Extreme point

In mathematics, an extreme point of a convex set

in a real or complex vector space is a point in

that does not lie in any open line segment joining two points of

In linear programming problems, an extreme point is also called vertex or corner point of

is a real or complex vector space.

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

of the vector space is called a support variety if

is not empty) and every open segment

[3] A 0-dimensional support variety is called an extreme point of

be a non-empty convex subset of a vector space

has no extreme points while any non-degenerate closed interval not equal to

does have extreme points (that is, the closed interval's endpoint(s)).

More generally, any open subset of finite-dimensional Euclidean space

The extreme points of the closed unit disk in

[2] The vertices of any convex polygon in the plane

sends the extreme points of a convex set

to the extreme points of the convex set

[2] This is also true for injective affine maps.

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

is the closed convex hull of its extreme points: In particular, such a set has extreme points.

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.

[4]) Theorem (Gerald Edgar) — Let

be a separable, closed, bounded, convex subset of

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] More generally, a point in a convex set

The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of

is a convex combination of extreme points.