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Can someone give me a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point?

I'd found this one but couln't understand it: Convex compact set must have extreme points

Thanks!

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  • $\begingroup$ @CaveJohnson So how could you define "extreme point" in a non-convex set? $\endgroup$ – Filburt Aug 4 '17 at 4:36
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Consider a sphere $S $ of radius $r$ where $K$ is inscribed in $S$.

Here inscribed implies that $K$ is in $r$-ball $B$ with a boundary $S$ and there is a point $x\in K$ s.t. $x$ is in $S$.

Then $x$ is an extreme point of $K$. If not, there is $p,\ q\neq p \in K$ s.t. $x$ is an interior point of a segment $[pq]$.

Note that $B$ can not contain a segment $[pq]$ so that $[pq]$ is not in $K$. It is a contradiction.

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    $\begingroup$ How do you know that $K$ can be inscribed in a sphere? $\endgroup$ – Robert Israel Aug 4 '17 at 6:34
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Take a point of $K$ that maximizes Euclidean distance from the origin. This exists since a continuous function on a compact set attains its supremum, and it is easily seen to be an extreme point (you may want to use the Parallelogram Law for this).

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