Unless I am very wrong, the following seems to be true:



> 
> If the angle between two vectors in $\mathbb{R}^{n}\_{++}$ is small, then the
>  value of the Hilbert projective metric between them is also small.
> 
> 
> 


I am looking for a reference to a precise statement of this notion.


P.S.
What I mean by the angle between $u$ and $v$ is $\arccos{\frac{\langle u,v \rangle}{||u|| \cdot ||v||}}$.


P.P.S.
If $u,v \in \mathbb{R}^{n}\_{++}$ then $d(u,v)=\log{\frac{\max{(u\_{i}/v\_{i})}}{\min{(u\_{i}/v\_{i})}}}$ is the Hilbert projective metric.



Angles and projective metric
This is equivalent to the following: in the Klein model of hyperbolic space, if two points are close together, then their hyperbolic distane is small. This is true locally (e.g., fix one point), but not true globally (in other words, no bound on Euclidean distance implies a fixed bound on hyperbolic distance -- the distortion grows as you approach the boundary).


