I am reading Theory of Orlicz spaces by M.Rao.
My question is that : why do we need a convex function to define Orlicz spaces ? Can't we take any other type of function?
Definition of Orlicz space:
L$^\phi$ ={ $f\colon A\to R$ such that $ \int_A \phi(|f|) \,d\nu < \infty$ }
here $\phi$ is a Young function defined as
$\phi\colon R\to R^+$ such that following holds
(1) $\phi(x)$ is a convex function.
(2) $\lim_{x \to \infty}\phi(x)=\infty$
(3)$\phi(-x)=\phi(x)$
(4)$\phi(0)=0$
In principle you can always define a space using any choice of $\phi$, it's just a question of what the properties of that space will have. For some $\phi$ it will be a space with many nice properties, for example when $\phi(x) = x^2$ it induces a Hilbert space which makes life very easy. For other $\phi$ you will have a space with essentially no standard structures which will make working in the space very difficult.
The main reason that you want $\phi$ to have the four properties listed is to ensure that $$||f||_\phi \triangleq \inf \left \{ t > 0:\int_A \phi\left (\frac{|f|}{t} \right ) d\nu \leq 1\right \}$$ defines a norm on the vector space $L^\phi$. In turn, this makes $L^\phi$ a normed vector space (and if you can get completeness too, then it's a Banach Space). Having a norm makes life much easier in many settings.
This is closely related to the idea of the Minkowski Functional and the norm that it can provide, in the sense that the conditions you provide ensure that the set $K$ that defines the Minkowski functional will be guaranteed to be convex.