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A cone in a Euclidean space is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245901.png" /> consisting of half-lines emanating from some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245902.png" />, the vertex of the cone. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245903.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245904.png" /> (consisting of half-lines called generators of the cone) is part of a [[Conical surface|conical surface]], and is sometimes also called a cone. Finally, the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245905.png" /> with a half-space containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245906.png" /> and bounded by a plane not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245907.png" /> 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 centre 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c0245909.png" /> is the height, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459010.png" /> is the radius of the base; the area of the lateral surface is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459014.png" /> are the radii of the base and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459015.png" /> is the height (the distance between the base); the area of the lateral surface is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459017.png" /> 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.

A cone over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459018.png" /> (the base of the cone) is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459019.png" /> obtained from the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459020.png" /> by collapsing the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459021.png" /> to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459022.png" /> (the vertex of the cone):

<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/c02459023.png" /></td> </tr></table>

In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459024.png" /> is the cylinder of the constant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459025.png" /> (see [[Cylindrical construction|Cylindrical construction]]) or the cone of the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459026.png" /> (see [[Mapping-cone construction|Mapping-cone construction]]). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459027.png" /> is contractible if and only if it is a retract of every cone over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459028.png" /> (cf. [[Retract of a topological space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459030.png" />, of an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459031.png" /> with common initial object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459032.png" /> is called a morphism cone with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459033.png" />. Dually one defines a morphism cocone as a set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459035.png" />, with common final object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459036.png" />. See , , .

''M.I. Voitsekhovskii''

A mapping cone is a topological space associated with a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459037.png" /> of topological spaces by the [[Mapping-cone construction|mapping-cone construction]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459038.png" /> be the cone of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459040.png" /> be the cone of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459041.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459042.png" /> is the mapping cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c02459043.png" />. Then the 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/c02459044.png" /></td> </tr></table>

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" />).

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>

is exact; all the terms in it starting from the fourth are groups and starting from the seventh, Abelian groups. See , .

''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.

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" />.

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.

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 , , .

''M.I. Voitsekhovskii''

====Comments====
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.

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" />).

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

<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).

A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024590/c024590138.png" /> is called normal if

<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>

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.

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 <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.

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''

====Comments====
A spatial cone (or wedge) is also called a spanning cone (spanning wedge).

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.

====References====
<table><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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Functional analysis'' , Math. Reference Library , Moscow  (1972)  pp. Chapt. 8  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,   "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Dold,   "Lectures on algebraic topology" , Springer  (1980)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><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><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><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><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><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></table>