I am looking for a simple proof of the following theorem — wasn't able to come up with one myself. Should be a use of the Bishop–Phelps theorem, in some way:

Let $X$ be a Banach space, $D \subset X^*$ a $w^*$-closed, convex set with nonempty interior. Then the $w^*$-support points of $D$ are dense in $\partial D$.

Note: The set of $w^*$-support points of $D$ is defined as: $\{f \in D : \text{$\exists x_0 \in X$, $x_0\neq0$, s.t $f(x_0)=\sup_{g\in D} g(x_0) $}\}$. Obviously, this set is contained in $\partial D$.