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.