Geometry of Compact Metric Space in Terms of Gromov–Hausdorff Distances to Regular Simplexes
In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov–Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are equal to each other. It turns out that these Gromov–Hausdorff distances depend on some geometrical characteristics of finite partitions of the compact metric spaces; some of the characteristics can be considered as a natural analogue of the lengths of edges of minimum spanning trees. As a consequence, we constructed an unexpected example of a continuum family of pairwise non-isometric finite metric spaces with the same distances to all simplexes.
In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov–Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are equal to each other. In  these distances were used to calculate the length of edges of a minimum spanning tree constructed on a finite metric space. In the present paper we generalize the results from  to the case of arbitrary compact metric spaces. It turns out that these Gromov–Hausdorff distances depend on some geometrical characteristics of finite partitions of compact metric spaces; some of the characteristics can be considered as a natural analogue of the lengths of edges of minimum spanning trees. We calculate the Gromov–Hausdorff distances from an arbitrary compact metric space to a simplex of sufficiently small or sufficiently large diameter, see Theorems 4.10, 4.14, and 4.15. For a finite -point metric space we find the distances to an arbitrary simplex consisting of at least points (Theorems 4.8, 4.1, and 4.16). Nevertheless, the general problem of calculating the distance from an arbitrary metric space to an arbitrary simplex remained unsolved yet. We demonstrate non-triviality of the problem by presenting a few examples in the end of the paper. In particular, we show that the set of all distances from a compact metric space to all simplexes is not a metric invariant, i.e., such collections can coincide for non-isometric finite metric spaces. Moreover, we construct an example of infinite (continuum) set of pairwise non-isometric finite metric spaces having the same collection of those distances.
In this paper we use the technique of irreducible optimal correspondences [1, 3, 4]. We show that to calculate the Gromov–Hausdorff distance from a compact metric space to an -point simplex, where is less than or equal to the cardinality of , one can consider only those correspondences which generates partitions of the space into nonempty disjoint subsets, see Theorem 4.2. This result has enabled us to advance essentially in calculations of concrete Gromov–Hausdorff distances.
For an arbitrary set by we denote its cardinality.
Let be an arbitrary metric space. The distance between its points and is denoted by . If are nonempty, then we put . If , then we write instead of .
For a point and a real number by we denote the open ball of radius centered at ; for any nonempty and we put .
1.1 Hausdorff and Gromov–Hausdorff distances
For nonempty we put
This value if called the Hausdorff distance between and . It is well-known  that the Hausdorff distance is a metric on the family of all nonempty closed bounded subsets of .
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 infimum of real numbers such that there exists a realization of the pair with . It is well-known  that the is a metric on the family of isometry classes of compact metric spaces.
For various calculations of the Gromov–Hausdorff distances, the technique of correspondences is useful.
Let and be arbitrary nonempty sets. Recall that a relation between the sets and is a subset of the Cartesian product . By we denote the set of all nonempty relations between and . Let us look at each relation as at a multivalued mapping, whose domain may be less than . Then, similarly with the case of mappings, for any and any their images and are defined, and for any and any their preimages and are also defined.
A relation is called a correspondence, if the restrictions of the canonical projections and onto are surjective. By we denote the set of all correspondences between and .
Let and be arbitrary metric spaces. The distortion of a relation is the value
Proposition 1.1 ().
For any metric spaces and we have
For finite metric spaces and the set is finite as well, therefore there always exists an such that . Every such correspondence is called optimal. Notice that the optimal correspondences exist also for any compact metric spaces and , see . The set of all optimal correspondences between and is denoted by . Thus, for compact metric spaces and we have .
The inclusion relation generates a partial order on : , iff . The relations minimal with respect to this order are called irreducible, and the remaining ones are referred as reducible. In  it is proved that for any compact metric spaces and there always exists an irreducible optimal correspondence . By we denote the set of all irreducible optimal correspondences between and . As it was mentioned above, .
The next well-known facts can be easily proved by means of the correspondences technique. For any metric space and any positive real let stand for the metric space which is obtained from by multiplication of all the distances by .
Proposition 1.2 ().
Let and be metric spaces. Then
If is a single-point metric space, then ;
If , then
, in particular, for bounded and it holds ;
For any and any we have . Moreover, for the unique invariant space is the single-point one. In other words, the multiplication of a metric by is a homothety of centered at the single-point space.
1.2 A few elementary relations
The next relations will be useful for concrete calculations of Gromov–Hausdorff distances.
For any nonnegative and the following inequality holds:
Indeed, if , then . If , then . ∎
Let be a nonempty bounded subset, and let . Then
Consider the segment . If is placed to the left side of the segment middle point, i.e., , then the value is achieved at the right end of the segment, i.e., it is equal to ; also, . Therefore, for such the proposition holds. One can similarly consider the case . ∎
Let be a nonempty bounded subset, , and let . Then
Let , then . By Proposition 1.4, we have
2 Minimum Spanning Trees
To calculate the Gromov–Hausdorff distance between finite metric spaces, minimum and maximum spanning trees turn out to be useful. Also, the edges lengths of these trees turn out to be closely related to some geometrical properties of various partitions of the ambient space.
