The $1-$norm in Euclidean geometry is $d_{geometric}(u,v)=\| u-v \|_1$. We can convert this in hyperbolic space using a Poincare disk model by:
$$ d_{hyperbolic}(u,v) = \operatorname{arcosh}\left( 1+ 2 \frac{\|u-v\|^2}{(1-\|u\|)^2(1-\|v\|)^2}\right) $$
My question: Can i generalise to $p$ dimensions as simply as:
$$ \|u-v\|_p =\operatorname{arcosh}\left( 1+ 2 \frac{\|u-v\|^2_p}{(1-\|u\|_p)^2(1-\|v\|_p)^2}\right)? $$
Furthermore, if the property does hold, then I assume it is also scalar invariant? i.e to convert $K \times d_{geometric}(u,v)$, we just write $K\times d_{hyperbolic}(u,v)$?