Extreme point

An extreme point of a convex set K of a real vector space is a point x from K that cannot be represented as a convex combination of two different points from K , i.e. it does not lie between any two other points from K. That means there are no points with for one . ${\displaystyle a\neq b\in K}$ ${\displaystyle x=\lambda a+(1-\lambda )b}$ ${\displaystyle 0<\lambda <1}$

Explanations and examples

Extreme points (red) of a convex set K (blue and red) cannot be represented as a convex combination of two different points from K.

1. A point is an extreme point of the convex set if and only if the remainder of the set is itself a convex set. ${\displaystyle x\in K}$ ${\displaystyle K}$ ${\displaystyle K\setminus \{x\}}$
2. A triangle is a convex set, the extremal points are exactly the corners of the triangle.
3. A completed ball in is convex, the extreme points are exactly the boundary points . This applies to all Hilbert spaces or, more generally, to all strictly convex spaces . An open sphere has no extremal points. ${\displaystyle \mathbb {R} ^{n}}$
4. The positive functionals with norm 1 of a commutative C * -algebra form a convex set. The extremal points are exactly the multiplicative functionals.
5. According to Birkhoff's and von Neumann's theorem , the permutation matrices are exactly the extremal points of the double-stochastic matrices.

Applications

• The extremal points of a polyhedron are called corners . They play an important role in the geometric interpretation of linear optimization .
• In many situations, extreme points can be characterized as objects with special properties, as in example 3. The theorem of Krein-Milman then leads to theorems about the existence of such objects.
• In the Choquet theory , the idea that a point of a convex set can be represented as the averaging over its extreme points is specified more precisely.
• The extreme points play an important role in analysis , linear optimization and the calculus of variations , as they considerably simplify the determination of extreme points of certain continuous real-valued functionals . This fact is described by Bauer's maximum principle .

Closing properties

The set of extreme points is generally not closed. A three-dimensional example is obtained by joining two oblique cones to form a double cone, so that the connecting line between the tips and (see adjacent sketch) runs on the lateral surfaces and meets the common circular line at a point . The set of extreme points of this double cone consists of the cone tips and and all points of the circular line without , because this point can be combined from and convex. but lies at the end of the extreme point set. ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle R}$ ${\displaystyle Q}$

In the infinite-dimensional case, the set of extreme points can be close. A simple example is the unit sphere in an infinite-dimensional Hilbert space with the weak topology (with respect to this it is compact). The set of extremal points is the set of all vectors with length 1. To see that the set of extremal points is dense in , let a vector with and be a weak neighborhood of . Then there are vectors and one with . Since is infinite dimensional, there is one to the orthogonal vector and then one such that the vector has length 1 and is therefore an extremal point. There follows . This shows that every weak neighborhood of a vector of length <1 contains an extreme point. Therefore, the termination of the extreme point set coincides with . ${\displaystyle U}$ ${\displaystyle H}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle x_{0}}$ ${\displaystyle \|x_{0}\|<1}$ ${\displaystyle V}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{1},\ldots ,y_{n}\in H}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \{x\in H;\,|\langle x-x_{0},y_{i}\rangle |<\varepsilon \,{\mbox{ für alle }}i=1,\ldots n\}\subset V}$ ${\displaystyle H}$ ${\displaystyle y_{i}}$ ${\displaystyle y}$ ${\displaystyle \lambda >0}$ ${\displaystyle x_{0}+\lambda y}$ ${\displaystyle \langle (x_{0}+\lambda y)-x_{0},y_{i}\rangle =\lambda \langle y,y_{i}\rangle =0}$ ${\displaystyle x_{0}+\lambda y\in V}$ ${\displaystyle U}$

Extremal quantities

The definition of an extremal point can be transferred naturally to sets: An extremal set is a subset of a convex set with the property that points from this set can only be represented as a convex combination of points from the convex set if these points are already in the subset itself are included. Formally:

Let be a vector space, convex and . Then there is an extremal set if: ${\displaystyle X}$ ${\displaystyle K\subset X}$ ${\displaystyle M\subset K}$ ${\displaystyle M}$

${\displaystyle \forall \,\alpha \in (0,1)~\forall \,x_{1},x_{2}\in K\!:~\alpha x_{1}+(1-\alpha )x_{2}\in M~\Rightarrow ~x_{1},x_{2}\in M.}$

Typical examples are sides or edges of polyhedra. An often used theorem is that extremal points of extremal sets are already extremal points of the surrounding convex set.

literature

• RV Kadison , JR Ringrose : Fundamentals of Operator Algebras , Academic Press 1983
• N. Bourbaki : Topological Vector Spaces: Chapters 1 - 5 . Springer Verlag , Berlin, Heidelberg, New York, London, Paris, Tokyo 1987, ISBN 3-540-13627-4 ( MR0910295 ).