Extreme point
In mathematics, an extreme point of a convex set in a real or complex vector space or affine space is a point in that does not lie in any open line segment joining two points of 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 of
Definition
Throughout, it is assumed that is a real or complex vector space or affine space.For any say that ' and if and there exists a such that
If is a subset of and then is called an ' of if it does not lie between any two points of That is, if there does exist and such that and The set of all extreme points of is denoted by
Generalizations
If is a subset of a vector space then a linear sub-variety of the vector space is called a if meets and every open segment whose interior meets is necessarily a subset of A 0-dimensional support variety is called an extreme point of
Characterizations
The ' of two elements and in a vector space is the vectorFor any elements and in a vector space, the set is called the ' or ' between and The ' or ' between and is when while it is when The points and 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 is equal to the convex hull of if So if is convex and then
If is a nonempty subset of and is a nonempty subset of then is called a ' of if whenever a point lies between two points of then those two points necessarily belong to
Examples
If are two real numbers then and are extreme points of the interval However, the open interval has no extreme points.Any open interval in has no extreme points while any non-degenerate closed interval not equal to does have extreme points. More generally, any open subset of finite-dimensional Euclidean space has no extreme points.
The extreme points of the closed unit disk in 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 are the extreme points of that polygon.
An injective linear map sends the extreme points of a convex set to the extreme points of the convex set This is also true for injective affine maps.
Properties
The extreme points of a compact convex set form a Baire space but this set may to be closed inTheorems
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.
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 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 is -extreme if it lies in the interior of a -dimensional convex set within but not a -dimensional convex set within Thus, an extreme point is also a -extreme point. If is a polytope, then the -extreme points are exactly the interior points of the -dimensional faces of More generally, for any convex set the -extreme points are partitioned into -dimensional open faces.The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of -extreme points. If is closed, bounded, and -dimensional, and if is a point in then is -extreme for some The theorem asserts that is a convex combination of extreme points. If then it is immediate. Otherwise lies on a line segment in which can be maximally extended. If the endpoints of the segment are and then their extreme rank must be less than that of and the theorem follows by induction.