Let $\rho\ge0$ be a uniform convex function on $R^n$, that is, $\rho(x)-\frac{c}2|x|^2$ is convex function for some $c>0$. Further assume that $\rho(0)=0$ and $\rho$ satisfies the $\Delta_2$ condition, that is, $\rho(2x)\le C\rho(x) \forall x\in R^n$, where $C>1$ is a constant.
For a bounded domain $U\subset R^m$, define the Orlicz space $L^\rho(U,R^n)$ as the collection of all $u=(u_1,\cdots,u_n)$ such that $\int_U \rho(u)\,dx<\infty$. The Luxemburg norm of $u$ is $$\|u\|_{L^\rho(U,R^n)}=\inf\left\{\lambda>0,\int_U\rho(\frac1\lambda u)\le1\right\}.$$
It is known that $L^\rho(U,R^n)$ is Banach space. When $n=1$, under above assumption on $\rho$, I know that $L^\rho(U,R)$ is a uniform convex Banach space. When $n\ge 2$, I do not know if $L^\rho(U,R^n)$ is also a uniform convex Banach space? Does anyone know this and some references on this?
Thanks.