Science Fair Project Encyclopedia
Suppose V is a real (or complex) vector space with a subset C. If 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 , then C is a cone with vertex at x0.
A proper cone is a cone that satisfies the following:
- C is convex;
- C is closed;
- C is solid, meaning it has nonempty interior;
- C is pointed, meaning .
A proper cone C induces a partial ordering "<=" on :
- In , the set x > 0 is a salient blunt cone.
- Suppose . Then for any , the set is an open cone. If , then .
Here, is the open ball at x with radius .
- 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
- for all λ > 0,
- For a convex pointed cone C, the set is the largest vector subspace contained in C.
- A pointed convex cone C is salient if and only if
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