In mathematics, an extreme point of a convex set ${\displaystyle S}$ in a real or complex vector space is a point in ${\displaystyle S}$ that does not lie in any open line segment joining two points of ${\displaystyle S.}$ In linear programming problems, an extreme point is also called vertex or corner point of ${\displaystyle S.}$^[1]

Definition

Throughout, it is assumed that ${\displaystyle X}$ is a real or complex vector space.

For any ${\displaystyle p,x,y\in X,}$ say that ${\displaystyle p}$ lies between^[2] ${\displaystyle x}$ and ${\displaystyle y}$ if ${\displaystyle x\neq y}$ and there exists a ${\displaystyle 0<t<1}$ such that ${\displaystyle p=tx+(1-t)y.}$

If ${\displaystyle K}$ is a subset of ${\displaystyle X}$ and ${\displaystyle p\in K,}$ then ${\displaystyle p}$ is called an extreme point^[2] of ${\displaystyle K}$ if it does not lie between any two distinct points of ${\displaystyle K.}$ That is, if there does not exist ${\displaystyle x,y\in K}$ and ${\displaystyle 0<t<1}$ such that ${\displaystyle x\neq y}$ and ${\displaystyle p=tx+(1-t)y.}$ The set of all extreme points of ${\displaystyle K}$ is denoted by ${\displaystyle \operatorname {extreme} (K).}$

Generalizations

If ${\displaystyle S}$ is a subset of a vector space then a linear sub-variety (that is, an affine subspace) ${\displaystyle A}$ of the vector space is called a support variety if ${\displaystyle A}$ meets ${\displaystyle S}$ (that is, ${\displaystyle A\cap S}$ is not empty) and every open segment ${\displaystyle I\subseteq S}$ whose interior meets ${\displaystyle A}$ is necessarily a subset of ${\displaystyle A.}$^[3] A 0-dimensional support variety is called an extreme point of ${\displaystyle S.}$^[3]

Characterizations

The midpoint^[2] of two elements ${\displaystyle x}$ and ${\displaystyle y}$ in a vector space is the vector ${\displaystyle {\tfrac {1}{2}}(x+y).}$

For any elements ${\displaystyle x}$ and ${\displaystyle y}$ in a vector space, the set ${\displaystyle =\{tx+(1-t)y:0\leq t\leq 1\}}$ is called the closed line segment or closed interval between ${\displaystyle x}$ and ${\displaystyle y.}$ The open line segment or open interval between ${\displaystyle x}$ and ${\displaystyle y}$ is ${\displaystyle (x,x)=\varnothing }$ when ${\displaystyle x=y}$ while it is ${\displaystyle (x,y)=\{tx+(1-t)y:0<t<1\}}$ when ${\displaystyle x\neq y.}$^[2] The points ${\displaystyle x}$ and ${\displaystyle y}$ are called the endpoints of these interval. An interval is said to be a non−degenerate interval or a proper interval if its endpoints are distinct. The midpoint of an interval is the midpoint of its endpoints.

The closed interval ${\displaystyle }$ is equal to the convex hull of ${\displaystyle (x,y)}$ if (and only if) ${\displaystyle x\neq y.}$ So if ${\displaystyle K}$ is convex and ${\displaystyle x,y\in K,}$ then ${\displaystyle \subseteq K.}$

If ${\displaystyle K}$ is a nonempty subset of ${\displaystyle X}$ and ${\displaystyle F}$ is a nonempty subset of ${\displaystyle K,}$ then ${\displaystyle F}$ is called a face^[2] of ${\displaystyle K}$ if whenever a point ${\displaystyle p\in F}$ lies between two points of ${\displaystyle K,}$ then those two points necessarily belong to ${\displaystyle F.}$

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