An isometry is a map between two metric spaces which preserves all distances: for all . (After typing a bunch of such formulas, one tends to prefer shorter notation: , with the metric inferred from contexts.) It’s a popular exercise to prove that every isometry from a compact metric space into itself is surjective.
A rough isometry allows additive distortion of distances: . (As contrasted with bi-Lipschitz maps, which allow multiplicative distortion). Rough isometries ignore small-scale features (in particular, they need not be continuous), but accurately capture the large scale structure of a space. This makes them convenient for studying hyperbolic spaces (trees and their relatives), where the large-scale structure is of interest.
If is an -rough isometry, then the Gromov-Hausdorff distance between and is at most . This follows from a convenient characterization of : it is equal to where the infimum is taken over all subsets that project surjectively onto each factor, and .
Since small-scale features are ignored, it is not quite natural to talk about rough isometries being surjective. A natural concept for them is -surjectivity, meaning that every point of is within distance of . When this happens, we can conclude that , because the Hausdorff distance between and is at most .
Recalling that an isometry of a compact metric space into is automatically onto, it is natural to ask whether -rough isometries of into are -surjective. This, however, turns out to be false in general.
First example, a finite space: with the metric (when ). Consider the backward shift (and ). By construction, this is a rough isometry with . Yet, the point is within distance from .
The above metric can be thought of a the distance one has to travel from to with a mandatory visit to . This makes it similar to the second example.
Second example, a geodesic space: is the graph shown below, a subset of with the intrinsic (path) metric, i.e., the length of shortest path within the set.
Define where means the positive part. Again, this is a -rough isometry. The omitted part, shown in red below, contains a ball of radius centered at . Of course, can be replaced by any positive integer.
Both of these examples are intrinsically high-dimensional: they cannot be accurately embedded into for low (using the restriction metric rather than the path metric). This raises a question: does there exist such that for every compact subset , equipped with the restriction metric, every -rough isometry is -surjective?