properties of extreme subsets of a closed convex set

Let $K$ be a closed convex subset (http://planetmath.org/ConvexSet) of a normed vector space

1. If $\{A_i:i\in I\}$ is a family of extreme subsets of $K$, such as ${\bigcap }_{i\in I}{A}_i\ne \mathrm{\varnothing }$then ${\bigcap }_{i\in I}{A}_i$ is extreme subset of $K$

2. $A\subset B\subset K$ such as $A,B$ are extreme subsets of $B$ and $K$ respectively. Then $A$ is an extreme subset of $K$.