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REVISITING THE RECTANGULAR CONSTANT IN BANACH SPACES

Published online by Cambridge University Press:  26 April 2021

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: [email protected]
P. L. PAPINI
Affiliation:
Via Martucci 19, 40136 Bologna, Italy e-mail: [email protected]
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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 $\mu _p(l_p)$ .

Type
Research Article
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

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