Local Structure of Gromov–Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Space
We investigate the geometry of the family of isometry classes of compact metric spaces, endowed with the Gromov–Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space , i.e., in the space with the norm . As a corollary, we get that each finite metric space can be isometrically embedded into in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number of points. If the initial space has points, then one can take as the least possible integer with .
By we denote the space of all compact metric spaces (considered up to an isometry) endowed with the Gromov–Hausdorff metric. It is well-known that is linear connected, complete, separable, but not proper. In a recent paper , A. Ivanov, N. Nikolaeva, and A. Tuzhilin have shown that is geodesic. There are many other open questions concerning geometrical properties of . S. Iliadis formulated the following problem: is it true that is universal for the family of compact metric spaces? The latter may be interpreted in weak and strong senses. The weak sense means that any compact metric space can be isometrically embedded into . The strong sense means in addition that each isometric embedding of a subspace of a compact metric space can be extended to an isometric embedding of the whole space. It is easy to see that is not universal in the strong sense (see below). Concerning the weak universality, we show the following: each finite metric space can be isometrically embedded into . Moreover, we construct such an embedding with its image belonging to the subspace of all finite metric spaces with points: if the initial space has points, then one can chose as the least possible integer such that .
The construction of such embedding is based on our results concerning the local geometry of the family of finite metric spaces with fixed number of points considered in sufficiently small neighborhoods of generic spaces. More precisely, we show that such neighborhoods are isometric to some neighborhoods of the corresponding points in the space , i.e., in the space with the norm .
Let be an arbitrary metric space. By we denote the distance between points and in . For every point and a real number by we denote the open ball of radius centered at ; for every nonempty and real number we put .
For nonempty , let us put
This value is called the Hausdorff distance between and . It is well-known  that the restriction of the Hausdorff distance to the family of all closed bounded subsets of is a metric.
Let and be metric spaces. A triple that consists of a metric space and its subsets and isometric to and , respectively, is called a realization of the pair . The Gromov–Hausdorff distance between and is the greatest lower bound of the real numbers such that there exists a realization of the pair with . It is well-known  that the restricted to the family of isometry classes of compact metric spaces is a metric.
Recall that a relation between sets and is a subset of the Cartesian product . By we denote the set of all nonempty relations between and . If and are the canonical projections, then their restrictions to each are denoted in the same manner.
We consider each relation as a multivalued mapping, whose domain may be less than the whole . By analogy with with mappings, for every its image is defined, and for every its preimage is defined also; for every its image is the union of the images of all the elements from , and, similarly, for every its preimage is the union of the preimages of all the elements from .
A relation between and is called a correspondence, if the restrictions of the canonical projections and onto are surjections. By we denote the set of all correspondences between and .
Let and be metric spaces, then for every relation its distortion is defined as
The following result is well-known.
Proposition 2.1 ().
For any metric spaces and we have
If and are finite metric spaces, then the set is finite, hence, there exists an such that . Every such correspondence is called optimal. Notice that optimal correspondences do also exist for any compact metric spaces and , see . By we denote the set of all optimal correspondences between and .
For arbitrary nonempty sets and a correspondence is called irreducible, if it is a minimal element of the set w.r.t. the order given by the inclusion relation. By we denote the set of all nonempty irreducible correspondences between and .
The next result describes the structure of irreducible correspondences.
Every irreducible correspondence generates partitions and , together with the partitions and of the sets and , respectively. Also, induces a bijection between the sets and such that iff either , or , or .
Figure 1 illustrated the latter theorem.
The next results (see also ) demonstrates the importance of the irreducible correspondences for calculating the Gromov–Hausdorff distances.
For every there exists an such that .
Now, let and be metric spaces. We put .
For any we have .
If is a metric space, then by we denote the diameter of , i.e., the value
The next result is well-known .
If stands for the one-point metric space, and is an arbitrary metric space, then
For any metric spaces and it holds
3 Strong Non-Universality of the Gromov–Hausdorff Space
Show that the two-point metric space with the distance may be isometrically embedded into in such a way that this embedding can not be extended to an isometric embedding of three-point metric space with distances and . This example demonstrates that the space is not universal in the strong sense.
4 Geometry of Gromov–Hausdorff Space in Neighborhoods of Generic Spaces
By we denote the subset of consisting of all metric spaces each of which has at most points; let stand for the subset of consisting of all -point metric spaces.
For every we define a mapping in the following way. For we consider all the distances between distinct points, arrange these values in ascending order, and let be the resulting vector from divided by . Let be the th coordinate of the vector .
