Global approximation of convex functions by differentiable convex functions on Banach spaces

We show that if $X$ is a Banach space whose dual $X^{\ast }$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\epsilon >0$, and for every continuous and convex function $f:U \rightarrow $\mathbb{R}$$ (not necessarily bounded on bounded sets) there exists a convex function $g:X\to \mathbb{R}$ of class $C^1(U)$ such that $f-\epsilon \le g\le f$ on $U\mathrm{.}$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions.