Definition (${\displaystyle \delta }$-hyperbolicity):

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

Definition (Gromov hyperbolicity):

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