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# 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, 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. 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. ###### 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*, 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.

## Mathematics Subject Classification

46-00*no label found*

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