In mathematics, an extreme point of a convex set in a real or complex vector space or affine space that does not lie in any open line segment The extreme points of a line segment are called its endpoints. In linear programming problems, an extreme point is also called vertex or corner point

Definition

is a real or complex vector space or affine space.

is called an if it does not lie between any two That is, if there does

Generalizations

is a subset of a vector space then a linear sub-variety (that is, an affine subspace of the vector space is called a

Characterizations

The

is called the or The or 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.

is equal to the convex hull

is called a

Examples

has no extreme points. Any open interval has no extreme points while any non-degenerate closed interval does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space has no extreme points.

The extreme points of the closed unit disk is the unit circle.

The perimeter of any convex polygon in the plane is a face of that polygon. are the extreme points of that polygon.

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

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

is -extreme-dimensional open faces.

and the theorem follows by induction.

See also

• Extreme set

• Exposed point

• Bang–bang control

Citations

Bibliography