Convex cones

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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===
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Let <math>\mathcal{K}</math> be a full closed pointed convex cone in some finite dimensional Euclidean space <math>\mathbb{R}^n</math>.
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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>,
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define <math>\,d_v(x)\,</math> to be the largest number <math>\,t^\star</math> such that <math>\,x-tv\!\in\!\mathcal{K}\,</math>.
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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 LaTeX: \mathcal{K} be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space LaTeX: \mathbb{R}^n.

For any vector LaTeX: \,v\, and a point LaTeX: \,x\!\in\!\mathcal{K}, define LaTeX: \,d_v(x)\, to be the largest number LaTeX: \,t^\star such that LaTeX: \,x-t^{}v\!\in\!\mathcal{K}\,.

Suppose LaTeX: \,x\, and LaTeX: \,y\, are points in LaTeX: \,\mathcal{K}\,.

Further, suppose that LaTeX: \,d_v(x)\!=_{\!}d_v(y)\, for every extreme direction LaTeX: \,v\, of LaTeX: \,\mathcal{K}\,.

Then LaTeX: \,x\, must be equal to LaTeX: \,y\,.

proof

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