View source for Convex cone

{{Short description|Mathematical set closed under positive linear combinations}}
{{CS1 config|mode=cs1}}
[[Image:Convex cone illust.svg|right|thumb|A convex cone (light blue). Inside of it, the light red convex cone consists of all points {{math|''αx'' + ''βy''}} with {{math|''α'', ''β'' > 0}}, for the depicted {{math|''x''}} and {{math|''y''}}. The curves on the upper right symbolize that the regions are infinite in extent.]]
In [[linear algebra]], a '''cone'''—sometimes called a '''linear cone''' for distinguishing it from other sorts of cones—is a subset of a real [[vector space]] that is [[closure (mathematics)|closed]] under positive scalar multiplication; that is, <math>C</math> is a cone if <math>x\in C</math> implies <math>sx\in C</math> for every {{nowrap|positive scalar <math>s</math>}}. This is a broad generalization of the standard [[cone]] in [[Euclidean space]].

A '''convex cone''' is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under [[linear combination]]s with positive coefficients. It follows that convex cones are [[convex set]]s.<ref>{{Cite book |last=Boyd |first=Stephen |url=http://dx.doi.org/10.1017/cbo9780511804441 |title=Convex Optimization |last2=Vandenberghe |first2=Lieven |date=2004-03-08 |publisher=Cambridge University Press |isbn=978-0-521-83378-3}}</ref>

The definition of a convex cone makes sense in a vector space over any [[ordered field]], although the field of [[real numbers]] is used most often.

== Definition ==
A subset <math>C</math> of a vector space is a cone if <math>x\in C</math> implies <math>sx\in C</math> for every <math>s>0</math>. Here <math>s>0</math> refers to (strict) positivity in the scalar field.

=== Competing definitions ===
Some other authors require <math>[0,\infty)C\subset C</math> or even <math>0\in C</math>.
Some require a cone to be convex and/or satisfy <math>C\cap-C\subset\{0\}</math>. 

The [[conical hull]] of a set <math>C</math> is defined as the smallest ''convex'' cone that contains <math>C\cup\{0\}</math>. Therefore, it need not be the smallest cone that contains <math>C\cup\{0\}</math>.

'''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.

=== Cone: 0 or not ===
A [[subset]] <math>C</math> of a vector space <math>V</math> over an [[ordered field]] <math>F</math> is a '''cone''' (or sometimes called a '''linear cone''') if for each <math>x</math> in <math>C</math> and positive scalar <math>\alpha</math> in <math>F</math>, the product <math>\alpha x</math> is in <math>C</math>.<ref name=":0">{{Cite book|url=https://books.google.com/books?id=x7isojLkDTcC|title=Matrix Mathematics: Theory, Facts, and Formulas|last=Bernstein|first=Dennis S.|date=2009-07-26|publisher=Princeton University Press|isbn=978-0691140391|pages=97|language=en|edition=Second}}</ref> Note that some authors define '''cone''' with the scalar <math>\alpha</math> ranging over all non-negative scalars (rather than all positive scalars, which does not include 0).<ref name="Zalinescu2002">{{cite book|author=C. Zalinescu|title=Convex Analysis in General Vector Spaces|url=https://books.google.com/books?id=a91q1q1djMwC|date=1 January 2002|publisher=World Scientific|isbn=978-981-238-067-8|page=1}}</ref> Some authors even require <math>0\in C</math>, thus excluding the empty set.<ref>{{Cite book |author=W. Fenchel |title=CONVEX CONES, SETS, AND FUNCTIONS |date=September 1953 |publisher=Princeton University, Department of Mathematics|page=1}}</ref>

Therefore, <math>[0,\infty)</math> is a cone, <math>\varnothing</math> is a cone only according to the 1st and 2nd definition above, and <math>(0,\infty)</math> is a cone only according to the 1st definition above. All of them are convex (see below).

