Inlinear algebra, acone—sometimes called alinear cone to distinguish it from other sorts of cones—is a subset of a realvector space that isclosed under positive scalar multiplication; that is,${\displaystyle C}$ is a cone if${\displaystyle x\in C}$ implies${\displaystyle sx\in C}$ for everypositive scalar${\displaystyle s}$. This is a broad generalization of the standardcone inEuclidean space.

Aconvex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed underlinear combinations with positive coefficients. It follows that convex cones areconvex sets.^[1]

The definition of a convex cone makes sense in a vector space over anyordered field, although the field ofreal numbers is used most often.

A subset${\displaystyle C}$ of a vector space is a cone if${\displaystyle x\in C}$ implies${\displaystyle sx\in C}$ for every${\displaystyle s>0}$. Here${\displaystyle s>0}$ refers to (strict) positivity in the scalar field.

Some other authors require${\displaystyle [0,\mathrm{\infty })C\subset C}$ or even${\displaystyle 0\in C}$.Some require a cone to be convex and/or satisfy${\displaystyle C\cap -C\subset \{0\}}$.

Theconical hull of a set${\displaystyle C}$ is defined as the smallestconvex cone that contains${\displaystyle C\cup \{0\}}$. Therefore, it need not be the smallest cone that contains${\displaystyle C\cup \{0\}}$.

Wedge may refer to what we call cones (when "cone" is reserved for something stronger), or just to a subset of them, depending on the author.

Asubset${\displaystyle C}$ of a vector space${\displaystyle V}$ over anordered field${\displaystyle F}$ is acone (or sometimes called alinear cone) if for each${\displaystyle x}$ in${\displaystyle C}$ and positive scalar${\displaystyle \alpha }$ in${\displaystyle F}$, the product${\displaystyle \alpha x}$ is in${\displaystyle C}$.^[2] Note that some authors definecone with the scalar${\displaystyle \alpha }$ ranging over all non-negative scalars (rather than all positive scalars, which does not include 0).^[3] Some authors even require${\displaystyle 0\in C}$, thus excluding the empty set.^[4]

Therefore,${\displaystyle [0,\mathrm{\infty })}$ is a cone,${\displaystyle \varnothing }$ is a cone only according to the 1st and 2nd definition above, and${\displaystyle (0,\mathrm{\infty })}$ is a cone only according to the 1st definition above. All of them are convex (see below).

A cone${\displaystyle C}$ is aconvex cone if${\displaystyle \alpha x+\beta y}$ belongs to${\displaystyle C}$, for any positive scalars${\displaystyle \alpha }$,${\displaystyle \beta }$, and any${\displaystyle x}$,${\displaystyle y}$ in${\displaystyle C}$.^[5][6] A cone${\displaystyle C}$ is convex if and only if${\displaystyle C+C\subseteq C}$.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over therational,algebraic, or (more commonly) thereal numbers. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to${\displaystyle C}$. Some authors use a definition that ensures the origin belongs to${\displaystyle C}$.^[7] Because of the scaling parameters${\displaystyle \alpha }$ and${\displaystyle \beta }$, cones are infinite in extent and not bounded.

If${\displaystyle C}$ is a convex cone, then for any positive scalar${\displaystyle \alpha }$ and any${\displaystyle x}$ in${\displaystyle C}$ the vector${\displaystyle \alpha x=\frac{\alpha }{2}x+\frac{\alpha }{2}x\in C.}$ So a convex cone is a special case of a linear cone as defined above.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed underconvex combinations, or just underaddition. More succinctly, a set${\displaystyle C}$ is a convex cone if and only if${\displaystyle \alpha C=C}$ for every positive scalar${\displaystyle \alpha }$ and${\displaystyle C+C=C}$.

