Inmathematics, anextreme point of aconvex set${\displaystyle S}$ in areal orcomplexvector space oraffine space is a point in${\displaystyle S}$ that does not lie in any openline segment joining two points of${\displaystyle S.}$ The extreme points of a line segment are called itsendpoints. Inlinear programming problems, an extreme point is also calledvertex orcorner point of${\displaystyle S.}$^[1]

Throughout, it is assumed that${\displaystyle X}$ is areal orcomplexvector space oraffine space.

For any${\displaystyle p,x,y\in X,}$ say that${\displaystyle p}$lies between^[2]${\displaystyle x}$ and${\displaystyle y}$ if${\displaystyle x\ne y}$ and there exists a 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 anextreme point^[2] of${\displaystyle K}$ if it does not lie between any twodistinct points of${\displaystyle K.}$ That is, if there doesnot exist${\displaystyle x,y\in K}$ and such that${\displaystyle x\ne y}$ and${\displaystyle p=tx+(1-t)y.}$ The set of all extreme points of${\displaystyle K}$ is denoted by${\displaystyle \mathrm{extreme}(K).}$

Generalizations

If${\displaystyle S}$ is a subset of a vector space then a linear sub-variety (that is, anaffine subspace)${\displaystyle A}$ of the vector space is called asupport 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]

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

For any elements${\displaystyle x}$ and${\displaystyle y}$ in a vector space, the set${\displaystyle [x,y]=\{tx+(1-t)y:0\le t\le 1\}}$ is called theclosed line segment orclosed interval between${\displaystyle x}$ and${\displaystyle y.}$ Theopen line segment oropen interval between${\displaystyle x}$ and${\displaystyle y}$ is${\displaystyle (x,x)=\varnothing }$ when${\displaystyle x=y}$ while it is when${\displaystyle x\ne y.}$^[2] The points${\displaystyle x}$ and${\displaystyle y}$ are called theendpoints of these interval. An interval is said to be anon−degenerate interval or aproper interval if its endpoints are distinct. Themidpoint of an interval is the midpoint of its endpoints.

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

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 aface^[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${\displaystyle F.}$

If 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] Anyopen interval in${\displaystyle \mathbb{R}}$ has no extreme points while any non-degenerateclosed interval not equal to${\displaystyle \mathbb{R}}$ does have extreme points (that is, the closed interval's endpoint(s)). More generally, anyopen subset of finite-dimensionalEuclidean space${\displaystyle {\mathbb{R}}^n}$ has no extreme points.

The extreme points of theclosed unit disk in${\displaystyle {\mathbb{R}}^2}$ is theunit 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.

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

TheKrein–Milman theorem is arguably one of the most well-known theorems about extreme points.

These theorems are forBanach spaces with theRadon–Nikodym property.

A theorem ofJoram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonemptyclosed andbounded set has an extreme point. (In infinite-dimensional spaces, the property ofcompactness is stronger than the joint properties of being closed and being bounded.^[4])

Edgar’s theorem implies Lindenstrauss’s theorem.

A closed convex subset of atopological vector space is calledstrictly convex if every one of its(topological) boundary points is an extreme point.^[6] Theunit ball of anyHilbert space is a strictly convex set.^[6]

More generally, a point in a convex set${\displaystyle S}$ is${\displaystyle k}$-extreme if it lies in the interior of a${\displaystyle k}$-dimensional convex set within${\displaystyle S,}$ but not a${\displaystyle k+1}$-dimensional convex set within${\displaystyle S.}$ Thus, an extreme point is also a${\displaystyle 0}$-extreme point. If${\displaystyle S}$ is a polytope, then the${\displaystyle k}$-extreme points are exactly the interior points of the${\displaystyle k}$-dimensional faces of${\displaystyle S.}$ More generally, for any convex set${\displaystyle S,}$ the${\displaystyle k}$-extreme points are partitioned into${\displaystyle k}$-dimensional open faces.

The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of${\displaystyle k}$-extreme points. If${\displaystyle S}$ is closed, bounded, and${\displaystyle n}$-dimensional, and if${\displaystyle p}$ is a point in${\displaystyle S,}$ then${\displaystyle p}$ is${\displaystyle k}$-extreme for some${\displaystyle k\le n.}$ The theorem asserts that${\displaystyle p}$ is a convex combination of extreme points. If${\displaystyle k=0}$ then it is immediate. Otherwise${\displaystyle p}$ lies on a line segment in${\displaystyle S}$ which can be maximally extended (because${\displaystyle S}$ is closed and bounded). If the endpoints of the segment are${\displaystyle q}$ and${\displaystyle r,}$ then their extreme rank must be less than that of${\displaystyle p,}$ and the theorem follows by induction.

• Extreme set
• Exposed point
• Choquet theory – Area of functional analysis and convex analysis
• Bang–bang control^[4]

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• Convex geometry
• Convex hulls
• Functional analysis
• Mathematical analysis

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