We say that are structural isomorphic if there exists a bijection preserving the order on the set of distances: iff . Every such we call a structural isomorphism. Notice that for a structural isomorphism and each the condition implies . In other words, for any , the th coordinates of vectors and are equal to half-distances between the pairs of points from and corresponding to each other under the structural isomorphism . Thus, if we consider as a correspondence between and , i.e., , then we get .
By we denote the subset of consisting of all the spaces such that all non-zero distances in them are distinct. Notice that if are structural isomorphic, then structural isomorphism is uniquely defined. Moreover, the structural isomorphism generates an equivalence relation on , therefore, is partitioned into the corresponding equivalence classes that are referred as classes of structural isomorphism. Notice the following obvious property: for any structural isomorphic , a space is structurally isomorphic to iff it is structurally isomorphic to . The latter proves the correctness of the concept of the closure of a structural isomorphism class as of the family of all which are structurally isomorphic to some (this definition does not depend on the choice of the representative in the structural isomorphism class). Thus, the space is covered by the closures of the structural isomorphism classes, and each two spaces from the same such closure are structurally isomorphic. It is not difficult to show that these closures can be defined as maximal subfamilies of such that any two their elements are structurally isomorphic.
The next result is not used in the present paper but seems to be self-important.
Let be the closure of a structural isomorphism class. Then the mapping is incompressible.
Let us choose arbitrary , and let be a structural isomorphism. Then , therefore,
By a generic space we mean every finite metric space such that all its non-zero distances are pairwise distinct, and all its triangle inequality hold strictly. By we denote the family of all the generic spaces. Clearly that is everywhere dense in . We put . By definition, , thus, is also partitioned into the structural isomorphism classes.
Let , . We define to be equal to the least of the following two numbers:
For we put .
If we put in the definition of the second minimum, then we get . Thus, for all . For this property holds as well.
It is easy to see that belongs to iff . Besides, for any such that for all we have , the and are structurally isomorphic, and the mapping is a structural isomorphism.
Let . We put , , , . Then the mapping is an isometry.
Choose an arbitrary . Let us show that . Indeed, suppose otherwise that , then for any there exists an such that for some distinct it holds , hence, . Therefore, , a contradiction.
So, . Choose an arbitrary . If is not a bijection, then we can apply the arguments we used just above and come to a contradiction again. Thus, has to be a bijection. Renumbering if necessary the elements from , without loss of generality we can assume that for all . Now, show that is a structural isomorphism. Notice that for any the relations
are valid, and hence, if , then we have , because .
Therefore, , thus, the mapping is an isometry.
It remains to show that is surjective. Choose an arbitrary , and for every put . Since , we have for all distinct . Choose arbitrary pairwise distinct , then
Besides . Thus, the values generates a metric on the set . Let be the metric space obtained in this way, and let be the bijection . Then
thus, . It remains to notice that . ∎
5 Isometric Embedding of a Finite Metric Space into
In this section we prove the weak universality of the Gromov–Hausdorff space for finite metric spaces.
Let be an arbitrary finite metric space consisting of points, and be the least integer such that . Then can be isometrically embedded into in such a way that its image belongs to .
Put . First assume that for some .
By we denote the isometric Kuratowski embedding: , see . Recall that the translations are isometries of .
Let stand for the diameter of . Consider a generic metric space such that . Put , then . As we mentioned above, the space with the metric induced from is isometric to .
Put and , then, by Theorem 4.1, is an isometry. Therefore, is isometric to .
Now, assume that for any . Consider the least such that . Extend upto a metric space consisting of points. We do it in such a way that the distances between points from are preserved, and all the remaining distances are equal to the diameter of the space (it is easy to verify that is a metric space). By the previous arguments, is isometric to a subspace of , thus, is isometric to the corresponding part of this subspace. ∎
- Ivanov A. O., Nikolaeva N. K., Tuzhilin A. A. The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic. arXiv:1504.03830, 2015.
- Burago D., Burago Yu., Ivanov S. A Course in Metric Geometry. Graduate Studies in Mathematics, vol.33. A.M.S., Providence, RI, 2001.
- Ivanov A.O., Iliadis S., Tuzhilin A.A. Realizations of Gromov-Hausdorff Distance. ArXiv e-prints, arXiv:1603.08850, 2016.
- Ivanov A.O., Tuzhilin A.A. Gromov–Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings. ArXiv e-prints, arXiv:1604.06116, 2016.
- Ivanov A.O., Tuzhilin A.A., “One-dimensional Gromov Minimal Filling,” ArXiv e-prints, arXiv:1101.0106, 2011.