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 say that**** p **** x and y if**** x *neq** y and** the**re** exi**st**s*** a 0 < t < 1 such**** that**** 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** **** The** set** of*** all extreme points**** of**** K is**** denoted by**** 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 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**** For any elements**** x and y in**** a vector**** space,**** the set is**** called**** the ' or ********************' bet**we**en**** x and y.**** The ' or ********************' bet**we**en**** x and y is**** **** when**** x = y while it**** is**** **** 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 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.****

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 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.

Theorems

Krein–Milman theorem

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

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. ) 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.

Citations