$l_{\infty }$ and interpolation between Banach lattices

We study the possibility of obtaining the $l_{\infty }$-norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in ${\mathbb {R}}^{n}$ such that the $l_{\infty }$-norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the $l_{\infty }$-norm by any method which guarantees interpolation of not only linear operators (= bilinear forms on ${\mathbb {R}}^{n}\times {\mathbb {R}}^{n})$ but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the $l_{\infty }$-norm, or both have to be weighted $l_{\infty }$-norms.