Definition:Extreme Set

Definition

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 $K$ if and only if:

whenever $x, y \in K$ and $t \in \openint 0 1$ have $t x + \paren {1 - t} y \in M$ we have $x, y \in M$.

Also known as

An extreme set might also be referred to as a face of $K$.

Also see

• Results about extreme sets can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $21.6$: The Krein-Milman Theorem