Weak nearly uniform smoothness of the $\psi$-direct sums $(X_1\oplus \cdots \oplus X_N{)}_{\psi }$

We shall characterize the weak nearly uniform smoothness of the $\psi$-direct sum $(X_1\oplus \cdots \oplus X_N{)}_{\psi }$ of $N$ Banach spaces $X_1,\dots ,X_N$, where $\psi$ is a convex function satisfying certain conditions on the convex set ${\mathrm{\Delta }}_N=\{(s_1,\dots ,s_{N-1})\in {\mathbb{R}}_{+}^{N-1}:\sum _{i=1}^{N-1}{s}_i\le 1$. To do this a class of convex functions which yield ${\ell }_1$-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular an example will be presented which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square.