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Let
. A metric space is hyperbolic if, for any figure in that is a geodesic triangle with respect to and for every
, there exists a point
such that
.
A hyperbolic metric space is a metric space that is hyperbolic for some
.
Although a metric space is hyperbolic if it is hyperbolic for some
, one usually tries to find the smallest value of for which a hyperbolic metric space is hyperbolic.
A simple example of a hyperbolic metric space is the real line under the usual metric. Given any three points
, we always have that
. Thus, for any
, we can take . Therefore, the real line is 0 hyperbolic. Similar reasoning can be used to show that every real tree is 0 hyperbolic.
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