It is known that the $\tilde j$-metric, half-apollonian metric and scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point) and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that preserves the Gromov hyperbolicity. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of metrics used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic.
In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that the doubling property regarding two classes of metrics on the Gromov boundary of hyperbolic spaces are coincided. Next, we obtain a characterization of unbounded Gromov hyperbolic domains via metric spaces sphericalization. Finally, we investigate the topological equivalence of Gromov hyperbolic $φ$-uniform domains between the Gromov boundary and the inner metric boundary.
Recently, strongly hyperbolic space as certain analytic enhancements of Gromov hyperbolic space was introduced by B. Nica and J. Spakula. In this note, we prove that the log-metric log(1+d) on a Ptolemy space (X,d) is a strongly hyperbolic metric. Using our result, we construct three metrics on a Ptolemy metric space and prove they are strongly hyperbolic.
The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces.
A real valued function $φ$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_φ= φ\circ d$ 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 $([0,\infty),|\cdot|_φ)$ 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 metri…
It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two points, metrics that are Gromov hyperbolic with parameter $\log 2$ and strongly hyperbolic with parameter $1$ with no further conditions on the open set. A class of examples is constructed on Hadamard manifolds showing these estimates of the parameters are sharp.
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 $0 < α< 1$ …
