PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
cone (Definition)
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)=\lbrace 0\rbrace$. 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\ge 0$ is a pointed salient cone.
  3. 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 $ 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.

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$. $ \qedsymbol$
  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.

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.



Anyone with an account can edit this entry. Please help improve it!

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

View style:

See Also: proper cone, generalized Farkas lemma

Also defines:  blunt cone, pointed cone, salient cone, cone with vertex, wedge, proper cone, generating

Attachments:
anti-cone (Definition) by stevecheng
Log in to rate this entry.
(view current ratings)

Cross-references: subspaces, obvious, convex combination, sum, iff, complemented lattice, complement, complete lattice, collection, union, unit sphere, intersection, normed vector space, radius, open ball, open, generated by, convex, vector subspace, contains, origin, subset, vector space, complex, real
There are 33 references to this entry.

This is version 13 of cone, born on 2005-10-26, modified 2007-05-07.
Object id is 7447, canonical name is Cone5.
Accessed 5566 times total.

Classification:
AMS MSC: 46-00 (Functional analysis :: General reference works )
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)