# Valid metric on a hyperbolic space

Note: cross-posted to mathoverflow.net

I'm looking at the distance that's defined in this paper on Poincaré Embeddings:

$$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \mathbf{u} - \mathbf{v} \right\|^2}{(1 - \left\| \mathbf{u} \right\|^2)(1 - \left\| \mathbf{v} \right\|^2)} \right)$$

for $$\mathbf{u}, \mathbf{v} \in \mathcal{B}^d$$ using the Poincaré ball model of hyperbolic space.

Does this define a valid metric? It's easy to see that

• $$d(\mathbf{u}, \mathbf{v}) \geq 0$$ since $$\left\| \mathbf{u} \right\| \leq 1$$ and $$\left\| \mathbf{v} \right\| \leq 1$$ and so the rhs inside the parentheses is positive ($$\operatorname{arccosh}(x)$$ is an increasing function for $$x \geq 0$$),
• $$\mathbf{u} = \mathbf{v}$$, then $$d(\mathbf{u}, \mathbf{v}) = 0$$, and
• $$d(\mathbf{u}, \mathbf{v}) = d(\mathbf{v}, \mathbf{u})$$

However I'm having a hard time trying to prove the triangle inequality in this case. I've tried using the logarithmic form but that didn't get me anywhere. I also found the identity $${\displaystyle \operatorname {arcosh} u\pm \operatorname {arcosh} v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)}$$ under "Addition Formulae (sic.)" on Wikipedia but that also seemed to be a dead end. Am I missing something obvious?

• I don't understand. If $u=v$ then the hyperbolic distance is zero and $f$ equal $1$ – user84976 Mar 2 at 23:48
• sorry should have been the original function $d$ - have updated the question – tdc Mar 3 at 9:43