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====Cone in an Euclidean space (by A.B. Ivanov)====
  
 
A cone in an [[Euclidean space]] is a set $K$ consisting of [[Half-line (ray)|half-lines]] emanating from some point $0$, the vertex of the cone. The boundary $\partial K$ of $K$ (consisting of half-lines called generators of the cone) is part of a [[conical surface]], and is sometimes also called a cone. Finally, the intersection of $K$ with a half-space containing $0$ and bounded by a plane not passing through $0$ is often called a cone. In this case the part of the plane lying inside the conical surface is called the base of the cone and the part of the conical surface between the base and the vertex is called the lateral surface of the cone.
 
A cone in an [[Euclidean space]] is a set $K$ consisting of [[Half-line (ray)|half-lines]] emanating from some point $0$, the vertex of the cone. The boundary $\partial K$ of $K$ (consisting of half-lines called generators of the cone) is part of a [[conical surface]], and is sometimes also called a cone. Finally, the intersection of $K$ with a half-space containing $0$ and bounded by a plane not passing through $0$ is often called a cone. In this case the part of the plane lying inside the conical surface is called the base of the cone and the part of the conical surface between the base and the vertex is called the lateral surface of the cone.
  
If the base of the cone is a disc, then the cone is called circular. A circular cone is called straight if the orthogonal projection of its vertex onto the plane of the base is the center of the base. The straight line passing through the vertex of a cone and perpendicular to the base is called the axis of the cone and the segment of it between the vertex and the base is the height of the cone. The volume of a straight circular cone is equal to $\frac{1}{3}\pi R^2 h$, where $h$ is the height, and $R$ is the radius of the base; the area of the lateral surface is equal to $\pi Rl$, where $l$ is the length of the segment of a generator between the vertex and the base. A subset of a cone contained between two parallel planes is called a truncated cone or a conical frustum. The frustum of a straight circular cone between planes parallel to the base has volume $\frac{1}{3}\pi (R^2 + r^2 + Rr) h$, where $R$, $r$ are the radii of the base and $h$ is the height (the distance between the base); the area of the lateral surface is $\pi (R+r)l$, where $l$ is the length of the segment of a generator.
 
  
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=====Circular cone=====
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If the base of the cone is a [[disc]], then the cone is called circular <ref group="comment" name="Right circular cone" />. A circular cone is called straight if the [[Orthogonal projector|orthogonal projection]] of its vertex onto the plane of the base is the center of the base. The [[straight line]] passing through the vertex of a cone and perpendicular to the base is called the axis of the cone and the segment of it between the vertex and the base is the height of the cone. The volume of a straight circular cone is equal to $\frac{1}{3}\pi R^2 h$, where $h$ is the height, and $R$ is the radius of the base; the area of the lateral surface is equal to $\pi Rl$, where $l$ is the length of the segment of a generator between the vertex and the base. A subset of a cone contained between two parallel planes is called a truncated cone or a conical frustum. The frustum of a straight circular cone between planes parallel to the base has volume $\frac{1}{3}\pi (R^2 + r^2 + Rr) h$, where $R$, $r$ are the radii of the base and $h$ is the height (the distance between the base); the area of the lateral surface is $\pi (R+r)l$, where $l$ is the length of the segment of a generator.
  
  
====Comments====
 
  
A right circular cone is also called a cone of revolution. Instead of truncated cone or conical frustum, the term frustum of a cone may be encountered.
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====Cone over a topological space (by M.I. Voitsekhovskii)====
  
 
A cone over a [[topological space]] $X$ (the base of the cone) is the space $CX$ obtained from the product $X\times [0,1]$ by collapsing the subspace $X\times\{0\}$ to a point $W$ (the vertex of the cone):
 
A cone over a [[topological space]] $X$ (the base of the cone) is the space $CX$ obtained from the product $X\times [0,1]$ by collapsing the subspace $X\times\{0\}$ to a point $W$ (the vertex of the cone):
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CX=(X\times [0,1])/(X\times\{0\})
 
CX=(X\times [0,1])/(X\times\{0\})
 
\end{equation}
 
\end{equation}
In other words, $CX$ is the cylinder of the constant mapping $X\rightarrow W$ (see [[Cylindrical construction]]) or the cone of the identity mapping $id\colon X\rightarrow X$ (see [[Mapping-cone construction]]). The space $X$ is contractible if and only if it is a retract of every cone over $X$ (cf. [[Retract of a topological space]]).
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In other words, $CX$ is the [[cylinder]] of the constant mapping $X\rightarrow W$ (see [[Cylindrical construction]]) or the cone of the identity mapping $id\colon X\rightarrow X$ (see [[Mapping-cone construction]]). The space $X$ is contractible if and only if it is a retract of every cone over $X$ (cf. [[Retract of a topological space]]).
  
