The Quasi-hyperbolicity Constant of a Metric Space

Abstract

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

Minor typos fixed and a motivational paragraph added. The "restricted quasi-hyperbolicity constant" in [v1] has been renamed with the more informative moniker "quadrilateral constant". To appear in "Mathematical Inequalities & Applications"