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?
Best Answer
Using Daniel Fisher comment as an answer.
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.