Let be an arbitrary (simple) graph with the vertex set and the edge set . If is a metric space, then the length of edge of the graph is defined as the distance between the ending vertices and of this edge; also, the length of the graph is defined as the sum of all its edges lengths.
Let be a finite metric space. We define the number as the length of the shortest tree of the form . This value is called the length of minimum spanning tree on ; a tree such that is called a minimum spanning tree on . Notice that for any there exists a minimum spanning tree on it. The set of all minimum spanning trees on is denoted by .
2.1 -spectrum of a finite metric space
Notice that minimum spanning tree may be defined not uniquely. For by we denote the vector, whose components are the lengths of edges of the tree , ordered descendingly. If it is clear which metric is used, then we write instead of . The next result is well-known.
For any we have .
Proposition 2.1 explains correctness of the following definition.
For any finite metric space , by we denote the vector for arbitrary , and we call this vector by the -spectrum of the space .
For any set by we denote the family of all possible partitions of the into its nonempty subsets. Suppose now that is a metric space and . Put
The next result is proved in .
Let be a finite metric space and . Then
2.2 -spectrum of an arbitrary metric space
Now we generalize the concept of -spectrum by means of Proposition 2.4.
For any metric space and we put if , and otherwise. We call the set by -spectrum of .
If , then for .
In , for metric spaces which can be connected by a finite length tree (see  for definitions), a necessary condition of minimum spanning trees existence is obtained. It follows from [6, Theorem 1] that for a metric space which can be connected by a minimum spanning tree of a finite length, all the edges of are exact in the following sense. For each edge and the corresponding vertex sets and of the trees forming the forest , the length of equals to the distance between and .
Moreover, for described above and any there are finitely many edges of , whose lengths are more than or equal to . This enables us to order the edges in such a way that their lengths decrease monotonically. Let be such an order, and put .
For any positive integer consider the partition of into vertex sets of the trees forming the forest . Then .
Each and , , are connected by a path in , and the path contains at least one of the edges , . Since is minimal, then , thus and, therefore, . On the other hand, if we choose and in such a way that connects them, then , because the edge is exact. ∎
Let be an arbitrary partition of , then .
Denote by the set of all edges of connecting different elements of partition . The set contains at least edges (otherwise, the graph is disconnected), hence . Let satisfy , and let and be those elements of which are connected by . Then . ∎
If there exists a minimum spanning tree connecting a metric space , and are the edges of this tree ordered descendingly, then .
3 Maximum Spanning Trees
Let be a finite metric space. Maximum spanning tree on is a longest tree among all trees of the form . By we denote the length of a maximum spanning tree on , and by we denote the set of all maximum spanning trees on .
The next construction is useful for description of relations between minimum and maximum spanning trees.
Let be an arbitrary not pointwise bounded metric space. Choose any and define on a new distance function: for any , and for any .
The function is a metric on .
Indeed, it is obvious that is positively definite and symmetric. To verify the triangle inequalities, choose any pairwise distinct points , then
If two of these points coincide, say, if , then and , hence
The set with the metric defined above is denoted by .
Let be a finite metric space, , and . Denote by the vector constructed from nonzero distances in , ordered descendingly.
If is a finite metric space, and , then .
The next Proposition describes a duality between minimum and maximum spanning trees.
Let be a finite metric space, and . Then a tree is a minimum spanning tree on , iff it is a maximum spanning tree on .
Let stand for the distance function on . Then . ∎
For , by we denote the vector constructed from the lengths of edges of the tree , ordered ascendingly. If it is clear which metric is in consideration, then we write instead of .
Proposition 3.3 implies the next result.
Let be a finite metric space, , , and . Let us denote also by the vector of the length , all whose components equal . Then and .
For any we have .
Proposition 3.5 motivates the next definition.
For any finite metric space by we denote the value for an arbitrary , and we call this value by -spectrum of the space .
Let be an arbitrary metric space, and be its nonempty subsets. Put
If the metric on is denoted by as well, we put .
For a set , let stand for the family of all coverings of the set consisting of nonempty subsets. Now, let be a metric space, and . Put . It it is clear which metric is in consideration, we write instead of .
Let be nonempty subsets of a bounded metric space, and be the metric of the space . Then .
For any we have .
Indeed, let be the metric of the space . Then, by Lemma 3.8, it holds
Up to the end of this section, stands for a finite metric space consisting of points, and .
The next result is an analogue of Proposition 2.4.
We have .
Under the above notations, it holds .
In the next two Propositions, we use , and the edges are supposed to be ordered in such a way that . Moreover, for the convenience reason, we put .
Let be a partition into the vertex sets of the trees forming the forest . Then for each we have .