=== Convex cone ===
A cone <math>C</math> is a '''convex cone''' if <math>\alpha x+\beta y</math> belongs to <math>C</math>, for any positive scalars <math>\alpha</math>, <math>\beta</math>, and any <math>x</math>, <math>y</math> in <math>C</math>.<ref>{{Cite book|url=https://books.google.com/books?id=cX-TGJb1gfkC|title=Linear Algebra|last=Nef|first=Walter|date=1988-01-01|publisher=Courier Corporation| isbn=9780486657721 |pages=35|language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=WHjO9K6xEm4C|title=Encyclopedic Dictionary of Mathematics| last=Itô|first=Kiyosi|date=1993-01-01|publisher=MIT Press|isbn=9780262590204|language=en}}</ref> 
A cone <math>C</math> is convex if and only if <math>C+C\subseteq C</math>.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the [[rational number|rational]], [[algebraic number|algebraic]], or (more commonly) the [[real number]]s. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to <math>C</math>. Some authors use a definition that ensures the origin belongs to <math>C</math>.<ref>{{Cite book|url=https://books.google.com/books?id=jzpzBwAAQBAJ|title=Convex Analysis|last=Rockafellar|first=Ralph Tyrell|date=2015-04-29|publisher=Princeton University Press|isbn=9781400873173|pages=13|language=en}}</ref> Because of the scaling parameters <math>\alpha</math> and <math>\beta</math>, cones are infinite in extent and not bounded.

If <math>C</math> is a convex cone, then for any positive scalar <math>\alpha</math> and any <math>x</math> in <math>C</math> the vector <math>\alpha x = \tfrac{\alpha}{2}x + \tfrac{\alpha}{2}x \in C.</math> 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 under [[convex combination]]s, or just under [[addition]]. More succinctly, a set <math>C</math> is a convex cone if and only if <math>\alpha C=C</math> for every positive scalar <math>\alpha</math> and <math>C+C=C</math>.

=== Face of a convex cone ===
A '''face''' of a convex cone <math>C</math> is a subset <math>F</math> of <math>C</math> such that <math>F</math> is also a convex cone, and for any vectors <math>x,y</math> in <math>C</math> with <math>x+y</math> in <math>F</math>, <math>x</math> and <math>y</math> must both be in <math>F</math>.{{sfn | Rockafellar| 1997 | p=162}} For example, <math>C</math> itself is a face of <math>C</math>. The origin <math>\{ 0\}</math> is a face of <math>C</math> if <math>C</math> contains no line (so <math>C</math> is "strictly convex", or "salient", as defined below). The origin and <math>C</math> are sometimes called the ''trivial faces'' of <math>C</math>. A [[Line_(geometry)#Ray|ray]] (the set of nonnegative multiples of a nonzero vector) is called an '''extremal ray''' if it is a face of <math>C</math>.

Let <math>C</math> be a [[closed set|closed]], strictly convex cone in <math>\R^n</math>. Suppose that <math>C</math> is more than just the origin. Then <math>C</math> is the [[convex hull]] of its extremal rays.{{sfn | Rockafellar| 1997 | p=166}}

== Examples ==
[[File:Circular-pyramid.png|alt=convex cone circular pyramid|thumb|Convex cone that is not a conic hull of finitely many generators.]]
[[File:Polyhedral-cone.png|thumb|Convex cone generated by the conic combination of the three black vectors.]]
[[File:Cone-not-convex.png|thumb|A cone (the union of two rays) that is not a convex cone.]]