The notion of a cone over a topological space can be generalized in the framework of category theory: A set of morphisms $\alpha_i\colon A\rightarrow A_i$, $i\in I$, of an arbitrary category $\mathfrak{A}$ with common initial object $A$ is called a morphism cone with vertex $A$. Dually one defines a morphism cocone as a set of morphisms $\beta_i\colon A_i\rightarrow A$, $i\in I$, with common final object $A$. See ''M.I. Voitsekhovskii''.
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The notion of a cone over a topological space can be generalized in the framework of [[category]] theory: A set of [[morphism]]s $\alpha_i\colon A\rightarrow A_i$, $i\in I$, of an arbitrary category $\mathfrak{A}$ with common initial object $A$ is called a morphism cone with vertex $A$. Dually one defines a morphism cocone as a set of morphisms $\beta_i\colon A_i\rightarrow A$, $i\in I$, with common final object $A$. See <ref name ="Dold" /> <ref name ="Spanier" /> <ref name ="Tsalenko" />.
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====Mapping cone (by A.F. Kharshildaze)====
  
 
A mapping cone is a topological space associated with a continuous mapping $f\colon X\rightarrow Y$ of topological spaces by the [[mapping-cone construction]]. Let $C_1$ be the cone of the imbedding $Y\subset C_f$, let $C_2$ be the cone of the imbedding $C_f\subset C_1$, etc., where $C_f$ is the mapping cone of $f$. Then the sequence
 
A mapping cone is a topological space associated with a continuous mapping $f\colon X\rightarrow Y$ of topological spaces by the [[mapping-cone construction]]. Let $C_1$ be the cone of the imbedding $Y\subset C_f$, let $C_2$ be the cone of the imbedding $C_f\subset C_1$, etc., where $C_f$ is the mapping cone of $f$. Then the sequence
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X\overset{f}{\rightarrow} Y\subset C_f\subset C_1\subset C_2\ldots
 
X\overset{f}{\rightarrow} Y\subset C_f\subset C_1\subset C_2\ldots
 
\end{equation}
 
\end{equation}
so obtained is called the Puppe sequence; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459046.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459047.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459048.png" />) is the [[Suspension|suspension]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459049.png" /> (respectively, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459050.png" />).
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so obtained is called the Puppe sequence; here $C_1\sim SX$, $X_2\sim SY$, etc., where $SX$ (respectively, $SY$) is the [[suspension]] over $X$ (respectively, over $Y$).
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One defines in an analogous way the reduced mapping cone $\tilde{C}_f$ of a mapping of [[pointed space]]s. Here, as for a [[cofibration]], for any pointed space $A$, the sequence of homotopy classes induced by the Puppe sequence
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\begin{equation}
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[X,A]\leftarrow[Y,A]\leftarrow[C_1,A]\leftarrow[C_2,A]\leftarrow\ldots
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\end{equation}
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is exact; all the terms in it starting from the fourth are groups and starting from the seventh, [[Abelian group]]s. See <ref name ="Dold" /> <ref name ="Spanier" />.
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One defines in an analogous way the reduced mapping cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459051.png" /> of a mapping of pointed spaces. Here, as for a cofibration, for any pointed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459052.png" />, the sequence of homotopy classes induced by the Puppe sequence
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459053.png" /></td> </tr></table>
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====Cone in a real vector space  (by M.I. Voitsekhovskii)====
  
is exact; all the terms in it starting from the fourth are groups and starting from the seventh, Abelian groups. See , .
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A cone in a real vector space $E$ is a set $K\in E$ such that $\lambda K\in K$ for any $\lambda >0$.
  
''A.F. Kharshiladze''
 
  
A cone in a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459054.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459056.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459057.png" />. A cone is called pointed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459058.png" /> and a pointed cone is called salient if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459059.png" /> contains no one-dimensional subspace. A non-salient cone is sometimes called a wedge.
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=====Pointed cone=====
  
A cone that is a convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459060.png" /> is called convex. Thus, a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459062.png" /> is a convex cone if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459063.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459065.png" />. In this case the vector subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459066.png" /> generated by the convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459067.png" /> is the same as the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459068.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459069.png" /> is pointed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459070.png" /> is the largest vector subspace contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459071.png" />. A pointed convex cone is salient if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459072.png" />.
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A cone is called pointed if $0\in K$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459073.png" /> is a (partially) ordered vector space, then the positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459074.png" /> is a salient pointed convex cone. Conversely, any such a convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459075.png" /> induces an order relation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459076.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459077.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459078.png" />.
 
