The Quasi-hyperbolicity Constant of a Metric Space

2019/08/12 by Dragomir, George, Nicas, Andrew
#46B20 #51F99 #51K05 (Primary) #51M10 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1908.04440

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 √(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.

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