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.