math lessons - 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 $C - x 0$ is a cone for some $x_0 \in V$ , then $C$ is a cone with vertex at $x 0$ .

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

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

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

Contents

1 Examples

In $\R^1$ , the set $x > 0$ is a salient blunt cone.
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

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

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

01-04-2007 01:18:14
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