For a Banach lattice X, its lattice Schäffer constant is defined by \(\begin{aligned} \lambda ^+(X)=\inf \{\max \{\Vert x+y\Vert ,\Vert x-y\Vert \}\,:\,\Vert x\Vert =\Vert y\Vert =1,x,y\ge \textbf{0}\}. \end{aligned}\) In this paper, we investigate this constant, as well as the companion parameter \(\begin{aligned} \beta (X)=\inf \{\Vert x\vee y\Vert \,:\, \Vert x\Vert =\Vert y\Vert =1, x,y\ge \textbf{0} \hbox { and } x\wedge y=\textbf{0}\}. \end{aligned}\) 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, X is q-concave for some \(q\in (1,\infty )\) ; (ii) \(\lambda ^+(X)=1\) if and only if X contains lattice-almost isometric copies of \(\ell _\infty ^2\) ; (iii) that \(\lambda ^+(X)=2\) if and only if X is an abstract L-space; and (iv) if X is a Banach lattice with p-convexity constant 1, then \(\lambda ^+(X)=2^{1/p}\) if and only if X is an abstract \(L_p\) -space. (2) We establish inequalities relating \(\lambda ^+(X)\) to the characteristics of monotonicity, \(\varepsilon _{0,m}(X)\) and \(\tilde{\varepsilon }_{0,m}(X)\) . Along the way, we compute \(\lambda ^+(X)\) and \(\beta (X)\) for various Banach lattices X.