* For a vector space <math>V</math>, every [[linear subspace]] of <math>V</math> is a convex cone. In particular, the space <math>V</math> itself and the origin <math>\{ 0\}</math> are convex cones in <math>V</math>. For authors who do not require a convex cone to contain the origin, the empty set <math>\emptyset</math> is also a convex cone.
* The [[conical hull]] of a finite or infinite set of vectors in <math>\R^n</math> is a convex cone.
* The [[tangent cone]]s of a convex set are convex cones.
* The set <math display="block">\left \{ x \in \R^2 \mid x_2 \geq 0, x_1 = 0 \right \} \cup \left \{ x \in \R^2 \mid x_1 \geq 0, x_2 = 0 \right \}</math> is a cone but not a convex cone.
* The norm cone <math display="block"> C = \left \{ (x, r) \in \R^{d+1} \mid \|x\| \leq r \right \}</math> is a convex cone. (For <math>d=2</math>, this is the round cone in the figure.) Each extremal ray of <math>C</math> is spanned by a vector <math>(x,1)</math> with <math>\|x\|=1</math> (so <math>x</math> is a point in the [[n-sphere|sphere]] <math>S^{d-1}</math>). These rays are in fact the only nontrivial faces of <math>C</math>.
* 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 map]]s. In particular, if <math>C</math> is a convex cone, so is its opposite <math>-C</math>, and <math>C \cap -C</math> is the largest linear subspace contained in <math>C</math>.
* The set of [[Positive semidefinite matrix|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.<ref>{{Cite book|url=https://books.google.com/books?id=hIYKBwAAQBAJ|title=Fundamentals of Convex Analysis|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|date=2012-12-06|publisher=Springer Science & Business Media|isbn=9783642564680|language=en}}</ref> A common example is translating a convex cone by a point {{nowrap|''p'': ''p'' + ''C''}}. Technically, such transformations can produce non-cones. For example, unless {{nowrap|''p'' {{=}} 0}}, {{nowrap|''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 <math>\{ x \in V \mid f(x) = c\}</math> where f is a [[Linear form|linear functional]] on the vector space V. A '''closed [[Half-space (geometry)|half-space]]''' is a set in the form <math>\{ x \in V \mid f(x) \leq c\}</math> or <math>\{ x \in V \mid f(x) \geq c\},</math> and likewise an open half-space uses strict inequality.<ref>{{Cite book|url=https://books.google.com/books?id=4hIq6ExH7NoC|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|last1=Aliprantis|first1=Charalambos D.|last2=Border|first2=Kim C.|date=2007-05-02|publisher=Springer Science & Business Media| isbn=9783540326960| pages =197 |language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=jzpzBwAAQBAJ|title=Convex Analysis|last=Rockafellar|first=Ralph Tyrell| date=2015-04-29|publisher=Princeton University Press|isbn=9781400873173|pages=10|language=en}}</ref>

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 ===

'''Polyhedral cones''' are special kinds of cones that can be defined in several ways:<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{Rp|256–257}}

* A cone <math>C</math> is polyhedral if it is the [[conical hull]] of finitely many vectors (this property is also called '''finitely-generated''').<ref name=":03">{{Cite book|last1=Loera|first1=Jesús A. De|url=https://books.google.com/books?id=eVHg7aECS7UC|title=Algebraic and Geometric Ideas in the Theory of Discrete Optimization|last2=Hemmecke|first2=Raymond|last3=Köppe|first3=Matthias|date=2012-01-01|publisher=SIAM|isbn=9781611972443|language=en}}</ref><ref name=":1">{{Cite book|last=Schrijver|first=Alexander|url=https://books.google.com/books?id=zEzW5mhppB8C|title=Theory of Linear and Integer Programming|date=1998-07-07|publisher=John Wiley & Sons|isbn=9780471982326|language=en}}</ref> I.e., there is a set of vectors  <math>\{v_1, \ldots, v_k\}</math> so that  <math> C = \{ a_1 v_1 + \cdots + a_kv_k \mid a_i \in \R_{\ge 0}, v_i \in \R^n \}</math>. 
* 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).<ref>{{cite journal
 | last = Weyl | first = H.
 | doi = 10.1007/BF01292722
 | journal = [[Commentarii Mathematici Helvetici]]
 | pages = 290–306
 | title = Elementare Theorie der konvexen Polyeder
 | volume = 7
 | year = 1935}}</ref><ref>{{cite journal
 | last = Mirkil | first = H.
 | doi = 10.4153/CJM-1957-001-5
 | journal = Canadian Journal of Mathematics
 | mr = 83761
 | pages = 1–4
 | title = New characterizations of polyhedral cones
 | volume = 9
 | year = 1957}}</ref>
* A cone <math>C</math> is polyhedral if there is some [[matrix (mathematics)|matrix]] <math>A\in \R^{m\times n}</math> such that <math>C = \{x \in \R^n \mid Ax \geq 0 \}</math>.
* 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 <math>A</math>. 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.<ref name=":03" />  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.<ref>{{cite book|last1=Bruns|first1=Winfried|last2=Gubeladze|first2=Joseph|title=Polytopes, Rings and K-Theory|url=https://archive.org/details/polytopesringskt00albr|url-access=limited|date=2009|publisher=Springer Monographs in Mathematics|isbn=9780387763552|page=[https://archive.org/details/polytopesringskt00albr/page/n14 3]|edition=1}}</ref>

