A new geometrical constant of Banach spaces and the uniform normal structure

We introduce and study a new geometrical constant \(\gamma_{X,\psi\) of a Banach space $X$, by using the notion of $\psi$-direct sum given in [Y. Takahashi, M. Kato and K.-S. Saito, J. Inequal. Appl. 7 (2002), 179-186]. At first, we characterize uniform non-squareness in terms of ${\gamma }_{X,\psi }$. Moreover, we consider Banach spaces having uniform normal structure.