Convex cone

In mathematics, a convex cone is a cone that is closed under linear combinations with non-negative coefficients. Convex cones play an important role in the linear optimization.

Definition

A subset is a convex cone if and always

Followed.

The term can be similarly defined for vector spaces over arranged bodies.

Cone over subsets of the sphere

For a subset of the unit sphere is called

The cone over.

Each cone is of the form for.

The convexity of cones can be described by the following equivalent geometric definition: A cone is then exactly a convex cone, if the average is contiguous with any great circle of the unit sphere.

Other terms

A cone is called regular if

The automorphism group of a cone

A cone is called homogeneous if the automorphism group transitive acts on.

It is called symmetric if, for every involution with as the only fixed point. Symmetric convex cones are always homogeneous.

A cone is called reducible if the form of the

With is irreducible usual

The dual cone is defined as to. Also, this definition can be analogously for vector spaces with scalar product formulated over a arranged body.

A cone is called self-dual if it is.

Characterization of symmetric convex cone: A convex cone is symmetric if it is open, regular, homogeneous and self-dual.

The positive taper of Jordan algebra, the amount of the elements with a positive spectrum. A Jordan algebra is called formally real if can not be represented as a nontrivial sum of squares. In a formally real Jordan algebra is an element if and only belongs to the positive cone if it is a square.

The set of Koecher - Vinberg states that the construction of the positive cone establishes a bijection between formally real Jordan algebras and symmetric convex cones.

Symmetric convex cones are therefore also called positivity area (English: domain of positivity ) refers.

Classification of symmetric convex cone

The irreducible symmetric convex cone are given by the following list:

• Lorentz cone
• The cone of positive symmetric matrices for
• The cone of positive Hermitian complex matrices for
• The cone of positive hermitian quaternionic matrices for
• And for the with cone.