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We shall characterize the weak nearly uniform smoothness of the $\psi$-direct sum $X{\oplus }_{\psi }Y$ of Banach spaces $X$ and $Y$. The Schur and WORTH properties will be also characterized. As a consequence we shall see in the ${\ell }_{\infty }$-sums of Banach spaces there are many examples of Banach spaces with the fixed point property which are not uniformly non-square.

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 $X_1,\cdots ,X_N$, where $\psi$ is a convex function satisfying certain conditions on the convex set ${\Delta }_N=\{(s_1,\cdots ,s_{N-1})\in {ℝ}_{+}^{N-1}:{\sum }_{i=1}^{N-1}{s}_i\le 1$. 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.

We shall characterize the uniform non-${\ell }_1^n$-ness of ${\ell }_{\infty }$-sums of Banach spaces ${(X_1\oplus \cdots \oplus X_m)}_{\infty }$. As applications, some results on super-reflexivity and the fixed point property for nonexpansive mappings will be presented.

We shall characterize the uniform non-${\ell }_1^n$-ness of the ${\ell }_1$-sum ${(X_1\oplus \cdots \oplus X_m)}_1$ of a finite number of Banach spaces $X_1,\cdots ,X_m$. Also we shall obtain that ${(X_1\oplus \cdots \oplus X_m)}_1$ is uniformly non-${\ell }_1^{m+1}$ if and only if all $X_1,\cdots ,X_m$ are uniformly non-square (note that ${(X_1\oplus \cdots \oplus X_m)}_1$ is not uniformly non-${\ell }_1^m$). Several related results will be presented.

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{NJ}\left(X\right)$, and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{NJ}\left(X\right)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{NJ}\left(X\right)$-constant, which implies that a Banach space with $C_{NJ}\left(X\right)$-constant less than 5/4 has the fixed point property.

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman's observation on Clarkson's inequalities and type. A similar investigation conceming the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several...