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A proper cone is a cone $C\subset\mathbb{R}^n$ 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 $\mathbb{R}^n$ :
This ordering has many nice properties, such as transitivity, reflexivity, and antisymmetry.
- 1
- S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
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"proper cone" is owned by dooder0001.
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Cross-references: antisymmetry, reflexivity, transitivity, properties, partial ordering, induces, interior, solid, closed, convex
There are 2 references to this entry.
This is version 4 of proper cone, born on 2004-09-20, modified 2004-09-21.
Object id is 6199, canonical name is ProperCone.
Accessed 2461 times total.
Classification:
| AMS MSC: | 52A20 (Convex and discrete geometry :: General convexity :: Convex sets in $n$ dimensions ) |
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Pending Errata and Addenda
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