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=== | ||
| - | 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 20:49, 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
.