Convex cone

In mathematics , a convex cone is a cone that is closed under linear combinations with positive coefficients (also called conical combinations ). Convex cones play an important role in conical optimization .

A convex cone (light blue). The violet set represents the linear combinations α x + β y with positive coefficients α, β> 0 for the points x and y . The curved lines on the right-hand edge should indicate that the areas are to be extended to infinity.

definition

Given is a set of a - vector space , where is an arranged body . Mostly is . ${\displaystyle C}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$

The set is a convex cone if one of the following definitions is true: ${\displaystyle C}$

• ${\displaystyle C}$is convex and a cone .
• ${\displaystyle C}$is a cone, and for any is again included in. ${\displaystyle x,y\in C}$ ${\displaystyle x+y}$ ${\displaystyle C}$
• For any and off is always in again . ${\displaystyle x,y\in C}$ ${\displaystyle \alpha ,\beta >0}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \alpha x+\beta y}$ ${\displaystyle C}$
• The set is complete regarding conical combinations . ${\displaystyle C}$

properties

• Intersections of families of convex cones are again convex cones. The convex cones thus form a system of envelopes .
• The conical hull (sometimes also called the positive hull) assigns the smallest convex cone containing that set to each set. Thus the conical hull is the hull operator to the hull system of convex cones. ${\displaystyle \operatorname {pos} (X)}$
• Each convex cone defines an order relation on the vector space in which it is located. The convex cone is then understood as an order cone .

Cones over subsets of the sphere

For a subset of the unit sphere is called ${\displaystyle \Omega \subset S^{n-1}}$ ${\displaystyle S^{n-1}=\left\{x\in \mathbb {R} ^{n}\colon \parallel x\parallel =1\right\}}$

${\displaystyle C(\Omega )=\left\{rv\colon v\in \Omega ,r\in \mathbb {R} _{>0}\right\}}$

the cone over . ${\displaystyle \Omega }$

Each cone is of the shape for . ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle K=C(\Omega )}$ ${\displaystyle \Omega =K\cap S^{n-1}}$

The convexity of cones can be described by the following equivalent geometric definition describe: a cone is precisely then a convex cone, if the average with each great circle of the unit sphere contiguous is. ${\displaystyle K\subset \mathbb {R} ^{n}}$

More terms

A cone is called a polyhedral cone if there is a matrix such that ${\displaystyle C\subset \mathbb {R} ^{n}}$ ${\displaystyle A}$

${\displaystyle C=\{x\in \mathbb {R} ^{n}|Ax\leq 0\}}$

is. A cone is a polyhedral cone if and only if it is generated by a finite set of vectors .

A cone is called regular if

${\displaystyle a\in K\wedge -a\in K\Longrightarrow a=0}$.

The automorphism group of a cone is ${\displaystyle K\subset \mathbb {R} ^{n}}$

${\displaystyle Aut(K)=\left\{A\in GL(n,\mathbb {R} ):AK=K\right\}}$.

A cone is called homogeneous if the automorphism group acts transitively on . ${\displaystyle K}$

It is called symmetric if there is an involution for each with the only fixed point. Symmetrical convex cones are always homogeneous. ${\displaystyle x\in K}$ ${\displaystyle A\in Aut(K)}$ ${\displaystyle x}$

A cone is called reducible if its shape ${\displaystyle K\subset \mathbb {R} ^{n}}$

${\displaystyle K=K_{1}+K_{2},K_{1}\subset R^{p}\times \left\{0\right\}^{n-p},K_{2}\subset \left\{0\right\}^{p}\times \mathbb {R} ^{n-p}}$

with is, otherwise irreducible . ${\displaystyle 0<p<n}$

The too dual cone is defined as . This definition can also be formulated analogously for vector spaces with a scalar product over an arranged body. ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle K^{*}=\left\{a\in \mathbb {R} ^{n}:\langle b,a\rangle \geq 0\ \forall b\in K\right\}}$

A cone is called self-dual if is. ${\displaystyle K=K^{*}}$

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

Koecher-Vinberg's theorem

The positive cone of a Jordan algebra is the set of elements with a positive spectrum . A Jordan algebra is formally called real if it can not be represented as a nontrivial sum of squares. In a formally real Jordan algebra, an element belongs to the positive cone if and only if it is a square. ${\displaystyle A}$ ${\displaystyle 0\in A}$

The set of quiver Vinberg states that the construction of the positive taper produces a bijection between formal real Jordan algebras and symmetrical convex cones.

Symmetrical convex cones are therefore also called positivity area (ger .: domain of positivity called).

Classification of symmetrical convex cones

In 1965, Max Koecher used the classification of formally real Jordan algebras to classify the symmetrical convex cones.

The irreducible symmetric convex cones in are given by the following list: ${\displaystyle \mathbb {R} ^{n}}$

• the Lorentz cone ${\displaystyle \Lambda _{n}=\left\{x\in \mathbb {R} ^{n}\colon x_{1}^{2}-x_{2}^{2}-\ldots -x_{n}^{2}>0,x_{1}>0\right\}}$
• the cone of positive symmetric matrices for ${\displaystyle \Pi _{m}(\mathbb {R} )}$ ${\displaystyle m\times m}$ ${\displaystyle n={\frac {m^{2}+m}{2}}}$
• the cone of the positive complex Hermitian matrices for ${\displaystyle \Pi _{m}(\mathbb {C} )}$ ${\displaystyle m\times m}$ ${\displaystyle n=m^{2}}$
• the cone of positive Hermitian quaternionic matrices for ${\displaystyle \Pi _{m}(\mathbb {H} )}$ ${\displaystyle m\times m}$ ${\displaystyle n=2m^{2}-m}$
• and for the cone with . ${\displaystyle n=27}$ ${\displaystyle \Pi _{3}(O)}$ ${\displaystyle Lie(Aut(\Pi _{3}(O)))={\mathfrak {e}}_{6(-26)}\oplus \mathbb {R} }$

literature

• Benoist, Yves: A survey on divisible convex sets. Geometry, analysis and topology of discrete groups, 1-18, Adv. Lect. Math. (ALM), 6, Int. Press, Somerville, MA 2008. pdf
• Koecher, Max: The Minnesota notes on Jordan algebras and their applications. Edited, annotated and with a preface by Aloys Krieg and Sebastian Walcher. Lecture Notes in Mathematics, 1710. Springer-Verlag, Berlin 1999, ISBN 3-540-66360-6 .