Local geometric properties in quasi-normed Orlicz spaces

Several local geometric properties of Orlicz space $L_\phi$ are presented for an increasing Orlicz function $\phi$ which is not necessarily convex, and thus $L_\phi$ does not need to be a Banach space. In addition to monotonicity of $\phi$ it is supposed that $\phi(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $\alpha_\phi>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality theory cannot be applied in this case. In this article we present complete criteria, in terms of growth conditions of $\phi$, for $L_\phi$ to have type $0<p\le2$, cotype $q\ge 2$, to be (order) $p$-convex or $q$-concave, to have an upper $p$-estimate or a lower $q$-estimate, for $0<p,q<\infty$. We provide detailed proofs of most results, avoiding appealing to general not necessary theorems.