Extreme point

An extreme point of a convex set K of a real vector space is a point x of K, which can not be represented as a convex combination of two distinct points in K, ie, between any two other points in K is. That is, there are no points with a.

Examples

Applications

• The extreme points of a polyhedron is called corners. They play an important role in the geometric interpretation of linear programming.
• In many situations characterizations of extremal succeed as objects with specific properties as in Example 3 The set of Krein - Milman then leads to theorems on the existence of such objects.
• In the Choquet theory is the idea that a point of a convex set can be represented as averaging over their extreme points, clarified.

Closure properties

The amount of the extreme points is generally not complete. A three-dimensional example is obtained by joining two oblique cone to a double cone, so that the link between the peaks and (see adjacent diagram) runs along the lateral surfaces and the common circle meets at a point. The set of extreme points of this double cone consists of the cone tips and and all points of the circle without, because this issue can be and combine convex yes. but is located in the financial statements of Extremalpunktmenge.

In the infinite-dimensional case, the set of extreme points are dense. A simple example is the unit sphere in an infinite-dimensional Hilbert space with the weak topology (with respect to this compact ). The Extremalpunktmenge is the set of all vectors of length 1 To see that the Extremalpunktmenge is dense in, either with a vector and a weak around. Then there are vectors and one with. There is infinite dimensional, there is a direction orthogonal to the vector and then, so that the vector has the length 1 and, consequently, is an extreme point. Since, follows. Thus we have shown that every weak neighborhood of a vector of length <1 contains an extreme point. Therefore, the conclusion of the Extremalpunktmenge falls along with.

Extremal quantities

The definition of a Extremalpunktes can be applied in a natural way to sets: An extremal set is a subset of a convex set with the property that points only can be represented as a convex combination of points of the convex set from this set, if these points already in the subset itself are included. Formal:

Typical examples are sides or edges of polyhedra. A frequently used phrase is that extreme points of extremal quantities already are extreme points of the surrounding convex set.