# Cone

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A cone in a Euclidean space is a set consisting of half-lines emanating from some point , the vertex of the cone. The boundary of (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 with a half-space containing and bounded by a plane not passing through 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 , where is the height, and is the radius of the base; the area of the lateral surface is equal to , where 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 , where are the radii of the base and is the height (the distance between the base); the area of the lateral surface is , where is the length of the segment of a generator.

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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 (the base of the cone) is the space obtained from the product by collapsing the subspace to a point (the vertex of the cone):

In other words, is the cylinder of the constant mapping (see Cylindrical construction) or the cone of the identity mapping (see Mapping-cone construction). The space is contractible if and only if it is a retract of every cone over (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 , , of an arbitrary category with common initial object is called a morphism cone with vertex . Dually one defines a morphism cocone as a set of morphisms , , with common final object . See , , .

M.I. Voitsekhovskii

A mapping cone is a topological space associated with a continuous mapping of topological spaces by the mapping-cone construction. Let be the cone of the imbedding , let be the cone of the imbedding , etc., where is the mapping cone of . Then the sequence

so obtained is called the Puppe sequence; here , , etc., where (respectively, ) is the suspension over (respectively, over ).

One defines in an analogous way the reduced mapping cone of a mapping of pointed spaces. Here, as for a cofibration, for any pointed space , the sequence of homotopy classes induced by the Puppe sequence

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

A cone in a real vector space is a set such that for any . A cone is called pointed if and a pointed cone is called salient if contains no one-dimensional subspace. A non-salient cone is sometimes called a wedge.

A cone that is a convex subset of is called convex. Thus, a subset of is a convex cone if and only if for any and . In this case the vector subspace of generated by the convex cone is the same as the set . If is pointed, then is the largest vector subspace contained in . A pointed convex cone is salient if and only if .

If is a (partially) ordered vector space, then the positive cone is a salient pointed convex cone. Conversely, any such a convex cone induces an order relation in : if .

A cone is said to be reproducing if any element can be expressed as a difference of elements of . For example, the cone of non-negative continuous (or summable) functions on the interval 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 provides the notion of a cone with a richer content enabling one to obtain non-trivial results. For example, suppose that is a separable locally convex space and that is a salient pointed convex cone in having a non-empty interior (such cones are called solid). Then every linear form on that is positive on is continuous ( is positive on if for ); if is a vector subspace of having a non-empty intersection with the interior of and is a linear form on that is positive on , then there exists on a linear form extending that is positive on . See , , .

M.I. Voitsekhovskii

A reproducing cone is also called a generating cone.

The theory of cones in Banach spaces is more thoroughly developed. Let be a cone in the Banach space inducing in an order relation . If the cone is closed, then the Archimedean principle holds for : If , if , , are numbers and if there exists a point such that for all , then . For a solid cone the converse also holds: If the Archimedean property holds for , then is closed.

Let be the dual wedge to , that is, the collection of all positive linear continuous functions on ( is positive if for any ). Then is a cone if and only if is spatial, that is, if the closure . If is closed, then for any (respectively, ) there exists an such that (respectively, ).

A cone is called unflattened if there exists for any elements such that

where is a constant.

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

A cone is called normal if

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

A cone is called regular (completely regular) if every sequence of elements of that is increasing and order bounded (norm bounded) converges. If is closed and regular, then it is normal; every completely-regular cone is normal and regular. If in fact is regular and solid, then it is completely regular. The regularity of a cone is related to the order continuity of the norm: If , that is, if the family is a decreasing directed set, and if , then . The regularity of a closed cone is equivalent to the property that the space is Dedekind complete and that the norm in is order continuous. The regularity of a solid cone implies the order continuity of the norm in .

A cone is called plasterable if there exists a cone and a number such that the ball for any . The plasterability of is equivalent to the existence in of an equivalent norm that is additive on . 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

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

 [a1] J.B. Hiriart-Urruty, "Tangent cones, generalized gradients and mathematical programming in Banach spaces" Mathematics of Operations Research , 4 (1979) pp. 79–97 [a2] R.B. Holmes, "Geometric functional analysis and its applications" , Springer (1975) [a3] A.L. Peressini, "Ordered topological vector spaces" , Harper & Row (1967) [a4] V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986)

#### References

 [1] , Functional analysis , Math. Reference Library , Moscow (1972) pp. Chapt. 8 (In Russian) [2] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) [3] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) [4] A. Dold, "Lectures on algebraic topology" , Springer (1980) [5] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) [6] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian) [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)
How to Cite This Entry:
Cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cone&oldid=14184
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article