Realizations of Gromov-Hausdorff Distance

Abstract

It is shown that for any two compact metric spaces there exists an "optimal" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in [1], where it is proved that the Gromov-Hausdorff space is geodesic. Notice that all proofs in the present paper are elementary and use no more than the idea of compactness.

Branching geodesics of the Gromov--Hausdorff distance

“…An element of CR opt (X, d, Y, e) is said to be optimal. The proof of the next lemma is presented in [2] and [8]. For a set X, we define ∆ X ∈ R(X, X) by ∆ X = { (x, x) | x ∈ X }, and we call it the trivial correspondence of X.…”

“…The following construction of geodesics using optimal closed correspondences is presented in [2] and [8].…”

“…Ivanov, Nikolaeva, and Tuzhilin [9] proved that (M, GH) is a geodesic space by showing the existence of the mid-point of all two points of M. Klibus [11] proved that the closed ball in the Gromov-Hausdorff space centered at the one-point metric space is a geodesic space. Chowdhury and Mémoli [2] constructed an explicit geodesic in (M, GH) using an optimal closed correspondence (see also [8]). They also showed that (M, GH) permits branching geodesics by constructing branching geodesics from the one-point metric space.…”

Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes

“…For finite metric spaces X and Y the set R(X, Y ) is finite as well, therefore there always exists an R ∈ R(X, Y ) such that d GH (X, Y ) = 1 2 dis R. Every such correspondence R is called optimal. Notice that the optimal correspondences exist also for any compact metric spaces X and Y , see [3]. The set of all optimal correspondences between X and Y is denoted by R opt (X, Y ).…”

“…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 X to an n-point simplex, where n is less than or equal to the cardinality of X, one can consider only those correspondences which generates partitions of the space X into n nonempty disjoint subsets, see Theorem 4.2.…”