A function f:XY between two metric spaces is called a quasi-isometry (or a near isometry) if there exists a constant C for which

1/C dX(a,b) ≤ dY(f(a),f(b)) ≤ C dX(a,b)
for all a,bX. That is, f distorts distances by no more than a factor of C.

A quasi-isometry is always one-to-one, but needn't be onto.

Isometry between metric spaces is a very strong condition indeed. Often, a weaker condition is useful. For instance, any two norms on Rd (d<∞) induce quasi-isometric metric spaces; except when the two norms are multiples of each other, they never induce isometry.

If f is a quasi-isometry which is also onto, then the topologies of the two metric spaces are equivalent: convergence in the one implies convergence in the other.

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