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# Cone (geometry)

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

If C - x0 is a cone for some $x_0 \in V$, then C is a cone with vertex at x0.

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

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

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

$a <= b\Leftrightarrow b-a\in C$.
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### Examples

1. In $\R^1$, the set x > 0 is a salient blunt cone.
2. Suppose $x\in \R^n$. Then for any $\varepsilon>0$, the set $C=\bigcup \{\, \lambda B_x(\varepsilon) \mid \lambda >0 \,\}$ is an open cone. If $|x| < \varepsilon$, then $C=\R^n$.

Here, $B_x(\varepsilon)$ is the open ball at x with radius $\varepsilon$.

## 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
$\lambda C \subset C,$ for all λ > 0,
$C+C\subset C.$
3. For a convex pointed cone C, the set $C\cap (-C)$ is the largest vector subspace contained in C.
4. A pointed convex cone C is salient if and only if $C\cap (-C)=\{0\}.$