Definition
Suppose that is a (not necessarily continuous) function from one metric space to a second metric space . Then is called a quasi-isometry from to if there exist constants, and such that the following two properties both hold:
- For every two points and in, the distance between their images is (up to the additive constant ) within a factor of of their original distance. More formally:
- Every point of is within the constant distance of an image point. More formally:
The two metric spaces and are called quasi-isometric if there exists a quasi-isometry from to .
Read more about this topic: Quasi-isometry
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