On the Exact $\mathcal{K}$-Monotonicity of Banach Couples

We present necessary and sufficient conditions for a Banach couple formed by the space $L_{\infty}$ and an arbitrary Lorentz space $\Lambda(\varphi)$ to be exact $\mathcal{K}$-monotone. The proof relies on the description of the set of extreme points of a $\mathcal{K}$-orbit for appropriate finite-dimensional couples. As a consequence of this description, we obtain a generalization of a well-known Markus theorem.