Let $(X,P)$ be a locally convex space, $K$ a compact, convex subset of $X$. A face $F$ of K is a nonempty, compact, convex subset of $K$ s.t. $$\forall y,z\in K \,\forall t\in(0,1) \left[ (1-t)y + tz \in F \rightarrow y,z\in F\right].$$
What is the geometrical idea behind a face and why is this construction so important. How should I think of it?
My idea: In some sense it is the opposite of a convex set because: Assume the convex combination is it the face, then also the points building up the convex combination are in the set. But I am not sure about that.
A face is just one of the 'outside boundaries' of a convex set, or the whole convex set itself.
You can see this from the definition as follows:
If there is a single point $p\in F$that is not on the boundary, then we can take a point $k\in K$, make a line through $p$, and then all points on this line on the opposite side of $p$ will also be in $F$, by the definition.
So the only faces of $K$ that are not all of $K$ are those where this construction fails. That is exactly when all of $F$ is on the 'boundary' of $K$.
Now if you have polytope, then the faces are just the corner point and the facets. If you have something like a sphere then the faces are just individual points on the boundary of the sphere.