Consider a continuous function $f : D \rightarrow \mathbb{R}^m$, where $D \subseteq \mathbb{R}^n$ is a compact convex set. I am in search of a result that helps me say something about the extreme points of the following set:

$$C= \{ z\ | \ \exists x \in D, z = f(x)\}. $$

This is the image of $f$ or the projection of the graph of $f$ on the range-space.

If $f$ were scalar valued ($m=1$), $C$ is an interval and its extreme points are the maximum and minimum value of $f$. However I would like to know: what is the generalization of this for vector-valued $f$? Is there a characterization, if not for general $f$, then at least for some special classes of $f$? The most simple case seems to be when $f$ is affine. So I would be interested knowing if anything more general is known for nonlinear $f$. I would appreciate anyone could help out with references or pointers.