Extreme point



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

Definition
Throughout, it is assumed that $$X$$ is a real or complex vector space.

For any $$p, x, y \in X,$$ say that $$p$$  $$x$$ and $$y$$ if $$x \neq y$$ and there exists a $$0 < t < 1$$ such that $$p = t x + (1-t) y.$$

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

Generalizations

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

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

For any elements $$x$$ and $$y$$ in a vector space, the set $$[x, y] = \{t x + (1-t) y : 0 \leq t \leq 1\}$$ is called the ' or ' between $$x$$ and $$y.$$ The ' or ' between $$x$$ and $$y$$ is $$(x, x) = \varnothing$$ when $$x = y$$ while it is $$(x, y) = \{t x + (1-t) y : 0 < t < 1\}$$ when $$x \neq y.$$ The points $$x$$ and $$y$$ are called the  of these interval. An interval is said to be a ' or a ' if its endpoints are distinct. The  is the midpoint of its endpoints.

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

If $$K$$ is a nonempty subset of $$X$$ and $$F$$ is a nonempty subset of $$K,$$ then $$F$$ is called a  of $$K$$ if whenever a point $$p \in F$$ lies between two points of $$K,$$ then those two points necessarily belong to $$F.$$

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

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

The perimeter of any convex polygon in the plane is a face of that polygon. The vertices of any convex polygon in the plane $$\R^2$$ are the extreme points of that polygon.

An injective linear map $$F : X \to Y$$ sends the extreme points of a convex set $$C \subseteq X$$ to the extreme points of the convex set $$F(X).$$ 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 to be closed in $$X.$$

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

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For Banach spaces
These theorems are for Banach spaces with the Radon–Nikodym property.

A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded. )

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Edgar’s theorem implies Lindenstrauss’s theorem.

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

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

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