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 of [[Polyhedron#Higher-dimensional polyhedra|polyhedra]]. For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as the [[Minkowski addition|Minkowski sum]] of a [[convex polytope]] and a polyhedral cone.<ref name=":12">{{Cite book|url=https://books.google.com/books?id=zEzW5mhppB8C|title=Theory of Linear and Integer Programming| last=Schrijver |first=Alexander |date=1998-07-07|publisher=John Wiley & Sons|isbn=9780471982326|pages=88–89|language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=antqBQAAQBAJ| title=Integer Programming|last1=Conforti|first1=Michele|last2=Cornuejols|first2=Gerard|last3=Zambelli|first3=Giacomo|date=2014-11-15| publisher=Springer |isbn=9783319110080|pages=111|language=en}}</ref> Polyhedral cones also play an important part in proving the related [[Convex polytope#Finite basis theorem|Finite Basis Theorem]] for polytopes which shows that every polytope is a polyhedron and every [[Bounded set|bounded]] polyhedron is a polytope.<ref name=":12" /><ref>{{Cite book|url=https://books.google.com/books?id=X-klBQAAQBAJ&q=finite+basis+theorem+polyhedra |title=Combinatorial Optimization: Theory and Algorithms| last1=Korte| first1=Bernhard| last2=Vygen| first2=Jens|date=2013-11-11|publisher=Springer Science & Business Media|isbn=9783662217115|pages=61|language=en}}</ref><ref>{{Cite book| url=https://books.google.com/books?id=t3oZBwAAQBAJ|title=Monomial Algebras, Second Edition|last=Villarreal|first=Rafael|date=2015-03-26|publisher=CRC Press| isbn=9781482234701|pages=9|language=en}}</ref>

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 <math>n</math>-by-<math>n</math> matrices with equal row and column sums. The inequality representation requires <math>n^2</math> inequalities and <math>2n-1</math> equations, but the vector representation requires <math>n!</math> vectors (see the [[Birkhoff-von Neumann Theorem]]). The opposite can also happen - the number of vectors may be polynomial while the number of inequalities is exponential.<ref name="lp" />{{Rp|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 as [[Farkas' 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 (convex hull)|Carathéodory's theorem]] guarantees that every vector in the cone can be represented by at most <math>d</math> defining vectors, where <math>d</math> is the dimension of the space.

=== {{Anchor|pointed|blunt}}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 '''{{visible anchor|pointed}}''' if '''0''' is in ''C'', and '''{{visible anchor|blunt}}''' if '''0''' is not in ''C''.<ref name=":0" /><ref>{{Cite book|url=https://books.google.com/books?id=eP9yuZMaJjQC| title= Optimality Conditions in Convex Optimization: A Finite-Dimensional View| last1=Dhara|first1=Anulekha|last2=Dutta|first2=Joydeep|date=2011-10-17| publisher =CRC Press|isbn=9781439868225|pages=243|language=en}}</ref> Some authors use "pointed" for <math>C\cap-C=\{0\}</math> or salient (see below).<ref>https://healy.econ.ohio-state.edu/kcb/Ec181/Lecture03.pdf</ref>

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''' (or '''strictly convex''') otherwise.<ref>{{Cite book|url=https://books.google.com/books?id=go59BgAAQBAJ|title=Optimization: A Theory of Necessary Conditions| last=Neustadt| first=Lucien W.|date=2015-03-08|publisher=Princeton University Press|isbn=9781400870530|pages=6|language=en}}</ref><ref>{{Cite book| url=https://books.google.com/books?id=fdhi90F0HvcC|title=Functional Analysis: Theory and Applications|last=Edwards|first=R. E.|date=2012-10-25|publisher=Courier Corporation|isbn=9780486145105|pages=135|language=en}}</ref> 
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'''}. 
A cone ''C'' is said to be '''generating''' if <math>C-C=\{x-y\mid x\in C, y\in C\}</math> equals the whole vector space.{{sfn | Schaefer|Wolff| 1999 | pp=205–209}}

