hyperbolic metric space

Let δ0. A metric space (X,d) is δ hyperbolic if, for any figure ABC in X that is a geodesic triangle with respect to d and for every PAB¯, there exists a point QAC¯BC¯ such that d(P,Q)δ.

A hyperbolic metric space is a metric space that is δ hyperbolic for some δ0.

Although a metric space is hyperbolic if it is δ hyperbolic for some δ0, one usually tries to find the smallest value of δ for which a hyperbolic metric space (X,d) is δ hyperbolic.

A example of a hyperbolic metric space is the real line under the usual metric. Given any three points A,B,C, we always have that AB¯AC¯BC¯. Thus, for any PAB¯, we can take Q=P. Therefore, the real line is 0 hyperbolic. reasoning can be used to show that every real tree is 0 hyperbolic.

Title hyperbolic metric space
Canonical name HyperbolicMetricSpace
Date of creation 2013-03-22 17:11:29
Last modified on 2013-03-22 17:11:29
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Definition
Classification msc 54E99
Classification msc 54E35
Classification msc 20F06
Defines δ hyperbolic