini@unibo.it
*
e-mail: baronti@dima.unige.it
Rights & Permissions [Opens in a new window]

Abstract

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Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

Keywords

hexagonal normorthogonal vectorsrectangular constantuniform nonsquareness

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B45: Banach sequence spaces 46B99: None of the above, but in this section 46C15: Characterizations of Hilbert spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 124 - 133
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021

References

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