Cone

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. 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.

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.

Cone over a topological space

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 $A$.

Mapping cone

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.

Cone in a real vector space

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 $P=\{x\colon x\in E, \Box x\geq 0\}$ 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 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.

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 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.

Comments

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 $K$.

The theory of cones in Banach spaces is more thoroughly developed. 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 , 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, $f(x_0)<0$).

Cones in Banach spaces are used in optimization theory. They can be used to define multi-valued derivatives of non-smooth mappings.

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.

References

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