hyperbolic metric space

Let $\delta \ge 0$. A metric space $(X,d)$ is $\delta$ hyperbolic if, for any figure $ABC$ in $X$ that is a geodesic triangle with respect to $d$ and for every $P\in \overline{AB}$, there exists a point $Q\in \overline{AC}\cup \overline{BC}$ such that $d(P,Q)\le \delta$.

A hyperbolic metric space is a metric space that is $\delta$ hyperbolic for some $\delta \ge 0$.

Although a metric space is hyperbolic if it is $\delta$ hyperbolic for some $\delta \ge 0$, one usually tries to find the smallest value of $\delta$ for which a hyperbolic metric space $(X,d)$ is $\delta$ hyperbolic.

A example of a hyperbolic metric space is the real line under the usual metric. Given any three points $A,B,C\in ℝ$, we always have that $\overline{AB}\subseteq \overline{AC}\cup \overline{BC}$. Thus, for any $P\in \overline{AB}$, 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	$\delta$ hyperbolic