In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars.
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Definition
A subset C of a real vector space V is a (linear) cone if and only if <math>\lambda x</math> belongs to C for any x in C and any positive scalar <math>\lambda</math> 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.
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 C is a cone, so is its opposite cone -C.
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.
The cone of a set
The (linear) cone of an arbitrary subset X of V is the set X<math>{}^*</math> of all vectors <math>\lambda</math>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<math>{}^*</math> will be always pointed.
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<math>\cap</math>(-C) <math>\subseteq</math> {0}.
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
- <math>C' = \{\, \frac{x}{|x|} \;:\; x \in C \wedge x \neq \mathbf{0} \,\}</math>
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
- <math>S = \{\, x \in V\;:\; |x| = 1 \,\}</math>
Indeed, the central projection C' is simply the spherical section of C, the set C<math>\cap</math>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' <math>\cap</math>(-C' ) = {}.
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.
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.
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.
See also
- Cone
- Ordered group with the concept of the "positive cone"
- Ordered vector space