Aface of a convex cone${\displaystyle C}$ is a subset${\displaystyle F}$ of${\displaystyle C}$ such that${\displaystyle F}$ is also a convex cone, and for any vectors${\displaystyle x,y}$ in${\displaystyle C}$ with${\displaystyle x+y}$ in${\displaystyle F}$,${\displaystyle x}$ and${\displaystyle y}$ must both be in${\displaystyle F}$.^[8] For example,${\displaystyle C}$ itself is a face of${\displaystyle C}$. The origin${\displaystyle \{0\}}$ is a face of${\displaystyle C}$ if${\displaystyle C}$ contains no line (so${\displaystyle C}$ is "strictly convex", or "salient", as defined below). The origin and${\displaystyle C}$ are sometimes called thetrivial faces of${\displaystyle C}$. Aray (the set of nonnegative multiples of a nonzero vector) is called anextremal ray if it is a face of${\displaystyle C}$.

Let${\displaystyle C}$ be aclosed, strictly convex cone in${\displaystyle {\mathbb{R}}^n}$. Suppose that${\displaystyle C}$ is more than just the origin. Then${\displaystyle C}$ is theconvex hull of its extremal rays.^[9]

• For a vector space${\displaystyle V}$, everylinear subspace of${\displaystyle V}$ is a convex cone. In particular, the space${\displaystyle V}$ itself and the origin${\displaystyle \{0\}}$ are convex cones in${\displaystyle V}$. For authors who do not require a convex cone to contain the origin, the empty set${\displaystyle \mathrm{\varnothing }}$ is also a convex cone.
• Theconical hull of a finite or infinite set of vectors in${\displaystyle {\mathbb{R}}^n}$ is a convex cone.
• Thetangent cones of a convex set are convex cones.
• The set$${\displaystyle \left\{x\in {\mathbb{R}}^2\mid x_2\ge 0,x_1=0\right\}\cup \left\{x\in {\mathbb{R}}^2\mid x_1\ge 0,x_2=0\right\}}$$ is a cone but not a convex cone.
• The norm cone$${\displaystyle C=\left\{(x,r)\in {\mathbb{R}}^{d+1}\mid \Vert x\Vert \le r\right\}}$$ is a convex cone. (For${\displaystyle d=2}$, this is the round cone in the figure.) Each extremal ray of${\displaystyle C}$ is spanned by a vector${\displaystyle (x,1)}$ with${\displaystyle \Vert x\Vert =1}$ (so${\displaystyle x}$ is a point in thesphere${\displaystyle S^{d-1}}$). These rays are in fact the only nontrivial faces of${\displaystyle C}$.
• 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 arbitrarylinear maps. In particular, if${\displaystyle C}$ is a convex cone, so is its opposite${\displaystyle -C}$, and${\displaystyle C\cap -C}$ is the largest linear subspace contained in${\displaystyle C}$.
• The set ofpositive semidefinite matrices.
• The set of nonnegative continuous functions is a convex cone.

Anaffine convex cone is the set resulting from applying an affine transformation to a convex cone.^[10] A common example is translating a convex cone by a pointp:p +C. Technically, such transformations can produce non-cones. For example, unlessp = 0,p +C is not a linear cone. However, it is still called an affine convex cone.

A (linear)hyperplane is a set in the form${\displaystyle \{x\in V\mid f(x)=c\}}$ where f is alinear functional on the vector space V. Aclosedhalf-space is a set in the form${\displaystyle \{x\in V\mid f(x)\le c\}}$ or${\displaystyle \{x\in V\mid f(x)\ge c\},}$ and likewise an open half-space uses strict inequality.^[11][12]

Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex coneC that is not the whole spaceV must be contained in some closed half-spaceH ofV; this is a special case ofFarkas' lemma.

