uniform convergences of convex functions

Question

Let $f_n(\cdot)$ be a sequence of continuous and convex function on $\mathbb{R}^d$, and be supported on a full dimensional compact convex set $D$. If $f_n(\cdot)$ converges point-wise to $f$ in the interior of $D$, and $f$ is a finite continuous and convex function on $D$, then can we conclude that $f_n$ converges uniformly to $f$ on the entire $D$?