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459079.png" /> is said to be reproducing if any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459080.png" /> can be expressed as a difference of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459081.png" />. For example, the cone of non-negative continuous (or summable) functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459082.png" /> is reproducing; so also is the set of positive operators in the space of bounded self-adjoint operators acting on a Hilbert space. However, the cone of non-negative non-decreasing continuous functions is not reproducing.
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=====Salient cone=====
  
The presence of a topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459083.png" /> provides the notion of a cone with a richer content enabling one to obtain non-trivial results. For example, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459084.png" /> is a separable locally convex space and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459085.png" /> is a salient pointed convex cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459086.png" /> having a non-empty interior (such cones are called solid). Then every linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459087.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459088.png" /> that is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459089.png" /> is continuous (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459090.png" /> is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459091.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459092.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459093.png" />); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459094.png" /> is a vector subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459095.png" /> having a non-empty intersection with the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459097.png" /> is a linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459098.png" /> that is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459099.png" />, then there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590100.png" /> a linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590101.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590102.png" /> that is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590103.png" />. See , , .
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A pointed cone is called salient if $K$ contains no one-dimensional subspace. A non-salient cone is sometimes called a [[Wedge (in a vector space)|wedge]].
  
''M.I. Voitsekhovskii''
 
  
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=====Convex cone=====
  
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A cone that is a convex subset of $E$ is called convex. Thus, a subset $K$ of $E$ is a [[convex cone]] if and only if $\lambda K\in K$ for any $\lambda >0$ and $K+K\subset K$. In this case the vector subspace of $E$ generated by the convex cone $K$ is the same as the set $K-K$. If $K$ is pointed, then $K\cap (-K)$ is the largest vector subspace contained in $K$. A pointed convex cone is salient if and only if $K\cap (-K)=0$.
  
====Comments====
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If $E$ is a (partially) ordered vector space, then the positive cone $P=\{x\colon x\in E,\; x\geq 0\}$ is a salient pointed convex cone. Conversely, any such a convex cone $K$ induces an [[order relation]] in $E$: $x_1\geq x_2$ if $x_1-x_2\in K$.
  
A reproducing cone is also called a generating cone.
 
  
The theory of cones in Banach spaces is more thoroughly developed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590104.png" /> be a cone in the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590105.png" /> inducing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590106.png" /> an order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590107.png" />. If the cone is closed, then the Archimedean principle holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590108.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590109.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590111.png" />, are numbers and if there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590112.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590113.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590114.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590115.png" />. For a solid cone the converse also holds: If the Archimedean property holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590116.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590117.png" /> is closed.
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=====Reproducing cone=====
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590118.png" /> be the dual wedge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590119.png" />, that is, the collection of all positive linear continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590120.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590121.png" /> is positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590122.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590123.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590124.png" /> is a cone if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590125.png" /> is spatial, that is, if the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590127.png" /> is closed, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590128.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590129.png" />) there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590130.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590131.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590132.png" />).
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A cone $K$ is said to be reproducing <ref group="comment" name="Reproducing cone" /> if any element $x\in E$ can be expressed as a difference of elements of $K$. For example, the cone of non-negative continuous (or summable) functions on the interval $[0,1]$ is reproducing; so also is the set of positive operators in the space of bounded [[self-adjoint operator]]s acting on a [[Hilbert space]]. However, the cone of non-negative non-decreasing continuous functions is not reproducing.
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590133.png" /> is called unflattened if there exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590134.png" /> elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590135.png" /> such that
+
The presence of a [[Topology, general|topology]] in a vector space $E$ provides the notion of a cone with a richer content enabling one to obtain non-trivial results. For example, suppose that $E$ is a separable locally convex space and that $K$ is a salient pointed convex cone in $E$ having a non-empty interior (such cones are called solid). Then every linear form $f$ on $E$ that is positive on $K$ is continuous ($f$ is positive on $K$ if $f(x)\geq 0$ for $x\in K$); if $M$ is a vector subspace of $E$ having a non-empty intersection with the interior of $K$ and $f$ is a linear form on $M$ that is positive on $K\cap M$, then there exists on $E$ a linear form $\tilde{f}$ extending $f$ that is positive on $K$. See <ref name ="MRE" /> <ref name ="Edwards" /> <ref name ="Schaefer" />.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590136.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590137.png" /> is a constant.
 
