Cone (geometry)

Suppose <math>V<math> is a real (or complex) vector space with a subset <math>C<math>. If <math>\lambda C \subset C<math> for any real <math>\lambda >0<math>, then <math>C<math> is a cone.

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

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

If <math>C-x_0<math> is a cone for some <math>x_0 \in V<math>, then <math>C<math> is a cone with vertex at <math>x_0<math>.

A proper cone is a cone <math>C \subset \R^n<math> that satisfies the following:

• <math>C<math> is convex;
• <math>C<math> is closed;
• <math>C<math> is solid, meaning it has nonempty interior;
• <math>C<math> is pointed, meaning <math>x, -x\in C\Rightarrow x=0<math>.

A proper cone <math>C<math> induces a partial ordering "<=" on <math>\R^n<math>:

<math>a <= b\Leftrightarrow b-a\in C <math>.

Examples

1. In <math>\R^1<math>, the set <math>x>0<math> is a salient blunt cone.
2. Suppose <math>x\in \R^n<math>. Then for any <math>\varepsilon>0<math>, the set <math>

C=\bigcup \{\, \lambda B_x(\varepsilon) \mid \lambda >0 \,\} <math> is an open cone. If <math>|x| < \varepsilon<math>, then <math>C=\R^n<math>. Here, <math>B_x(\varepsilon)<math> is the open ball at <math>x<math> with radius <math>\varepsilon<math>.

Properties

1. The union and intersection of a collection of cones is a cone.
2. A set <math>C<math> in a real (or complex) vector space is a convex cone if and only if

   <math>\lambda C \subset C,<math> for all <math>\lambda>0,<math>

   <math>C+C\subset C.<math>

3. For a convex pointed cone <math>C<math>, the set <math>C\cap (-C)<math> is the largest vector subspace contained in <math>C<math>.
4. A pointed convex cone <math>C<math> is salient if and only if <math>C\cap (-C)=\{0\}.<math>

See also

• Cone
  • Cone (solid)
  • Cone (topology)

References

• This article incorporates material from Cone on PlanetMath, which is licensed under the GFDL.
• This article incorporates material from proper cone on PlanetMath, which is licensed under the GFDL.