hyperbolic metric space

Let $\\delta\\geq 0$ . A metric space $(X,d)$ is $\\delta$ hyperbolic if, for any figure $A B C$ 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 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.

单词	HyperbolicMetricSpace
释义	hyperbolic metric space Let $\\delta\\geq 0$ . A metric space $(X,d)$ is $\\delta$ hyperbolic if, for any figure $A B C$ 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 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.
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