  
If a cone is closed and reproducing, then it is unflattened (the Krein–Shmul'yan theorem).
+
====Cone in a Banach space (by B.Z. Vulikh)====
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590138.png" /> is called normal if
+
The theory of cones in [[Banach space]]s is more thoroughly developed <ref group="comment" name="Cones in Banach spaces"/>. Let $K$ be a cone in the Banach space $E$ inducing in $E$ an order relation $\geq$. If the cone is closed, then the [[Archimedean axiom|Archimedean principle]] holds for $E$: If $x\in E$, if $\lambda_n>0$, $\lambda_n\rightarrow\infty$, are numbers and if there exists a point $y$ such that $\lambda_n x\leq y$ for all $n$, then $x\leq 0$. For a solid cone the converse also holds: If the Archimedean property holds for $E$, then $K$ is closed.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590139.png" /></td> </tr></table>
+
Let $K'$ be the dual wedge to $K$, that is, the collection of all positive linear continuous functions on $E$ ($f$ is positive if $f(x)\geq 0$ for any $x\in K$). Then $K'$ is a cone if and only if $K$ is spatial <ref group="comment" name ="Spatial cone" />, that is, if the closure $\overline{K-K}=E$. If $K$ is closed, then for any $x_0>0$ (respectively, $x_0\notin K$) there exists an $f\in K'$ such that $f(x_0)>0$ (respectively, $f(x_0)<0$).
  
Normality of a cone is equivalent to semi-monotonicity of the norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590140.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590142.png" /> is a constant. In order that a wedge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590143.png" /> be reproducing in the dual space, it is necessary and sufficient that the cone be normal (Krein's theorem). Dually: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590144.png" /> is the normal cone corresponding to a closed cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590145.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590146.png" /> is reproducing. There exists a one-to-one linear continuous mapping of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590147.png" /> with a normal cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590148.png" /> into a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590149.png" /> of continuous functions on some compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590150.png" /> under which the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590151.png" />, and only these, are taken to non-negative functions.
+
=====Unflattened cone=====
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590152.png" /> is called regular (completely regular) if every sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590153.png" /> that is increasing and order bounded (norm bounded) converges. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590154.png" /> is closed and regular, then it is normal; every completely-regular cone is normal and regular. If in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590155.png" /> is regular and solid, then it is completely regular. The regularity of a cone is related to the order continuity of the norm: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590156.png" />, that is, if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590157.png" /> is a decreasing directed set, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590158.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590159.png" />. The regularity of a closed cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590160.png" /> is equivalent to the property that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590161.png" /> is Dedekind complete and that the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590162.png" /> is order continuous. The regularity of a solid cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590163.png" /> implies the order continuity of the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590164.png" />.
+
A cone $K$ is called unflattened if there exists for any $x\in E$ elements $u,v\in K$ such that
 +
\begin{equation}
 +
x=u-v,\quad\lVert u\rVert,\lVert v\rVert\leq M\lVert x\rVert,
 +
\end{equation}
 +
where $M$ is a constant.
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590165.png" /> is called plasterable if there exists a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590166.png" /> and a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590167.png" /> such that the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590168.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590169.png" />. The plasterability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590170.png" /> is equivalent to the existence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590171.png" /> of an equivalent norm that is additive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590172.png" />. A plasterable cone is completely regular.
+
If a cone is closed and reproducing, then it is unflattened (the Krein–Shmul'yan theorem).
  
The theory of cones has also been developed for arbitrary normed spaces. However, in the general case, some of the above-mentioned results no longer hold, for example, the Krein–Shmul'yan theorem is no longer true, and the regularity of a closed cone no longer implies its normality. See , , , , .
 
  
''B.Z. Vulikh''
+
=====Normal cone=====
  
 +
A cone $K$ is called normal if
 +
\begin{equation}
 +
\inf\{\lVert x+y\rVert\colon x,y\in K,\lVert x\rVert=\lVert y\rVert=1 \}>0.
 +
\end{equation}
  
 +
Normality of a cone is equivalent to semi-monotonicity of the norm: $0\leq y\leq x$ implies $\lVert y\rVert\leq M\lVert x\rVert$, where $M$ is a constant. In order that a wedge $K'$ be reproducing in the dual space, it is necessary and sufficient that the cone be normal (Krein's theorem). Dually: If $K'$ is the normal cone corresponding to a closed cone $K$, then $K$ is reproducing. There exists a one-to-one linear [[continuous mapping]] of a space $E$ with a normal cone $K$ into a subspace of the space $C(Q)$ of continuous functions on some compactum $Q$ under which the elements of $K$, and only these, are taken to non-negative functions.
  