Polyhedral cones are special kinds of cones that can be defined in several ways:^{[13]: 256–257}

• A cone${\displaystyle C}$ is polyhedral if it is theconical hull of finitely many vectors (this property is also calledfinitely-generated).^[14][15] I.e., there is a set of vectors${\displaystyle \{v_1,\dots ,v_k\}\subset {\mathbb{R}}^n}$ so that${\displaystyle C=\{a_1{v}_1+\cdots +a_k{v}_k\mid a_i\in {\mathbb{R}}_{\ge 0}\}}$.
• A cone is polyhedral if it is the intersection of a finite number of half-spaces which have 0 on their boundary (the equivalence between these first two definitions was proved by Weyl in 1935).^[16][17]
• A cone${\displaystyle C}$ is polyhedral if there is somematrix${\displaystyle A\in {\mathbb{R}}^{m\times n}}$ such that${\displaystyle C=\{x\in {\mathbb{R}}^n\mid Ax\ge 0\}}$.
• A cone is polyhedral if it is the solution set of a system of homogeneous linear inequalities. Algebraically, each inequality is defined by a row of the matrix${\displaystyle A}$. Geometrically, each inequality defines a halfspace that passes through the origin.

Every finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone.^[14] Every polyhedral cone has a unique representation as a conical hull of its extremal generators, and a unique representation of intersections of halfspaces, given each linear form associated with the halfspaces also define a support hyperplane of a facet.^[18]

Each face of a polyhedral cone is spanned by some subset of its extremal generators. As a result, a polyhedral cone has only finitely many faces.

Polyhedral cones play a central role in the representation theory ofpolyhedra. For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as theMinkowski sum of aconvex polytope and a polyhedral cone.^[19][20] Polyhedral cones also play an important part in proving the relatedFinite Basis Theorem for polytopes which shows that every polytope is a polyhedron and everybounded polyhedron is a polytope.^[19][21][22]

The two representations of a polyhedral cone - by inequalities and by vectors - may have very different sizes. For example, consider the cone of all non-negative${\displaystyle n}$-by-${\displaystyle n}$ matrices with equal row and column sums. The inequality representation requires${\displaystyle n^2}$ inequalities and${\displaystyle 2n-1}$ equations, but the vector representation requires${\displaystyle n!}$ vectors (see theBirkhoff-von Neumann Theorem). The opposite can also happen - the number of vectors may be polynomial while the number of inequalities is exponential.^[13]: 256

The two representations together provide an efficient way to decide whether a given vector is in the cone: to show that it is in the cone, it is sufficient to present it as a conic combination of the defining vectors; to show that it is not in the cone, it is sufficient to present a single defining inequality that it violates. This fact is known asFarkas' lemma.

A subtle point in the representation by vectors is that the number of vectors may be exponential in the dimension, so the proof that a vector is in the cone might be exponentially long. Fortunately,Carathéodory's theorem guarantees that every vector in the cone can be represented by at most${\displaystyle d}$ defining vectors, where${\displaystyle d}$ is the dimension of the space.

According to the above definition, ifC is a convex cone, thenC ∪ {0} is a convex cone, too. A convex cone is said to bepointed if0 is inC, andblunt if0 is not inC.^[2][23] Some authors use "pointed" for${\displaystyle C\cap -C=\{0\}}$ or salient (see below).^[24]

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 calledflat if it contains some nonzero vectorx and its opposite −x, meaningC contains a linear subspace of dimension at least one, andsalient (orstrictly convex) otherwise.^[25][26] A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex coneC is salient if and only ifC ∩ −C ⊆ {0}. A coneC is said to begenerating if${\displaystyle C-C=\{x-y\mid x\in C,y\in C\}}$ equals the whole vector space.^[27]

Some authors require salient cones to be pointed.^[28] 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 spaceV, or what is called a salient cone).^[29][30][31] The termproper (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.^[32][33][34] Because of these varying definitions, the context or source should be consulted for the definition of these terms.

A type of cone of particular interest to pure mathematicians is thepartially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming.".^[35] This object arises when we study cones in${\displaystyle {\mathbb{R}}^d}$ together with thelattice${\displaystyle {\mathbb{Z}}^d}$. A cone is calledrational (here we assume "pointed", as defined above) whenever its generators all haveinteger coordinates, i.e., if${\displaystyle C}$ is a rational cone, then${\displaystyle C=\{a_1{v}_1+\cdots +a_k{v}_k\mid a_i\in {\mathbb{R}}_{+}\}}$ for some${\displaystyle v_i\in {\mathbb{Z}}^d}$.

