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?
The last question is very nice. As you probably already considered, one might guess that it holds for all doubling metric spaces K, with the constant C depending only on the doubling constant. Both your examples get worse and worse doubling constants as they grow. But I don’t see a proof (of either the Euclidean case or the general doubling case) at the moment.