====Comments====
 
  
A spatial cone (or wedge) is also called a spanning cone (spanning wedge).
+
=====Regular cone=====
 +
 
 +
A cone $K$ is called regular (completely regular) if every sequence of elements of $K$ that is increasing and order bounded (norm bounded) converges. If $K$ is closed and regular, then it is normal; every completely-regular cone is normal and regular. If in fact $K$ is regular and solid, then it is completely regular. The regularity of a cone is related to the order continuity <ref group="comment" name="Order continuity" /> of the norm: If $x_\alpha\downarrow 0$, that is, if the family $\{x_\alpha\}$ is a decreasing [[directed set]], and if $\inf x_\alpha=0$, then $\lVert x_\alpha\rVert\rightarrow 0$. The regularity of a closed cone $K$ is equivalent to the property that the space $E$ is [[Dedekind completion|Dedekind complete]] and that the norm in $E$ is order continuous. The regularity of a solid cone $K$ implies the order continuity of the norm in $E$.
  
Order continuity is sometimes called monotone continuity.
 
  
Cones in Banach spaces are used in optimization theory. They can be used to define multi-valued derivatives of non-smooth mappings.
+
=====Plasterable cone=====
  
 +
A cone $K$ is called plasterable if there exists a cone $K_1\subset X$ and a number $\delta>0$ such that the ball $S(x; \delta\lVert x\rVert)\subset K_1$ for any $x\in K$. The plasterability of $K$ is equivalent to the existence in $E$ of an equivalent norm that is additive on $K$. A plasterable cone is completely regular.
  
 +
The theory of cones has also been developed for arbitrary normed spaces. However, in the general case, some of the above-mentioned results no longer hold, for example, the Krein–Shmul'yan theorem is no longer true, and the regularity of a closed cone no longer implies its normality. See <ref name ="MRE" /> <ref name ="Krasnosel'skii" /> <ref name ="Vulikh" /> <ref name ="VulikhA" /> <ref name ="Krein" />.
  
====References====
 
  
<table>
+
====Comments====
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Hiriart-Urruty,  "Tangent cones, generalized gradients and mathematical programming in Banach spaces"  ''Mathematics of Operations Research'' , '''4'''  (1979)  pp. 79–97</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.B. Holmes,  "Geometric functional analysis and its applications" , Springer  (1975)</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.L. Peressini,  "Ordered topological vector spaces" , Harper &amp; Row  (1967)</TD></TR>
 
<TR><TD valign="top">[a4]</TD> <TD valign="top">  V. Barbu,  Th. Precupanu,  "Convexity and optimization in Banach spaces" , Reidel  (1986)</TD></TR>
 
</table>
 
  
General references for this article can be found below.
+
<references group="comment">
 +
<ref name="Right circular cone">A right circular cone is also called a cone of revolution. Instead of truncated cone or conical frustum, the term frustum of a cone may be encountered.</ref>
 +
<ref name="Reproducing cone">A reproducing cone is also called a generating cone.</ref>
 +
<ref name="Cones in Banach spaces">Cones in Banach spaces are used in optimization theory. They can be used to define multi-valued derivatives of non-smooth mappings.</ref>
 +
<ref name ="Spatial cone">A spatial cone (or wedge) is also called a spanning cone (or wedge).</ref>
 +
<ref name="Order continuity">Order continuity is sometimes called monotone continuity.</ref>
 +
</references>
  
  
Line 104: Line 124:
 
