A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point.

Question

A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point.

How do you prove this result? Can you give me sketch?

Thanks

Answer 1

Just consider the map $x\rightarrow ||x||^2$, which has a maximum (Why?), say $x'$. Now suppose on the contrary that $x'$ is not an extremum point so that there are two points, say $y,z$, that forms $x'$ (i.e., $x'=1/2\times y+z\times 1/2$). Now take its norm and arrive at a contradiction.