A convex set in light blue, and its extreme points in red.

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]

Examples

If ${\displaystyle a<b}$ are two real numbers then ${\displaystyle a}$ and ${\displaystyle b}$ are extreme points of the interval ${\displaystyle [a,b].}$ However, the open interval ${\displaystyle (a,b)}$ has no extreme points.^[2] Any open interval in ${\displaystyle \mathbb {R} }$ has no extreme points while any non-degenerate closed interval not equal to ${\displaystyle \mathbb {R} }$ does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ has no extreme points.

The extreme points of the closed unit disk in ${\displaystyle \mathbb {R} ^{2}}$ is the unit circle.

The perimeter of any convex polygon in the plane is a face of that polygon.^[2] The vertices of any convex polygon in the plane ${\displaystyle \mathbb {R} ^{2}}$ are the extreme points of that polygon.

An injective linear map ${\displaystyle F:X\to Y}$ sends the extreme points of a convex set ${\displaystyle C\subseteq X}$ to the extreme points of the convex set ${\displaystyle F(X).}$^[2] This is also true for injective affine maps.

Properties

The extreme points of a compact convex set form a Baire space (with the subspace topology) but this set may fail to be closed in ${\displaystyle X.}$^[2]

Theorems

Related notions

A closed convex subset of a topological vector space is called strictly convex if every one of its (topological) boundary points is an extreme point.^[6] The unit ball of any Hilbert space is a strictly convex set.^[6]

See also

• Choquet theory – Area of functional analysis and convex analysis
• Bang–bang control^[7]

Citations

Bibliography

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