\begin{definition}[Definition:Extreme Set]
Let $X$ be a vector space over $\R$. 
Let $K$ be a convex subset of $X$. 
Let $M \subseteq K$ be a non-empty closed set.
We say that $M$ is an '''extreme set''' in $K$ {{iff}}:
:whenever $x, y \in K$ and $t \in \openint 0 1$ have $t x + \paren {1 - t} y \in M$ we have $x, y \in M$.
\end{definition}