The lattice Schäffer constant

For a Banach lattice $X$, its lattice Schäffer constant is defined by: \begin{gather*} λ^+(X)=\inf{\max{|x+y|,|x-y|},\colon,|x|=|y|=1,x,y\geq{\bf0}}. \end{gather*} In this paper, we investigate this constant, as well as the companion parameter \begin{gather*} β(X)=\inf{|x\vee y|,\colon,\mbox{$\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }y\mathrm{\parallel }=1$, $x,y\ge \mathbf{0}$ and $x\wedge y=\mathbf{0}$}}. \end{gather*} Our main results fall into two groups. (1) We link the behavior of the parameters ${\lambda }^{+}$ and $\beta$ to the global properties of the lattice $X$. For instance, we prove that (i) if ${\lambda }^{+}(X)>1$, then the Banach lattice $X$ is a KB-space, and moreover, it satisfies a lower $q$-estimate for some $q\in (1,\mathrm{\infty })$; (ii) ${\lambda }^{+}(X)=1$ if and only if $X$ contains lattice-almost isometric copies of ${\mathrm{\ell }}_{\mathrm{\infty }}^2$; and (iii) that ${\lambda }^{+}(X)=2$ if and only if $X$ is an abstract $L$-space. (2) We establish inequalities relating ${\lambda }^{+}(X)$ to the characteristics of monotonicity, ${\epsilon }_{0,m}(X)$ and ${\tilde{\epsilon }}_{0,m}(X)$. Along the way, we compute ${\lambda }^{+}(X)$ and $\beta (X)$ for various Banach lattices $X$.