LetC ⊂V be a set, not necessarily a convex set, in a real vector spaceV equipped with aninner product. The (continuous or topological)dual cone toC is the set

${\displaystyle C^{\ast }=\{v\in V\mid \mathrm{\forall }w\in C,⟨w,v⟩\ge 0\},}$

which is always a convex cone. Here,${\displaystyle ⟨w,v⟩}$ is theduality pairing betweenC andV, i.e.${\displaystyle ⟨w,v⟩=v(w)}$.

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

${\displaystyle C^{\ast }:=\left\{v\in V^{\ast }\mid \mathrm{\forall }w\in C,v(w)\ge 0\right\}.}$

In other words, ifV* is thealgebraic dual space ofV,C* is the set of linear functionals that are nonnegative on the primal coneC. If we takeV* to be thecontinuous dual space then it is the set of continuous linear functionals nonnegative onC.^[36] This notion does not require the specification of an inner product onV.

In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous,^[37] and every continuous linear functional in an inner product space induces a linear isomorphism (nonsingular linear map) fromV* toV, and this isomorphism will take the dual cone given by the second definition, inV*, onto the one given by the first definition; see theRiesz representation theorem.^[36]

IfC is equal to its dual cone, thenC is calledself-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.

• Given a closed, convex subsetK ofHilbert spaceV, the outward normal cone to the setK at the pointx inK is given by

  ${\displaystyle N_K(x)=\left\{p\in V:\mathrm{\forall }{x}^{\ast }\in K,⟨p,x^{\ast }-x⟩\le 0\right\}.}$

• Given a closed, convex subsetK ofV, thetangent cone (orcontingent cone) to the setK at the pointx is given by

  ${\displaystyle T_K(x)=\overline{\bigcup _{h>0}\frac{K-x}{h}}.}$

• Given a closed, convex subsetK of Hilbert spaceV, thetangent cone to the setK at the pointx inK can be defined aspolar cone to outwards normal cone${\displaystyle N_K(x)}$:

  ${\displaystyle T_K(x)=N_K^o(x)\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{y\in V\mid \mathrm{\forall }\xi \in N_K(x):⟨y,\xi ⟩⩽0\}}$

Both the normal and tangent cone have the property of being closed and convex.

They are important concepts in the fields ofconvex optimization,variational inequalities andprojected dynamical systems.

IfC is a non-empty convex cone inX, then the linear span ofC is equal toC -C and the largest vector subspace ofX contained inC is equal toC ∩ (−C).^[38]

A pointed and salient convex coneC induces apartial ordering "≥" onV, defined so that${\displaystyle x\ge y}$ if and only if${\displaystyle x-y\in C.}$ (If the cone is flat, the same definition gives merely apreorder.) 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 anordered vector space. Examples include theproduct order on real-valued vectors,${\displaystyle {\mathbb{R}}^n,}$ and theLoewner order on positive semidefinite matrices. Such an ordering is commonly found insemidefinite programming.

• Cone (disambiguation)
  • Cone (geometry)
  • Cone (topology)
• Farkas' lemma
• Bipolar theorem
• Ordered vector space

• Bourbaki, Nicolas (1987).Topological Vector Spaces. Elements of Mathematics. Berlin, New York:Springer-Verlag.ISBN 978-3-540-13627-9.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Rockafellar, R. T. (1997) [1970].Convex Analysis. Princeton, NJ: Princeton University Press.ISBN 1-4008-7317-7.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
• Zălinescu, C. (2002).Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific.ISBN 981-238-067-1.MR 1921556.

• Convex analysis
• Convex geometry
• Geometric shapes
• Linear algebra

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