# cone

###### Definition 1.

Suppose $V$ is a real (or complex) vector space with a subset $C$.

1. 1.

If $\lambda C\subset C$ for any real $\lambda>0$, then $C$ is called a cone.

2. 2.

If the origin belongs to a cone, then the cone is said to be pointed. Otherwise, the cone is blunt.

3. 3.

A pointed cone is salient, if it contains no $1$-dimensional vector subspace of $V$.

4. 4.

If $C-x_{0}$ is a cone for some $x_{0}$ in $V$, then $C$ is a cone with vertex at $x_{0}$.

5. 5.

A convex pointed cone is called a wedge.

6. 6.

A proper cone is a convex cone $C$ with vertex at $0$, such that $C\cap(-C)=\{0\}$. A slightly more specific definition of a proper cone is this entry (http://planetmath.org/ProperCone), but it requires the vector space to be topological.

7. 7.

A cone $C$ is said to be generating if $V=C-C$. In this case, $V$ is said to be generated by $C$.

## Examples

1. 1.

In $\mathbb{R}$, the set $x>0$ is a blunt cone.

2. 2.

In $\mathbb{R}$, the set $x\geq 0$ is a pointed salient cone.

3. 3.

Suppose $x\in\mathbb{R}^{n}$. Then for any $\varepsilon>0$, the set

 $C=\bigcup\{\,\lambda B_{x}(\varepsilon)\mid\lambda>0\,\}$

is an open cone. If $|x|<\varepsilon$, then $C=\mathbb{R}^{n}$. Here, $B_{x}(\varepsilon)$ is the open ball at $x$ with radius $\varepsilon$.

4. 4.

In a normed vector space, a blunt cone $C$ is completely determined by the intersection of $C$ with the unit sphere.

## Properties

1. 1.

The union and intersection of a collection of cones is a cone. In other words, the set of cones forms a complete lattice.

2. 2.

The complement of a cone is a cone. This means that the complete lattice of cones is also a complemented lattice.

3. 3.

A cone $C$ is convex iff $C+C\subseteq C$.

###### Proof.

If $C$ is convex and $a,b\in C$, then $\frac{1}{2}a,\frac{1}{2}b\in C$, so their sum, being the convex combination of $a,b$, is in $C$, and therefore $a+b=2(\frac{1}{2}a+\frac{1}{2}b)\in C$ also. Conversely, suppose a cone $C$ satisfies $C+C\subseteq C$, and $a,b\in C$. Then $\lambda a,(1-\lambda)b\in C$ for $\lambda>0$ (the case when $\lambda=0$ is obvious). Therefore their sum is also in $C$. ∎

4. 4.

A cone containing $0$ is a cone with vertex at $0$. As a result, a wedge is a cone with vertex at $0$.

5. 5.

The only cones that are subspaces at the same time are wedges.

## References

 Title cone Canonical name Cone1 Date of creation 2013-03-22 15:32:58 Last modified on 2013-03-22 15:32:58 Owner matte (1858) Last modified by matte (1858) Numerical id 16 Author matte (1858) Entry type Definition Classification msc 46-00 Related topic ProperCone Related topic GeneralizedFarkasLemma Defines blunt cone Defines pointed cone Defines salient cone Defines cone with vertex Defines wedge Defines proper cone Defines generating