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. 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.