====References====
 
====References====
  
<table>
+
<references>
<TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Functional analysis'' , Math. Reference Library , Moscow  (1972)  pp. Chapt. 8  (In Russian)</TD></TR>
+
<ref name ="MRE">''Functional analysis'' , Mathematical Reference Library , Moscow  (1972)  pp. Chapt. 8  (In Russian)</ref>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR>
+
<ref name ="Edwards">R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</ref>
<TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR>
+
<ref name ="Schaefer">H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</ref>
<TR><TD valign="top">[4]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR>
+
<ref name ="Dold">A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</ref>
<TR><TD valign="top">[5]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR>
+
<ref name ="Spanier">E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</ref>
<TR><TD valign="top">[6]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR>
+
<ref name ="Tsalenko">M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</ref>
<TR><TD valign="top">[7]</TD> <TD valign="top"> M.A. Krasnosel'skii,  "Positive solutions of operator equations" , Wolters-Noordhoff  (1964)  (Translated from Russian)</TD></TR>
+
<ref name ="Krasnosel'skii">M.A. Krasnosel'skii,  "Positive solutions of operator equations" , Wolters-Noordhoff  (1964)  (Translated from Russian)</ref>
<TR><TD valign="top">[8]</TD> <TD valign="top">  B.Z. Vulikh,  "Introduction to the theory of cones in normed spaces" , Kalinin  (1977)  (In Russian)</TD></TR>
+
<ref name ="Vulikh">B.Z. Vulikh,  "Introduction to the theory of cones in normed spaces" , Kalinin  (1977)  (In Russian)</ref>
<TR><TD valign="top">[9]</TD> <TD valign="top">  B.Z. Vulikh,  "Special questions in the geometry of cones in normed spaces" , Kalinin  (1977)  (In Russian)</TD></TR>
+
<ref name ="VulikhA">B.Z. Vulikh,  "Special questions in the geometry of cones in normed spaces" , Kalinin  (1977)  (In Russian)</ref>
<TR><TD valign="top">[10]</TD> <TD valign="top">  M.G. Krein,  M.A. Rutman,  "Linear operators leaving invariant a cone in a Banach space"  ''Uspekhi Mat. Nauk'' , '''3''' :  1(23)  (1948)  pp. 3–95  (In Russian)</TD></TR>
+
<ref name ="Krein">M.G. Krein,  M.A. Rutman,  "Linear operators leaving invariant a cone in a Banach space"  ''Uspekhi Mat. Nauk'' , '''3''' :  1(23)  (1948)  pp. 3–95  (In Russian)</ref>
</table>
+
</references>
 +
<ol start="11">
 +
<li>J.B. Hiriart-Urruty,  "Tangent cones, generalized gradients and mathematical programming in Banach spaces"  ''Mathematics of Operations Research'' , '''4'''  (1979)  pp. 79–97</li>
 +
<li>R.B. Holmes,  "Geometric functional analysis and its applications" , Springer  (1975)</li>
 +
<li>A.L. Peressini,  "Ordered topological vector spaces" , Harper &amp; Row  (1967)</li>
 +
<li>V. Barbu,  Th. Precupanu,  "Convexity and optimization in Banach spaces" , Reidel  (1986)</li>
 +
</ol>

Latest revision as of 13:05, 10 June 2016


Cone in an Euclidean space (by A.B. Ivanov)

A cone in an Euclidean space is a set $K$ consisting of half-lines emanating from some point $0$, the vertex of the cone. The boundary $\partial K$ of $K$ (consisting of half-lines called generators of the cone) is part of a conical surface, and is sometimes also called a cone. Finally, the intersection of $K$ with a half-space containing $0$ and bounded by a plane not passing through $0$ is often called a cone. In this case the part of the plane lying inside the conical surface is called the base of the cone and the part of the conical surface between the base and the vertex is called the lateral surface of the cone.


Circular cone

If the base of the cone is a disc, then the cone is called circular [comment 1]. A circular cone is called straight if the orthogonal projection of its vertex onto the plane of the base is the center of the base. The straight line passing through the vertex of a cone and perpendicular to the base is called the axis of the cone and the segment of it between the vertex and the base is the height of the cone. The volume of a straight circular cone is equal to $\frac{1}{3}\pi R^2 h$, where $h$ is the height, and $R$ is the radius of the base; the area of the lateral surface is equal to $\pi Rl$, where $l$ is the length of the segment of a generator between the vertex and the base. A subset of a cone contained between two parallel planes is called a truncated cone or a conical frustum. The frustum of a straight circular cone between planes parallel to the base has volume $\frac{1}{3}\pi (R^2 + r^2 + Rr) h$, where $R$, $r$ are the radii of the base and $h$ is the height (the distance between the base); the area of the lateral surface is $\pi (R+r)l$, where $l$ is the length of the segment of a generator.


Cone over a topological space (by M.I. Voitsekhovskii)

A cone over a topological space $X$ (the base of the cone) is the space $CX$ obtained from the product $X\times [0,1]$ by collapsing the subspace $X\times\{0\}$ to a point $W$ (the vertex of the cone): \begin{equation} CX=(X\times [0,1])/(X\times\{0\}) \end{equation} In other words, $CX$ is the cylinder of the constant mapping $X\rightarrow W$ (see Cylindrical construction) or the cone of the identity mapping $id\colon X\rightarrow X$ (see Mapping-cone construction). The space $X$ is contractible if and only if it is a retract of every cone over $X$ (cf. Retract of a topological space).

The notion of a cone over a topological space can be generalized in the framework of category theory: A set of morphisms $\alpha_i\colon A\rightarrow A_i$, $i\in I$, of an arbitrary category $\mathfrak{A}$ with common initial object $A$ is called a morphism cone with vertex $A$. Dually one defines a morphism cocone as a set of morphisms $\beta_i\colon A_i\rightarrow A$, $i\in I$, with common final object $A$. See [1] [2] [3].


