Let $K$ a non-empty closed convex subset of a normed vector space. A set $A\subseteq K$ is called an extreme subset of $K$ if $A$ is closed, convex and satisfies the condition $\colon$ for any $x,y \in K$ and $tx+(1-t)y \in A, t\in (0,1)$ then $x, y \in A$ .

For example let $K=[0,1]\times[0,1]$ then $K$ , sides of $K$ , included the endpoints, and $\{(1,1),(0,1),(1,0),(0,0)\}$ are extreme subsets of $K$ .