Revisiting the rectangular constant in banach spaces

Let X be a real Banach space. The rectangular constant and some generalisations of it, for, 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, correcting some statements appearing in the literature, and extend to some characterisations of uniformly nonsquare spaces, known only for. We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning and.