cone

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

1.

If $\lambda C\subset C$ for any real $\lambda>0$ , then $C$ is called a cone.
2.

If the origin belongs to a cone, then the cone is said to be pointed. Otherwise, the cone is blunt.
3.

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

A convex pointed cone is called a wedge.
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.

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.

In $\mathbb{R}$ , the set $x>0$ is a blunt cone.
2.

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

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

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.

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

A cone containing $0$ is a cone with vertex at $0$ . As a result, a wedge is a cone with vertex at $0$ .
5.

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

1 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
2 J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company, 1966.
3 R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
4 I.M. Glazman, Ju.I. Ljubic, Finite-Dimensional Linear Analysis, A systematic Presentation in Problem Form, Dover Publications, 2006.