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We said that the Krein Milman theorem is valid in a LCS $X$ for non-empty convex and compact sets $K$ and it tells us that:

1.) ex(K) $\neq \emptyset$

2.) $K = \overline{co}(ex(K))$.

3.) If $K = \overline{co}(B)$ then $ex(K) \subset \overline{B}.$

The second one tells us that the set of extreme points is large enough to recover the set again, i.e. this implies one as the set of extreme points must not be trivial.

However, I don't have a good feeling for what the third statement is actually telling me ( I mean, I can read what it says, but I don't know how to get some intuition for this statement).

Maybe this is a somewhat strange question, but is there anybody who could try to make this statement plausible?

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  • $\begingroup$ No point that can be written as a convex combination of points of $B$ can be an extreme point of the convex hull generated by $B$. $\endgroup$ – DisintegratingByParts Mar 5 '15 at 16:49
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(3) in some sense says that you need the extreme points to "recover" the whole set $K$, at least approximately. For instance, if $B \subseteq \mathrm{ex}\, K$ is a closed, proper subset, then $\overline{\rm co}\, B \ne K$ (by (3)). So you need all the extreme points in $B$, to be more exact, you need $B$ to approximate all extreme points (that is $\overline B$ containing them).

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  • $\begingroup$ Dankeschön... :-) $\endgroup$ – user159356 Mar 5 '15 at 15:02
  • $\begingroup$ Bitte, gern. :-) $\endgroup$ – martini Mar 5 '15 at 15:03

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