Cone (linear algebra)

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

Definition

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

The definition makes sense for any vector space V which allows the notion of "positive scalar", such as spaces over the rationas and (more commonly) real numbers . The concept can also be extended for any vector space V whose scalars are a superset of the real numbers (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.

Pointed and blunt cones

A cone C is said to be pointed if it includes the null vector (vector space) (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) real numbers.

The cone of a set

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

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

Salient and convex cones

A cone X is salient if it contains no one-dimensional subspace of V; that is, if x, -x $\in$ C implies x = 0.

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

Spherical section

Let |·| be any norm for V. By definition, a nonzero vector x belongs to a cone C if and only if the unit-norm vector (1/|x|)x belongs to C. Therefore, a blunt (or pointed) cone C is completely specified by its central projection of C onto the sphere S; that is,

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 a norm of V 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.

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.

References

Cone at PlanetMath.