Cone (linear algebra)

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In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars.

1 Definition
2 Boolean, additive and linear closure
3 Pointed and blunt cones
4 The cone of a set
5 Salient cone
6 Spherical section and projection
7 Convex cone
8 Affine cone
9 Proper cone
10 See also
11 References

[edit] Definition

A subset C of a real vector space V is a (linear) cone if and only if $λ x$ belongs to C for any x in C and any positive scalar $λ$ of V.

The condition can be written more succinctly as "λC = C for any positive scalar λ of V".

The definition makes sense for any vector space V which allows the notion of "positive scalar", such as spaces over the rational, algebraic, or (more commonly) real numbers . The concept can also be extended for any vector space V whose scalar field is a superset of those fields (such as the complex numbers, quaternions, etc.), to the extent that such a space can be viewed as a real vector space of higher dimension.

[edit] Boolean, additive and linear closure

Linear cones are closed under Boolean operations (set intersection, union, and complement). They are also closed under addition (if C and D are cones, so is C + D) and arbitrary linear maps. In particular, if Cis a cone, so is its opposite cone -C.

[edit] Pointed and blunt cones

A cone C is said to be pointed if it includes the null vector (origin) 0 of the vector space; otherwise C is said to be blunt. Note that a pointed cone is closed under multiplication by arbitrary non-negative (not just positive) scalars.

[edit] The cone of a set

The (linear) cone of an arbitrary subset X of V is the set X $*$ of all vectors $λ$ x where x belongs to X and λ is a positive real number.

With this definition, the cone of X is pointed or blunt depending on whether X contains the origin 0 or not. If "positive" is replaced by "non-negative" in the defitions, the cone X $*$ will be always pointed.

[edit] Salient cone

A cone X is salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if C $\cap$ (-C) $\subseteq$ {0}.

[edit] Spherical section and projection

Let |·| be any norm for V, with the property that the norm of any vector is a scalar of V. By definition, a nonzero vector x belongs to a cone C of V if and only if the unit-norm vector x/|x| belongs to C. Therefore, a blunt (or pointed) cone C is completely specified by its central projection onto the sphere S; that is, by the set

$C' = \{\, \frac{x}{|x|} \;:\; x \in C \wedge x \neq \mathbf{0} \,\}$

It follows that there is a one-to-one correspondence between blunt (or pointed) cones and subsets of the unit-norm sphere of V, the set

$S = \{\, x \in V\;:\; |x| = 1 \,\}$

Indeed, the central projection C' is simply the spherical section of C, the set C $\cap$ S of its unit-norm elements.

A cone C is closed with respect to the norm |·| if it is a closed set in the topology induced by that norm. That is the case if and only if C is pointed and its spherical section is a closed subset of S.

Note that the cone C is salient if and only if its spherical section does not contain two opposite vectors; that is, C' $\cap$ (-C' ) = {}.

[edit] Convex cone

A convex cone is a cone that is closed under convex combinations, i.e. if and only if αx + βy belongs to C for any non-negative scalars α, β with α + β = 1.

[edit] Affine cone

If C - v is a cone for some v in V, then C is said to be an (affine) cone with vertex v.

[edit] Proper cone

The term proper cone is variously defined, depending on the context. It often means a salient and convex cone, or a cone that is contained in an open halfspace of V.