Extreme set

Remarks

A point $x \in A$ is an extreme point of $A$ exactly when $\set{x}$ is an extreme set of $A$. The sets $\emptyset$ and $A$ are always extreme sets of $A$, and an extreme set of an extreme set of $A$ is an extreme set of $A$.

Example: for $A = [0, 1]^2$, the bottom edge $B = [0, 1] \times \set{0}$ is an extreme set. Let $x \in B$ lie in $(x_1, x_2)$ with $x_1, x_2 \in A$. The second coordinates of $x_1, x_2$ are $\ge 0$, and an average of them with positive weights is $0$. So both are $0$, and $x_1, x_2 \in B$.

Non-example: the diagonal $D = \set{(s, s) : 0 \le s \le 1}$ of the same square. The point $(1/2, 1/2) \in D$ lies in the open segment between $(1, 0)$ and $(0, 1)$, which are in $A$ but not in $D$.