# Convex cone

In linear algebra, a **convex cone** is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

## Definition

A subset *C* of a vector space *V* is a **cone** (or sometimes called a **linear cone**) if for each *x* in *C* and positive scalars *α*, the product *αx* is in *C*.[1]

A cone *C* is a **convex cone** if *αx* + *βy* belongs to *C*, for any positive scalars *α*, *β*, and any *x*, *y* in *C*.[2][3]

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C. Some authors use a definition that ensures the origin belongs to *C*.[4] Because of the scaling parameters *α* and *β*, cones are infinite in extent and not bounded.

If *C* is a convex cone, then for any positive scalar *α* and any *x* in *C* the vector It follows that a convex cone *C* is a special case of a linear cone.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set *C* is a convex cone if and only if *αC* = *C* and *C* + *C* = *C*, for any positive scalar *α*.

## Examples

- For a vector space
*V*, the empty set, the space*V*, and any linear subspace of*V*are convex cones. - The conical combination of a finite or infinite set of vectors in is a convex cone.
- The tangent cones of a convex set are convex cones.
- The set

- is a cone but not a convex cone.

- The norm cone

- is a convex cone.

- The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one.
- The class of convex cones is also closed under arbitrary linear maps. In particular, if
*C*is a convex cone, so is its opposite and is the largest linear subspace contained in*C*. - The set of positive semidefinite matrices.
- The set of nonnegative continuous functions is a convex cone.

## Special Examples

### Affine convex cones

An **affine convex cone** is the set resulting from applying an affine transformation to a convex cone.[5] A common example is translating a convex cone by a point *p*: *p+C*. Technically, such transformations can produce non-cones. For example, unless *p*=0, *p*+*C* is not a linear cone. However, it is still called an affine convex cone.

### Half-spaces

A (linear) **hyperplane** is a set in the form where f is a linear functional on the vector space V. A **closed half-space** is a set in the form or and likewise an open half-space uses strict inequality.[6][7]

Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex cone *C* that is not the whole space *V* must be contained in some closed half-space *H* of *V*; this is a special case of Farkas' lemma.

### Polyhedral and finitely generated cones

A cone is called **polyhedral** if there is some matrix such that , and a cone is called **finitely generated** if it is the conic combination of finitely many vectors.[8][9]

Every polyhedral cone in can be represented in two different ways; as an intersection of inequalities or the conical hull of vectors. In the inequality description, the polyhedral cone can be given by a matrix such that . Geometrically, each system of linear inequalities defines a halfspace that passes through the origin, and their intersection defines the cone.

In the conical combination description, can be represented by a finite set of vectors such that is the conical combination of , or . Every finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone.[8] Every polyhedral cone has a unique representation as a conical hull of its extremal generators, and every polyhedral cone has a unique representation of intersections of halfspaces, given each linear form associated with the halfspaces also define a support hyperplane of a facet. [10]

Polyhedral cones play a central role in the representation theory of polyhedra. For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as the Minkowski sum of a convex polytope and a polyhedral cone.[11][12] Polyhedral cones also play an important part in proving the related Finite Basis Theorem for polytopes which shows that every polytope is a polyhedron and every bounded polyhedron is a polytope.[11][13][14]

### Blunt, pointed, flat, salient, and proper cones

According to the above definition, if *C* is a convex cone, then *C* ∪ {**0**} is a convex cone, too. A convex cone is said to be **pointed** if **0** is in *C*, and **blunt** if **0** is not in *C*.[1][15] Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.

A cone is called **flat** if it contains some nonzero vector *x* and its opposite -*x,* meaning *C* contains a linear subspace of dimension at least one, and **salient** otherwise.[16][17] A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone *C* is salient if and only if *C* ∩ −*C* ⊆ {**0**}.

Some authors require salient cones to be pointed.[18] The term "pointed" is also often used to refer to a closed cone that contains no complete line (i.e., no nontrivial subspace of the ambient vector space *V*, or what is called a salient cone).[19][20][21] The term **proper** (**convex**) **cone** is variously defined, depending on the context and author. It often means a cone that satisfies other properties like being convex, closed, pointed, salient, and full-dimensional.[22][23][24] Because of these varying definitions, the context or source should be consulted for the definition of these terms.

### Rational Cones

A type of cone of particular interest to pure mathematicians is the partially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming." [25]. This object arises when we study cones in together with the lattice . A cone is called **rational** (here we assume "pointed", as defined above) whenever its generators all have integer coordinates, i.e., if is a rational cone, then .

## Dual cone

Let *C* ⊂ *V* be a set, not necessary a convex set, in a real vector space *V* equipped with an inner product. The (continuous or topological) **dual cone** to *C* is the set

which is always a convex cone.

More generally, the (algebraic) dual cone to *C* ⊂ *V* in a linear space V is a subset of the dual space *V** defined by:

In other words, if *V** is the algebraic dual space of *V*, it is the set of linear functionals that are nonnegative on the primal cone *C*. If we take *V** to be the continuous dual space then it is the set of continuous linear functionals nonnegative on *C*.[26] This notion does not require the specification of an inner product on *V*.

In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous,[27] and every continuous linear functional in a inner product space induces a linear isomorphism (nonsingular linear map) from *V** to *V*, and this isomorphism will take the dual cone given by the second definition, in *V**, onto the one given by the first definition; see the Riesz representation theorem.[26]

If *C* is equal to its dual cone, then *C* is called **self-dual**. A cone can be said to be self-dual without reference to any given inner product, if there exists an inner product with respect to which it is equal to its dual by the first definition.

## Partial order defined by a convex cone

A pointed and salient convex cone *C* induces a partial ordering "≤" on *V*, defined so that if and only if (If the cone is flat, the same definition gives merely a preorder.) Sums and positive scalar multiples of valid inequalities with respect to this order remain valid inequalities. A vector space with such an order is called an ordered vector space. Examples include the product order on real-valued vectors, and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in positive semidefinite programming.

## See also

## Notes

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*Encyclopedic Dictionary of Mathematics*. MIT Press. ISBN 9780262590204. - Rockafellar, Ralph Tyrell (2015-04-29).
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*Algebraic and Geometric Ideas in the Theory of Discrete Optimization*. SIAM. ISBN 9781611972443. - Schrijver, Alexander (1998-07-07).
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*Integer Programming*. Springer. p. 111. ISBN 9783319110080. - Korte, Bernhard; Vygen, Jens (2013-11-11).
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*Monomial Algebras, Second Edition*. CRC Press. p. 9. ISBN 9781482234701. - Dhara, Anulekha; Dutta, Joydeep (2011-10-17).
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