A generalization of the isosceles constant in Banach spaces

N. Gastinel and J.L. Joly defined the rectangular constant in Banach spaces using the notion of orthogonality according to Birkho and its generalization p, with p 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 contant Hp that generalizes the isosceles constat H as well p generalizes . 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.