The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points.

Is there a "simple argument" that the extreme points is not empty that avoids going through the full Krein-Milman theorem?

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    $\begingroup$ In the proof of the Krein-Milman theorem, one shows that a minimal extreme set of a compact convex set is a singleton. Whether one views that as a "simple argument" depends. If you're used to using the nonemptiness of a nested intersection of nonempty compact sets, and to Hahn-Banach arguments, it is simple. Otherwise, not so much. $\endgroup$ – Daniel Fischer Aug 5 '15 at 8:01
  • $\begingroup$ Similar question on MO: mathoverflow.net/questions/15654/… $\endgroup$ – Martin Sleziak Aug 5 '15 at 8:16

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