cone
Definition 1.
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 $V=CC$. 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 $x\ge 0$ is a pointed salient cone.

3.
Suppose $x\in {\mathbb{R}}^{n}$. Then for any $\epsilon >0$, the set
$$C=\bigcup \{\lambda {B}_{x}(\epsilon )\mid \lambda >0\}$$ is an open cone. If $$, then $C={\mathbb{R}}^{n}$. Here, ${B}_{x}(\epsilon )$ is the open ball at $x$ with radius $\epsilon $.

4.
In a normed vector space^{}, a blunt cone $C$ is completely determined by the intersection^{} of $C$ with the unit sphere.
Properties

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.
References
 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, AddisonWesley Publishing Company, 1966.
 3 R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
 4 I.M. Glazman, Ju.I. Ljubic, FiniteDimensional Linear Analysis, A systematic Presentation^{} in Problem Form, Dover Publications, 2006.
Title  cone 
Canonical name  Cone1 
Date of creation  20130322 15:32:58 
Last modified on  20130322 15:32:58 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  16 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 4600 
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 