What is the minimal $\delta$ such that the hyperbolic plane is $\delta$-hyperbolic, in the sense of the four point definition of Gromov?


**Four point definition of Gromov:** A metric space $(X, d)$ is $\delta$-hyperbolic if, for all $w, x, y, z \in X$,
$$ d(w, x) + d(y, z) \leq \text{max}\{d(x, y) + d(w, z), d(x, z) + d(w, y) \} +2\delta. $$


Empirically, the minimal value seems to be approximately $0.693$.


There is a [related question](https://mathoverflow.net/questions/23061/it-is-well-known-that-hyperbolic-space-is-delta-hyperbolic-but-what-is-delta), but this concerns the optimal $\delta$ in the $\delta$-slim definition. While this implies a bound on the $\delta$ of the four point definition, it hasn't yet helped me to derive the minimal value.


Any help (or a reference) would be greatly appreciated!



Optimal $\delta$ for Gromov's $\delta$-hyperbolicity of the hyperbolic plane
Indeed, the hyperbolic plane is $\log(2)$-hyperbolic (with the 4-point definition of hyperbolicity) and this is the optimal constant. The result is nontrivial and first appeared as Corollary 5.4 in


*Nica, Bogdan; Špakula, Ján*, [**Strong hyperbolicity**](http://dx.doi.org/10.4171/GGD/372), Groups Geom. Dyn. 10, No. 3, 951-964 (2016). [ZBL1368.20057](https://zbmath.org/?q=an:1368.20057).


