Extreme Set in Compact Convex Set contains Extreme Point

Theorem
Let $X$ be a vector space over $\R$.

Let $K$ be a non-empty compact convex subset of $X$.

Let $E$ be an extreme set of $K$.

Then $E$ contains an extreme point of $K$.

Proof
From Closed Convex Set is Extreme Set of itself:


 * $K$ is an extreme set of $K$.

Let $P$ be the set of extreme sets in $K$.

Define a relation $\preceq$ on $P$ by $A \preceq B$ $B \subseteq A$.

Lemma 1
We show that every non-empty chain in $\struct {P, \preceq}$ has an upper bound.

We will then invoke Zorn's Lemma.

Lemma 2
From Zorn's Lemma, we have that $P$ has a maximal element.

Let $E$ be a maximal element of $P$.

Then $E$ is an extreme set of $K$ such that:


 * whenever $E_\ast \in P$ has $E \preceq E_\ast$ we have $E = E_\ast$.

That is:


 * whenever an extreme set $E_\ast$ of $K$ has $E_\ast \subseteq E$, we have $E = E_\ast$.

So $E$ does not properly contain any other extreme set.

With view to apply Point in Convex Set is Extreme Point iff Singleton is Extreme Set, we show that $E$ is a singleton.

Suppose that $a, b \in E$ with $a \ne b$.

Then from Normed Dual Space Separates Points, there exists an $f \in X^\ast$ with:


 * $\map f a \ne \map f b$

where $X^\ast$ is the normed dual of $X$.

So either:


 * $\map f a > \map f b$

or:


 * $\map f a < \map f b$

suppose that $\map f a < \map f b$.

Otherwise, swap $a$ and $b$.

Define:


 * $\ds E^f = \set {x \in E : \map f x = \max_{y \in E} \map f y}$

From Preimage of Maximum of Bounded Linear Functional in Extreme Set in Convex Compact Set is Extreme Set, we have that:


 * $E^f$ is an extreme set in $K$.

However, from the definition of maximum, we have:


 * $\ds \map f b \le \max_{y \in E} \map f y$

So we have:


 * $\ds \map f a < \max_{y \in E} \map f y$

So:


 * $a \not \in E^f$

Since we have:


 * $E^f \subseteq E$

this gives that:


 * $E^f$ is a proper subset of $E$.

So we have a contradiction and $E$ is a singleton, say:


 * $E = \set x$

for $x \in X$.

From Point in Convex Set is Extreme Point iff Singleton is Extreme Set, we have:


 * $x$ is an extreme point in $K$.