# Convex cones

### From Wikimization

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(New page: ==Nonorthogonal projection on extreme directons of convex cone== ===pseudo coordinates=== Let <math>\mathcal{K}</math> be a full closed pointed convex cone in some finite dimensional Eucli...) |
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==Nonorthogonal projection on extreme directons of convex cone== | ==Nonorthogonal projection on extreme directons of convex cone== | ||

===pseudo coordinates=== | ===pseudo coordinates=== | ||

- | Let <math>\mathcal{K}</math> be a full closed pointed convex cone in | + | Let <math>\mathcal{K}</math> be a full-dimensional closed pointed convex cone |

+ | in finite-dimensional Euclidean space <math>\mathbb{R}^n</math>. | ||

For any vector <math>\,v\,</math> and a point <math>\,x\!\in\!\mathcal{K}</math>, | For any vector <math>\,v\,</math> and a point <math>\,x\!\in\!\mathcal{K}</math>, | ||

- | define <math>\,d_v(x)\,</math> to be the largest number <math>\,t^\star</math> such that <math>\,x- | + | define <math>\,d_v(x)\,</math> to be the largest number <math>\,t^\star</math> such that <math>\,x-t^{}v\!\in\!\mathcal{K}\,</math>. |

Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>. | Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>. |

## Revision as of 19:49, 28 August 2008

## Nonorthogonal projection on extreme directons of convex cone

### pseudo coordinates

Let be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space .

For any vector and a point , define to be the largest number such that .

Suppose and are points in .

Further, suppose that for every extreme direction of .

Then must be equal to .