Hyperbolic Geometry/Gromov hyperbolicity

Definition ($\delta$-hyperbolicity):

Let $\delta >0$. A metric space $(M,d)$ is said to be $\delta$-hyperbolic iff every triangle in $M$ has a $\delta$-center.

Definition (Gromov hyperbolicity):

A metric space $(M,d)$ is called Gromov hyperbolic iff it is $\delta$-hyperbolic for some $\delta >0$.