Point in Convex Set is Extreme Point iff Singleton is Extreme Set

Theorem

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

Let $K$ be a convex subset of $X$.

Let $a \in K$.

Then $a$ is an extreme point of $K$ if and only if:

$\set a$ is an extreme set in $K$.

Proof

We have:

$a$ is an extreme point of $K$.

if and only if:

whenever $a = t x + \paren {1 - t} y$ for some $x, y \in K$ and $t \in \openint 0 1$, we have $x = y = a$.

We can rewrite this:

whenever $t x + \paren {1 - t} y \in \set a$ for some $x, y \in K$ and $t \in \openint 0 1$, we have $x, y \in \set a$.

This is precisely the criteria for $\set a$ being an extreme set in $K$.

$\blacksquare$

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $21.6$: The Krein-Milman Theorem