Extreme point

Remarks

In $\R^2$, let $K = [0, 1]^2$. The corner $(0, 0)$ is extreme: if $(0, 0) = t y + (1 - t) z$ with $y, z \in K$ and $0 < t < 1$, then each coordinate is a combination of nonnegative numbers with positive weights equal to $0$, so $y = z = (0, 0)$. The point $(1/2, 0)$ is not extreme, because it is the midpoint of $(0, 0)$ and $(1, 0)$, which lie in $K$ and differ from it. Equivalently, $x$ is extreme when $x$ is not the midpoint of two distinct points of $K$. An open ball in a nonzero normed space has no extreme points. The closed unit ball of the sequence space $c_0$ also has no extreme points.