Cone (Geometry)

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

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Examples

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

Properties

1. The union and intersection of a collection of cones is a cone.
2. A set $C$ in a real (or complex) vector space is a convex cone if and only if
3. :$\backslash lambda\; C\; \backslash subset\; C,$ for all $\backslash lambda>0,$
4. :$C+C\backslash subset\; C.$
5. For a convex pointed cone $C$, the set $C\backslash cap\; (-C)$ is the largest vector subspace contained in $C$.
6. A pointed convex cone $C$ is salient if and only if $C\backslash cap\; (-C)=\backslash \{0\backslash \}.$

See also

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

References

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