# Optimal \$delta\$ for Gromov’s \$delta\$-hyperbolicity of the hyperbolic plane

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 $$w, x, y, z in X$$,
$$d(w, x) + d(y, z) leq text{max}{d(x, y) + d(w, z), d(x, z) + d(w, y) } +2delta.$$

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

There is a related question, 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!