Some authors require salient cones to be pointed.<ref>{{Cite book|url=https://books.google.com/books?id=y_IJuH1bs8YC|title=Generalized Convexity and Generalized Monotonicity: Proceedings of the 6th International Symposium on Generalized Convexity/Monotonicity, Samos, September 1999| last1=Hadjisavvas| first1=Nicolas|last2=Martinez-Legaz |first2=Juan E.|last3=Penot|first3=Jean-Paul|date=2001-04-10|publisher=Springer Science & Business Media| isbn= 9783540418061| pages=238|language=en}}</ref> 
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).<ref>{{Cite book|url=https://books.google.com/books?id=cxL3jL7ONjQC|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces|last1=Bauschke|first1=Heinz H.|last2=Combettes|first2=Patrick L.|date=2011-04-19| publisher=Springer Science & Business Media|isbn=9781441994677|pages=88|language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=fGg7AAAAIAAJ|title=Introduction to Linear and Convex Programming|last=Cameron|first=Neil|date=1985-09-05|publisher=CUP Archive| isbn=9780521312073| pages=32| language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=7s_gBwAAQBAJ|title=Linear Programming: Mathematics, Theory and Algorithms| last=Panik|first=M. J.|date=2013-12-01|publisher=Springer Science & Business Media|isbn=9781461334347|pages=40|language=en}}</ref> 
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.<ref>{{Cite book|url=https://books.google.com/books?id=VxuRF4NhjqkC|title=Convex Optimization & Euclidean Distance Geometry|last=Dattorro|first=Jon|date=2005-01-01|publisher=Meboo Publishing USA|isbn=9780976401308|pages=96|language=en}}</ref><ref>{{Cite book| url=https://books.google.com/books?id=Zm3mCAAAQBAJ|title=Mainstream Mathematical Economics in the 20th Century|last=Nicola|first=PierCarlo|date=2013-03-14| publisher=Springer Science & Business Media|isbn=9783662042380|pages=125|language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=f966BQAAQBAJ| title=Harmonic Analysis on Exponential Solvable Lie Groups|last1=Fujiwara|first1=Hidenori|last2=Ludwig|first2=Jean|date=2014-12-05| publisher =Springer |isbn=9784431552888|pages=246|language=en}}</ref> 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.".<ref>{{cite journal|last1=Gubeladze|first1=Joseph|last2=Michałek|first2=Mateusz|title=The poset of rational cones|journal=Pacific Journal of Mathematics|date=1 January 2018|volume=292|issue=1|pages=103–115|doi=10.2140/pjm.2018.292.103|arxiv=1606.02083|s2cid=119639952}}</ref> This object arises when we study cones in <math>\R^d</math> together with the [[Lattice (group)|lattice]] <math>\Z^d</math>. A cone is called '''rational''' (here we assume "pointed", as defined above) whenever its generators all have [[integer]] coordinates, i.e., if <math>C</math> is a rational cone, then <math>C = \{ a_1 v_1 + \cdots + a_k v_k \mid a_i \in \R_+ \} </math> for some <math>v_i \in \Z^d</math>.

==Dual cone==
{{Main article|dual cone}}
Let ''C'' ⊂ ''V'' be a set, not necessarily 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

:<math> C^* = \{ v\in V \mid \forall w\in C, \langle w,v \rangle \ge 0 \}, </math>

which is always a convex cone. Here, <math>\langle w,v \rangle</math> is the [[dual system|duality pairing]] between ''C'' and ''V'', i.e. <math>\langle w,v\rangle = v(w)</math>.

