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# hyperbolic metric space

Let $\delta\geq 0$. 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 $d(P,Q)\leq\delta$.

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

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

A simple example of a hyperbolic metric space is the real line under the usual metric. Given any three points $A,B,C\in\mathbb{R}$, 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. Similar reasoning can be used to show that every real tree is 0 hyperbolic.

## Mathematics Subject Classification

54E99*no label found*54E35

*no label found*20F06

*no label found*

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