A question on convex

Question

I am thinking about this without solutions. I would like you to give hints. Let $Q$ be a polyhedron with $Q=convex.hull (X)$ for some $X \subset R^n$. Let $E$ be a face of $Q$. Prove that $E \cap X \ne \empty$ and that $E=convex.hull (E \cap X)$. Thank you.

Answer 1

It's general fact that a convex compact closed set in a Hausdorff locally convex topological vector space is generated by its extreme points, this is Krein-Milman theorem.

In particular, your X has to contain all the extreme points of Q, this is by the very definition of extreme points. Now the extreme points for a face( considered as a convex set) is the same as an extreme point for X that lies in the face. Apply Krein-Milman theorem again to the face gives you the desired result.

The wikipedia link for K-M theorem is too short, you may consult chapter section 1.? in

Fundamentals of the theory of operator algebra, Volume 1,by Kadison and Ringrose.