# Convex cones

(Difference between revisions)

Ranjelin (Talk | contribs)
(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...)
Next diff →

## Nonorthogonal projection on extreme directons of convex cone

### pseudo coordinates

Let $LaTeX: \mathcal{K}$ be a full closed pointed convex cone in some 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 $LaTeX: \,x-tv\!\in\!\mathcal{K}\,$.

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 $LaTeX: \,y\,$.