No 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. Gromov hyperbolicity, and also the lack thereof, has attracted considerable interest in the theory of networks. The quasi-hyperbolicity constant for an unbounded space lies in the closed interval [1,2] . It is equal to one for an unbounded Gromov hyperbolic space. For a CAT (0) -space, it is bounded from above by √ 2 . The quasi-hyperbolicity constant of a Banach space that is at least two dimensional is bounded from below by √ 2 , and for a non-trivial L p -space it is exactly max{2 1/p ,2 1−1/p } . If 0 < α < 1 then the quasi-hyperbolicity constant of the α -snowflake of any metric space is bounded from above by 2 α . We give an exact calculation in the case of the α -snowflake of the Euclidean real line.

A real valued function ϕ of one variable is called a metric transform if for every metric space (X, d) the composition dϕ = ϕ • d is also a metric on X. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms ϕ such that the transformed Euclidean half line ([0, ∞), | · |ϕ) is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if (X, d) is any metric space containing a rough geodesic ray and ϕ is an approximately nondecreasing, unbounded metric transform such that the transformed space (X, dϕ) is Gromov hyperbolic and roughly geodesic then ϕ is an approximate dilation and the original space (X, d) is Gromov hyperbolic and roughly geodesic.