# proof of properties of extreme subsets of a closed convex set

For the first claim, it is obvious that $\bigcap_{i\in I}A_{i}$ is closed convex subset (http://planetmath.org/ConvexSet) of $K$. Let $z\in A$ and $0, $x,y\in K$ such as $z=tx+(1-t)y$. Then $z\in A_{i}$, for all $i\in I$ so we have that $x,y\in A_{i}$ for all $i\in I$. Therefore $x,y\in\bigcap_{i\in I}A_{i}.$
For the second claim suppose $x,y\in K$, $t\in(0,1)$ and $z\in A$ such as $z=tx+(1-t)y$. From the hypothesis $A\subset B$ we have that $z\in B$ and since $B$ is an extreme subset of $K$, $x,y\in B$. Analogously from the hypothesis that $A$ is an extreme subset of $B$, we have that $x,y\in A$.

Title proof of properties of extreme subsets of a closed convex set ProofOfPropertiesOfExtremeSubsetsOfAClosedConvexSet 2013-03-22 15:25:12 2013-03-22 15:25:12 georgiosl (7242) georgiosl (7242) 4 georgiosl (7242) Proof msc 52A99