Must every convex compact set have extreme points?

Question

I am reading a paper and there is such description as title. Why?

I have an example: $(0,1)$. This is a convex set but not closed, so I cannot find an extreme point. However if convex and compact,

I read some related problems:

1. Exposed point of a compact convex set
   There must be at least one exposed point. But an extreme point is not necessary equal to an exposed point.
2. Convex hull of extreme points
   A convex hull $P$ of finite points. Then $P$ is the convex hull of its extreme points.

It seems there is a requirement "finite points" to guarantee the topic?

Answer 1

In a finite dimensional space (which is the case here, according to a comment), the existence of an extreme point of a compact convex set $K$ is easy to prove. Take any point $x\in K$ at which the norm $\|x\|$ is maximized. If there is $y\ne 0$ such that $x\pm y \in K$, then $$ 2\|x\|^2 \ge \|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2+2\|y\|^2> 2\|x\|^2 $$ a contradiction.