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

:<math> C^* := \left \{ v\in V^* \mid \forall w\in C, v(w) \ge 0 \right \}. </math>

In other words, if ''V*'' is the [[algebraic dual space]] of ''V'', ''C*'' 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''.<ref name=":2">{{Cite book| url =https://books.google.com/books?id=oOYQVeHmNk4C|title=Applied Analysis|last1=Hunter|first1=John K.|last2=Nachtergaele|first2=Bruno|date=2001-01-01| publisher =World Scientific| isbn=9789810241919|pages=116|language=en}}</ref> 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,<ref>{{Cite book|url=https://books.google.com/books?id=pec4r3EsIiMC|title=A Short Course on Banach Space Theory|last=Carothers|first=N. L.|date=2005-01-01 |publisher =Cambridge University Press|isbn=9780521603720|language=en}}</ref> and every continuous linear functional in an 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]].<ref name=":2" />

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.

== Constructions ==
* Given a closed, convex subset ''K'' of [[Hilbert space]] ''V'', the ''' outward normal cone''' to the set ''K'' at the point ''x'' in ''K'' is given by
*: <math>N_K(x) = \left\{ p \in V \colon \forall x^* \in K, \left\langle p, x^* - x \right\rangle \leq 0 \right\}.</math>
* Given a closed, convex subset ''K'' of ''V'', the '''[[tangent cone]]''' (or '''contingent cone''') to the set ''K'' at the point ''x'' is given by
*: <math>T_K(x) = \overline{\bigcup_{h>0} \frac{K - x}{h}}.</math>
* Given a closed, convex subset ''K'' of Hilbert space ''V'', the '''tangent cone''' to the set ''K'' at the point ''x'' in ''K'' can be defined as [[dual cone and polar cone#Polar cone|polar cone]] to outwards normal cone <math>N_K(x)</math>:
*: <math>T_K(x) = N_K^o(x) \ \overset{\underset{\mathrm{def}}{}}{=}\  \{y \in V \mid \forall \xi \in N_K(x): \langle y, \xi \rangle \leqslant 0 \}</math>
 
Both the normal and tangent cone have the property of being closed and convex.

They are important concepts in the fields of [[convex optimization]], [[variational inequality|variational inequalities]] and [[projected dynamical system]]s.

== Properties ==

If ''C'' is a non-empty convex cone in ''X'', then the linear span of ''C'' is equal to ''C'' - ''C'' and the largest vector subspace of ''X'' contained in ''C'' is equal to ''C'' ∩ (−''C'').{{sfn |Narici|Beckenstein | 2011 | pp=149-153}}

==Partial order defined by a convex cone==
A pointed and salient convex cone ''C'' induces a [[partial ordering]] "≥" on ''V'', defined so that <math>x\geq y</math> if and only if <math>x-y\in C.</math> (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, <math>\R^n,</math> and the [[Loewner order]] on positive semidefinite matrices. Such an ordering is commonly found in [[semidefinite programming]].

== See also ==

* [[Cone (disambiguation)]]
** [[Cone (geometry)]]
** [[Cone (topology)]]
* [[Farkas' lemma]]
* [[Bipolar theorem]]
* [[Ordered vector space]]

==Notes==
{{reflist|30em}}

==References==

* {{Cite book| last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | location= Berlin, New York | series=Elements of Mathematics | isbn=978-3-540-13627-9 | year=1987 | url=https://books.google.com/books?id=S4wnAQAAIAAJ }}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{cite book | last=Rockafellar | first=R. T. | author-link=R. Tyrrell Rockafellar | title=Convex Analysis |publisher=Princeton University Press | location=Princeton, NJ | orig-year=1970 | year=1997 | isbn=1-4008-7317-7 |url=https://books.google.com/books?id=1TiOka9bx3sC }}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* {{cite book | last=Zălinescu | first=C. | title=Convex Analysis in General Vector Spaces | publisher=World Scientific | location= River Edge,&nbsp;NJ | year=2002 |isbn=981-238-067-1 | mr=1921556 | url=https://books.google.com/books?id=jcbUCgAAQBAJ }}

{{Functional analysis}}
{{Topological vector spaces}}

{{DEFAULTSORT:Convex Cone}}
[[Category:Convex analysis]]
[[Category:Convex geometry]]
[[Category:Geometric shapes]]
[[Category:Linear algebra]]