# Extreme point

A point in the closure of a set, say $LaTeX: S$, that is not the midpoint of any open line segment with end points in closure of $LaTeX: S$. Equivalently, $LaTeX: x$ is an extreme point of a closed set, $LaTeX: S$, if there do not exist $LaTeX: y, z \in S \backslash {x}$ for which $LaTeX: \textstyle x = (y + z)/2$. When $LaTeX: S$ is a polyhedron of the standard form, $LaTeX: S=\{x: Ax=b, \; x \ge 0\}$, with $LaTeX: A$ of full row rank, we have one of the fundamental theorems of linear programming that underlies the simplex method:
$LaTeX: x$ is an extreme point of the feasible region if, and only if, $LaTeX: x$ is a basic feasible solution.