Mapping cone (by A.F. Kharshildaze)

A mapping cone is a topological space associated with a continuous mapping $f\colon X\rightarrow Y$ of topological spaces by the mapping-cone construction. Let $C_1$ be the cone of the imbedding $Y\subset C_f$, let $C_2$ be the cone of the imbedding $C_f\subset C_1$, etc., where $C_f$ is the mapping cone of $f$. Then the sequence \begin{equation} X\overset{f}{\rightarrow} Y\subset C_f\subset C_1\subset C_2\ldots \end{equation} so obtained is called the Puppe sequence; here $C_1\sim SX$, $X_2\sim SY$, etc., where $SX$ (respectively, $SY$) is the suspension over $X$ (respectively, over $Y$).

One defines in an analogous way the reduced mapping cone $\tilde{C}_f$ of a mapping of pointed spaces. Here, as for a cofibration, for any pointed space $A$, the sequence of homotopy classes induced by the Puppe sequence \begin{equation} [X,A]\leftarrow[Y,A]\leftarrow[C_1,A]\leftarrow[C_2,A]\leftarrow\ldots \end{equation} is exact; all the terms in it starting from the fourth are groups and starting from the seventh, Abelian groups. See [1] [2].


Cone in a real vector space (by M.I. Voitsekhovskii)

A cone in a real vector space $E$ is a set $K\in E$ such that $\lambda K\in K$ for any $\lambda >0$.


Pointed cone

A cone is called pointed if $0\in K$.


Salient cone

A pointed cone is called salient if $K$ contains no one-dimensional subspace. A non-salient cone is sometimes called a wedge.


Convex cone

A cone that is a convex subset of $E$ is called convex. Thus, a subset $K$ of $E$ is a convex cone if and only if $\lambda K\in K$ for any $\lambda >0$ and $K+K\subset K$. In this case the vector subspace of $E$ generated by the convex cone $K$ is the same as the set $K-K$. If $K$ is pointed, then $K\cap (-K)$ is the largest vector subspace contained in $K$. A pointed convex cone is salient if and only if $K\cap (-K)=0$.

If $E$ is a (partially) ordered vector space, then the positive cone $P=\{x\colon x\in E,\; x\geq 0\}$ is a salient pointed convex cone. Conversely, any such a convex cone $K$ induces an order relation in $E$: $x_1\geq x_2$ if $x_1-x_2\in K$.


Reproducing cone

A cone $K$ is said to be reproducing [comment 2] if any element $x\in E$ can be expressed as a difference of elements of $K$. For example, the cone of non-negative continuous (or summable) functions on the interval $[0,1]$ is reproducing; so also is the set of positive operators in the space of bounded self-adjoint operators acting on a Hilbert space. However, the cone of non-negative non-decreasing continuous functions is not reproducing.

The presence of a topology in a vector space $E$ provides the notion of a cone with a richer content enabling one to obtain non-trivial results. For example, suppose that $E$ is a separable locally convex space and that $K$ is a salient pointed convex cone in $E$ having a non-empty interior (such cones are called solid). Then every linear form $f$ on $E$ that is positive on $K$ is continuous ($f$ is positive on $K$ if $f(x)\geq 0$ for $x\in K$); if $M$ is a vector subspace of $E$ having a non-empty intersection with the interior of $K$ and $f$ is a linear form on $M$ that is positive on $K\cap M$, then there exists on $E$ a linear form $\tilde{f}$ extending $f$ that is positive on $K$. See [4] [5] [6].


Cone in a Banach space (by B.Z. Vulikh)

The theory of cones in Banach spaces is more thoroughly developed [comment 3]. Let $K$ be a cone in the Banach space $E$ inducing in $E$ an order relation $\geq$. If the cone is closed, then the Archimedean principle holds for $E$: If $x\in E$, if $\lambda_n>0$, $\lambda_n\rightarrow\infty$, are numbers and if there exists a point $y$ such that $\lambda_n x\leq y$ for all $n$, then $x\leq 0$. For a solid cone the converse also holds: If the Archimedean property holds for $E$, then $K$ is closed.

Let $K'$ be the dual wedge to $K$, that is, the collection of all positive linear continuous functions on $E$ ($f$ is positive if $f(x)\geq 0$ for any $x\in K$). Then $K'$ is a cone if and only if $K$ is spatial [comment 4], that is, if the closure $\overline{K-K}=E$. If $K$ is closed, then for any $x_0>0$ (respectively, $x_0\notin K$) there exists an $f\in K'$ such that $f(x_0)>0$ (respectively, $f(x_0)<0$).

Unflattened cone

A cone $K$ is called unflattened if there exists for any $x\in E$ elements $u,v\in K$ such that \begin{equation} x=u-v,\quad\lVert u\rVert,\lVert v\rVert\leq M\lVert x\rVert, \end{equation} where $M$ is a constant.

If a cone is closed and reproducing, then it is unflattened (the Krein–Shmul'yan theorem).


Normal cone

A cone $K$ is called normal if \begin{equation} \inf\{\lVert x+y\rVert\colon x,y\in K,\lVert x\rVert=\lVert y\rVert=1 \}>0. \end{equation}

Normality of a cone is equivalent to semi-monotonicity of the norm: $0\leq y\leq x$ implies $\lVert y\rVert\leq M\lVert x\rVert$, where $M$ is a constant. In order that a wedge $K'$ be reproducing in the dual space, it is necessary and sufficient that the cone be normal (Krein's theorem). Dually: If $K'$ is the normal cone corresponding to a closed cone $K$, then $K$ is reproducing. There exists a one-to-one linear continuous mapping of a space $E$ with a normal cone $K$ into a subspace of the space $C(Q)$ of continuous functions on some compactum $Q$ under which the elements of $K$, and only these, are taken to non-negative functions.


Regular cone

A cone $K$ is called regular (completely regular) if every sequence of elements of $K$ that is increasing and order bounded (norm bounded) converges. If $K$ is closed and regular, then it is normal; every completely-regular cone is normal and regular. If in fact $K$ is regular and solid, then it is completely regular. The regularity of a cone is related to the order continuity [comment 5] of the norm: If $x_\alpha\downarrow 0$, that is, if the family $\{x_\alpha\}$ is a decreasing directed set, and if $\inf x_\alpha=0$, then $\lVert x_\alpha\rVert\rightarrow 0$. The regularity of a closed cone $K$ is equivalent to the property that the space $E$ is Dedekind complete and that the norm in $E$ is order continuous. The regularity of a solid cone $K$ implies the order continuity of the norm in $E$.


Plasterable cone

A cone $K$ is called plasterable if there exists a cone $K_1\subset X$ and a number $\delta>0$ such that the ball $S(x; \delta\lVert x\rVert)\subset K_1$ for any $x\in K$. The plasterability of $K$ is equivalent to the existence in $E$ of an equivalent norm that is additive on $K$. A plasterable cone is completely regular.

The theory of cones has also been developed for arbitrary normed spaces. However, in the general case, some of the above-mentioned results no longer hold, for example, the Krein–Shmul'yan theorem is no longer true, and the regularity of a closed cone no longer implies its normality. See [4] [7] [8] [9] [10].


Comments

  1. A right circular cone is also called a cone of revolution. Instead of truncated cone or conical frustum, the term frustum of a cone may be encountered.
  2. A reproducing cone is also called a generating cone.
  3. Cones in Banach spaces are used in optimization theory. They can be used to define multi-valued derivatives of non-smooth mappings.
  4. A spatial cone (or wedge) is also called a spanning cone (or wedge).
  5. Order continuity is sometimes called monotone continuity.


References

  1. 1.0 1.1 A. Dold, "Lectures on algebraic topology" , Springer (1980)
  2. 2.0 2.1 E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
  3. M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
  4. 4.0 4.1 Functional analysis , Mathematical Reference Library , Moscow (1972) pp. Chapt. 8 (In Russian)
  5. R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
  6. H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
  7. M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian)
  8. B.Z. Vulikh, "Introduction to the theory of cones in normed spaces" , Kalinin (1977) (In Russian)
  9. B.Z. Vulikh, "Special questions in the geometry of cones in normed spaces" , Kalinin (1977) (In Russian)
  10. M.G. Krein, M.A. Rutman, "Linear operators leaving invariant a cone in a Banach space" Uspekhi Mat. Nauk , 3 : 1(23) (1948) pp. 3–95 (In Russian)
  1. J.B. Hiriart-Urruty, "Tangent cones, generalized gradients and mathematical programming in Banach spaces" Mathematics of Operations Research , 4 (1979) pp. 79–97
  2. R.B. Holmes, "Geometric functional analysis and its applications" , Springer (1975)
  3. A.L. Peressini, "Ordered topological vector spaces" , Harper & Row (1967)
  4. V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986)
How to Cite This Entry:
Cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cone&oldid=38826
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article