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 $\colon$ 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 $K$.

Title extreme subset of convex set ExtremeSubsetOfConvexSet 2013-03-22 15:24:43 2013-03-22 15:24:43 georgiosl (7242) georgiosl (7242) 7 georgiosl (7242) Definition msc 52A99 ConvexSet