On the Class of Banach Spaces with James Constant $${\sqrt{2}}$$ 2 : Part II

Abstract

The class of two-dimensional normed spaces with James constant \({\sqrt{2}}\) is studied. It is shown that, if the norm is \({\pi/2}\)-rotation invariant, then its James constant is \({\sqrt{2}}\) if and only if the norm is \({\pi/4}\)-rotation invariant. We also present a characterization of \({\pi/4}\)-rotation invariant norms using some properties of certain convex functions on the unit interval, which allow us to easily construct norms the James constant of which are \({\sqrt{2}}\). Moreover, two important examples are given, which show that neither absoluteness, symmetry nor \({\pi/2}\)-rotation invariance can be a characteristic property of the norms with James constant \({\sqrt{2}}\).