Monotonicities of quasi-normed Calderón–Lozanovskiĭ spaces with applications to approximation problems

We consider the geometric structure of quasi-normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi-norm and the quasi-modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well-known normed case, we develop a number of new techniques and methods, among which the conditions ${\Delta }_{\epsilon }$ and ${\Delta }_{2-str}$ play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi-normed Orlicz spaces, which are new even in this particular context. We consider both the function and the sequence case as well as we admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi-Banach lattices.