I am given this definition of face from Convexity: An analytic viewpoint:
Definition: A face of a convex set $P$ is a set $F\subsetneq \overline{P}$ such that for every $x,y\in\overline{P}$ and for every $t\in(0,1)$, $tx+(1-t)y\in F \Rightarrow x,y\in F$.
But, I'm a little upset by this definition. We should hope that $F$ would be convex in every case. But, there's an easy counterexample. Take $F$ to be the extreme points of a triangle. Note that this is a valid face, as none of the points in $F$ are ever written as interior points. So the face property of this set is true by vacuity. Am I missing something here? Can we fix this by simply augmenting the definition to insist that $F$ is convex?