# Convex cones

(Difference between revisions)
 Revision as of 19:39, 28 August 2008 (edit) (New page: ==Nonorthogonal projection on extreme directons of convex cone== ===pseudo coordinates=== Let $\mathcal{K}$ be a full closed pointed convex cone in some finite dimensional Eucli...)← Previous diff Revision as of 19:49, 28 August 2008 (edit) (undo)Next diff → Line 1: Line 1: ==Nonorthogonal projection on extreme directons of convex cone== ==Nonorthogonal projection on extreme directons of convex cone== ===pseudo coordinates=== ===pseudo coordinates=== - Let $\mathcal{K}$ be a full closed pointed convex cone in some finite dimensional Euclidean space $\mathbb{R}^n$. + Let $\mathcal{K}$ be a full-dimensional closed pointed convex cone + in finite-dimensional Euclidean space $\mathbb{R}^n$. For any vector $\,v\,$ and a point $\,x\!\in\!\mathcal{K}$, For any vector $\,v\,$ and a point $\,x\!\in\!\mathcal{K}$, - define $\,d_v(x)\,$ to be the largest number $\,t^\star$ such that $\,x-tv\!\in\!\mathcal{K}\,$. + define $\,d_v(x)\,$ to be the largest number $\,t^\star$ such that $\,x-t^{}v\!\in\!\mathcal{K}\,$. Suppose $\,x\,$ and $\,y\,$ are points in $\,\mathcal{K}\,$. Suppose $\,x\,$ and $\,y\,$ are points in $\,\mathcal{K}\,$.

## 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 $LaTeX: \,y\,$.