# 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 $A,B,C\in \mathbb{R}$, 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. 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 |