A Generalization of the Isosceles Constant in Banach Spaces

Abstract

N. Gastinel and J.L. Joly defined the rectangular constant \(\mu \) in Banach spaces using the notion of orthogonality according to Birkhoff and its generalization \(\mu _p\), with \(p\ge 1\). Recently, M. Baronti, E. Casini, and P.L. Papini defined a new constant, the isosceles constant H, in Banach spaces in a very similar way to the rectangular constant, but in this case using the isosceles orthogonality defined by James. In this paper, first of all, we generalize such constant, by defining a new constant \(H_p\) that generalizes the isosceles constant H as well \(\mu _p\) generalizes \(\mu \). After that, we explain its properties, and we give a characterization of Hilbert spaces in terms of it. Moreover a partial characterization of uniformly non-square spaces is given. We conclude by a conjecture about the characterization of uniformly non-square spaces.