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Concept
Extreme point
Summary
In mathematics, an extreme point of a convex set in a real or complex vector space is a point in which 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
Throughout, it is assumed that is a real or complex vector 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 (that is, an affine subspace) of the vector space is called a if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of A 0-dimensional support variety is called an extreme point of
The of two elements and in a vector space is the vector
For 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 (and only 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
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 (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 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.
Official source
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