On the von Neumann-Jordan constant for Banach spaces

Let $C_{\mathrm {NJ}} (E)$ be the von Neumann-Jordan constant for a Banach space $E$. It is known that $1\le C_{\mathrm {NJ}}(E)\le 2$ for any Banach space $E$; and $E$ is a Hilbert space if and only if $C_{\mathrm {NJ}} (E)=1$. We show that: (i) If $E$ is uniformly convex, $C_{\mathrm {NJ}} (E)$ is less than two; and conversely the condition $C_{\mathrm {NJ}} (E)<2$ implies that $E$ admits an equivalent uniformly convex norm. Hence, denoting by $\widetilde C_{\mathrm {NJ}} (E)$ the infimum of all von Neumann-Jordan constants for equivalent norms of $E$, $E$ is super-reflexive if and only if $\widetilde C_{\mathrm {NJ}} (E)<2$. (ii) If $\widetilde C_{\mathrm {NJ}} (E)=2^{2/p-1}$, $1<p\le 2$ (the same value as that of $L_p$-space), $E$ is of Rademacher type $r$ and cotype $r'$ for any $r$ with $1\le r<p$, where $1/r+1/r'=1$; the converse holds if $E$ is a Banach lattice and $l_p$ is finitely representable in $E$ or $E'$.