The Gromov-Hausdorff distance is realized #
In this file, we construct of a good coupling between nonempty compact metric spaces, minimizing their Hausdorff distance. This construction is instrumental to study the Gromov-Hausdorff distance between nonempty compact metric spaces.
Given two nonempty compact metric spaces
Y, we define
OptimalGHCoupling X Y as a
compact metric space, together with two isometric embeddings
OptimalGHCoupling X Y. The main property of the optimal
coupling is that the Hausdorff distance between
OptimalGHCoupling X Y is smaller
than the corresponding distance in any other coupling. We do not prove completely this fact in this
file, but we show a good enough approximation of this fact in
will suffice to obtain the full statement once the Gromov-Hausdorff distance is properly defined,
The key point in the construction is that the set of possible distances coming from isometric
Y in metric spaces is a set of equicontinuous functions. By Arzela-Ascoli,
it is compact, and one can find such a distance which is minimal. This distance defines a premetric
space structure on
X ⊕ Y. The corresponding metric quotient is
OptimalGHCoupling X Y.
candidates give rise to elements of
We will then choose the candidate minimizing the Hausdorff distance. Except that we are not
in a metric space setting, so we need to define our custom version of Hausdorff distance,
HD, and prove its basic properties.
Explicit bound on
HD (dist). This means that when looking for minimizers it will
be sufficient to look for functions with
HD(f) bounded by this bound.
With the optimal candidate, construct a premetric space structure on
X ⊕ Y, on which the
predistance is given by the candidate. Then, we will identify points at
to obtain a genuine metric space.
The optimal coupling between two compact spaces
Y is still a compact space
For any candidate
HD(f) is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that
HD of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality