extreme subset of convex set

Let $K$ a non-empty closed convex subset (http://planetmath.org/ConvexSet) 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 $:$ 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 $K$.