Is closed convex set with finite number of extreme points convex polyhedron

Question

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question:

Is closed convex set with finite number of extreme points convex polyhedron

If not, can you please give me some hint to find the counter-example of that statement? Thanks a lot. I really appreciate

Answer 1

You additionally need that your set is bounded, otherwise it may have too few extreme points:

• any convex, closed cone has only one extreme point
• For any closed, convex $C$, consider $C \times \mathbb R$: no extreme points.

If your set is bounded, it is (assuming that the ambient space is finite-dimensional) compact. By Krein-Milman, the set is the convex hull of its extreme points, hence, a polyhedron.