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cone

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Definition 1 Suppose is a real (or complex) vector space with a subset .

If $\lambda C \subset C$ for any real $\lambda >0$ , then is called a cone.
If the origin belongs to a cone, then the cone is said to be pointed. Otherwise, the cone is blunt.
A pointed cone is salient, if it contains no -dimensional vector subspace of .
If is a cone for some in , then is a cone with vertex at .
A convex pointed cone is called a wedge.
A proper cone is a convex cone with vertex at 0, such that $C\cap (-C)=\lbrace 0\rbrace$ . A slightly more specific definition of a proper cone is this entry, but it requires the vector space to be topological.
A cone is said to be generating if . In this case, is said to be generated by .

In $\mathbb{R}$ , the set is a blunt cone.
In $\mathbb{R}$ , the set $x\ge 0$ is a pointed salient cone.
Suppose $x\in \mathbb{R}^n$ . Then for any $\varepsilon>0$ , the set
$\displaystyle C=\bigcup \{\, \lambda B_x(\varepsilon) \mid \lambda >0 \,\}$
is an open cone. If $\vert x\vert < \varepsilon$ , then $C=\mathbb{R}^n$ . Here, $B_x(\varepsilon)$ is the open ball at with radius $\varepsilon$ .
In a normed vector space, a blunt cone is completely determined by the intersection of with the unit sphere.

The union and intersection of a collection of cones is a cone. In other words, the set of cones forms a complete lattice.
The complement of a cone is a cone. This means that the complete lattice of cones is also a complemented lattice.
A cone is convex iff $C+C\subseteq C$ .
Proof. If 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 , is in , and therefore $a+b=2(\frac{1}{2}a+\frac{1}{2}b)\in C$ also. Conversely, suppose a cone 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 . $\qedsymbol$
A cone containing 0 is a cone with vertex at 0. As a result, a wedge is a cone with vertex at 0.
The only cones that are subspaces at the same time are wedges.

Bibliography

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.

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"cone" is owned by matte. [ full author list (3) ]

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Also defines:

blunt cone, pointed cone, salient cone, cone with vertex, wedge, proper cone, generating

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