cone


Definition 1.

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

  1. 1.

    If λCC for any real λ>0, then C is called a cone.

  2. 2.

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

  3. 3.

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

  4. 4.

    If C-x0 is a cone for some x0 in V, then C is a cone with vertex at x0.

  5. 5.

    A convex pointed cone is called a wedge.

  6. 6.

    A proper cone is a convex cone C with vertex at 0, such that C(-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. 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. 1.

    In , the set x>0 is a blunt cone.

  2. 2.

    In , the set x0 is a pointed salient cone.

  3. 3.

    Suppose xn. Then for any ε>0, the set

    C={λBx(ε)λ>0}

    is an open cone. If |x|<ε, then C=n. Here, Bx(ε) is the open ball at x with radius ε.

  4. 4.

    In a normed vector spacePlanetmathPlanetmath, a blunt cone C is completely determined by the intersectionMathworldPlanetmath of C with the unit sphere.

Properties

  1. 1.

    The union and intersection of a collectionMathworldPlanetmath of cones is a cone. In other words, the set of cones forms a complete latticeMathworldPlanetmath.

  2. 2.

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

  3. 3.

    A cone C is convex iff C+CC.

    Proof.

    If C is convex and a,bC, then 12a,12bC, so their sum, being the convex combination of a,b, is in C, and therefore a+b=2(12a+12b)C also. Conversely, suppose a cone C satisfies C+CC, and a,bC. Then λa,(1-λ)bC for λ>0 (the case when λ=0 is obvious). Therefore their sum is also in C. ∎

  4. 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. 5.

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

References

Title cone
Canonical name Cone1
Date of creation 2013-03-22 15:32:58
Last modified on 2013-03-22 15:32:58
Owner matte (1858)
Last modified by matte (1858)
Numerical id 16
Author matte (1858)
Entry type Definition
Classification msc 46-00
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