The quasi-hyperbolicity constant of a metric space

We introduce the quasi-hyperbolicity constant of a metric space, rough isometry invariant that measures how space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in closed interval $[1,2]$. The an hyperbolic is equal to one. For CAT$(0)$-space, it bounded above by $\sqrt{2}$. Banach at least two dimensional below $\sqrt{2}$, and non-trivial $L_p$-space exactly $\max\{2^{1/p},2^{1-1/p}\}$. If $0 < \alpha 1$ then $\alpha$-snowflake any $2^\alpha$. give exact calculation case Euclidean real line.