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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

1 Answers 1

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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.