Suppose we are working a real vector space $X$ with convex hull operator $\mathsf{con}$. Let $C$ be a convex set and $E$ an extreme subset of $C$. It is easy to check that if $e \in E$ and $c \in C \smallsetminus E$, then for all $x \in \mathsf{con}(\{ c, e \} ) \smallsetminus \{ c, e \} $ we have $x \in C \smallsetminus E$. Informally, if you look at $E$ from inside $C \smallsetminus E$ the set $E$ appears to be one element thick.
Also notice that if $e_{1}, e_{2} \in E$ are distinct points and there exists an $x \in X \smallsetminus \{ e_{1}, e_{2} \}$ such that $e_{2} \in \mathsf{con}(\{ e_{1}, x \} )$ then $x \notin C$. Informally $E$ is as wide as possible.
Is there commonly used terminology for these properties?
