extreme points of the image of a nonlinear vector-valued function

Question

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.

Answer 1

Let $K$ be the convex hull of $C$. The extreme points of $K$ are in $C$. Since they are extreme points of a compact convex set, they have a supporting hyperplane, i.e. if $w = f(p)$ is such a point there is some $v \ne 0\in {\mathbb R}^m$ such that $v \cdot w = \max_{x \in D} v \cdot f(x)$. On the other hand, for any $v \ne 0$, there is at least one extreme point $w = f(p)$ of $K$ such that $v \cdot w = \max_{x \in D} v \cdot f(x)$.