Choose arbitrary . If , then by the order we have chosen on . If , then consider the unique path in connecting and . Since is the set of vertices of a tree from the forest , then for each edge of this path it holds , and, therefore, . Since the tree is maximal, then for some , hence . ∎
Let be a partition into the vertex sets of the trees forming the forest . Then for every we have .
Indeed, consider arbitrary , , and let , be arbitrary points. Consider the unique path in the tree connecting and . This path contains at least one of the edges thrown out. Let it be . If is not an edge in , then contains a unique cycle , and the maximality of implies that each edges of the path is not shorter than . In particular, . If is an edge of , then it coincides with an edge we threw out, say, with , and again we have . Thus, . ∎
3.2 -spectrum of an arbitrary metric space
For any metric space and any we put if , and otherwise. The set we call the -spectrum of .
If , then for .
4 Calculation of Distances between Compact Metric Space and Finite Simplexes
We call a metric space a simplex, if all its nonzero distances are the same. Notice that a simplex is compact, iff it is finite. A simplex consisting of vertices on the distance from each other is denoted by . For the space is denoted by for short.
4.1 Distances from a finite metric space to simplexes with greater numbers of points
Let be a finite metric space, . Then for every , , and we have
Choose an arbitrary . Since , then there exists such that , hence and, therefore, .
Put and let be the correspondence . Then
where the second equality follows from Proposition 1.5. This implies that .
If , then , hence , q.e.d.
If , then . Choose a pair such that . Take an arbitrary . Then one of the following conditions holds:
there exists such that , but then ;
there exist such that , and then .
Thus, for any we have , therefore, in the case under consideration the equality is valid. ∎
4.2 Distances from a compact metric space to simplexes with no greater number of points
Let be a compact metric space. Then for every , , and there exists an such that the family is a partition of . In particular, if , then one can take a bijection as an optimal correspondence .
Let be an arbitrary irreducible correspondence. Since is irreducible, the condition for some implies for some . In particular, if for some , then does not intersect any , . Let us introduce the following notation: for we put to be equal to the number of pairs such that and . Clearly that for the condition is equivalent to that the family forms a partition .
Suppose that the family is not a partition. We show that in this case one can always reconstruct the correspondence in such a way that the resulting correspondence becomes an irreducible one with and . If , then we put , and repeat this procedure. After a finite number of steps we will get a correspondence such that and , q.e.d.
So, let for some it holds . Since , then there exists such that , hence and does not intersect any other . Choose an arbitrary point , and construct a new correspondence which coincides with at all elements of the simplex, except , , , and
Clearly that since is uniquely defined on all elements of the simplex, and each element of goes to at least one element of the simplex. Besides that, the correspondence is still irreducible, which can be verified directly.
Further, to estimate let us notice that among all only and are changed, thus, it is sufficient to investigate how the intersections with these two sets changes. Since does not intersect the remaining , and since , then and does not intersect the remaining . Besides that, . Thus, the number of the remaining intersecting with is the same as the number of the remaining intersecting with (and it is equal ). Concerning , the intersection is nonempty (and, perhaps, there are some other nonempty intersections with ). However, does not intersect any of the remaining , hence .
Let us prove that .
Clearly that and for all . To complete the proof it remains to show that for any and the inequality holds.
If and are not contained in , then .
Now, suppose that one of the indices and , say , is not contained in , but the remaining one is contained. In this case,
, because and ;
, because ;
, because .
Finally, consider the case . We have
, because and ;
, because and ;
where the second inequality holds according to Proposition 1.3.
Thus, all the values from the expression for do not exceed , hence , q.e.d. ∎
Theorem 4.2 helps to get a useful formula for Gromov–Hausdorff distance between a compact metric space and a finite simplex such that the number of points in the simplex does not exceed the cardinality of .
For an arbitrary metric space , , and we put . Notice that for any which differs from by renumbering of the elements of the partition , we have .
For let us put
Let be an arbitrary metric space, and , . Then for any and it holds
Let . By definition of distortion,
By Proposition 1.4, we have
Let be a compact metric space. Then for every , , and it holds
By Theorem 4.2, there exists an such that the family is a partition of . Thus, is achieved at some , . ∎
Let be a compact metric space, and , . Then for any we have
4.3 Distance from finite metric space to simplexes with the same number of points
For any metric space we put
Notice that , and the equality holds, iff is a simplex. Besides that, if is a finite metric space consisting of points, and , then .
Let be a finite metric space, , , , . Then
More exactly, if , then , otherwise .
For each it holds . Besides that, for all such we have and . It remains to apply Corollary 4.7. ∎
4.4 Distance from a compact metric space to simplexes having at most the same number of points
In  the following result is proved.
Proposition 4.9 ().
Let be a finite metric space, , and . Then for all .
The next theorem generalizes Proposition 4.9 to the case of compact metric spaces and also weaken the restrictions on the parameter .
Let be a compact metric space, be the -spectrum of , and . Then for all , .
Put and choose an arbitrary partition . Due to Proposition 4.5, we have
Notice that and . Further, since , then , and hence