Cone (geometry)

Suppose V is a real (or complex) vector space with a subset C.

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If lambda C subset C for any real lambda >0,

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then C is a cone.

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If the origin belongs to a cone, then the cone is called pointed.

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Otherwise, the cone is called blunt.

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A pointed cone is salient, if it contains no

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1-dimensional vector subspace of V.

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If C-x_0 is a cone for some x_0 in V,

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then C is a cone with vertex at x_0.

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

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

A proper cone C induces a partial ordering "

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:a

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Cone (geometry): Examples ►