Some Properties of Gromov–Hausdorff Distances
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Abstract
The Gromov–Hausdorff distance between metric spaces appears to be a useful tool for modeling some object matching procedures. Since its conception it has been mainly used by pure mathematicians who are interested in the topology generated by this distance, and quantitative consequences of the definition are not very common. As a result, only few lower bounds for the distance are known, and the stability of many metric invariants is not understood. This paper aims at clarifying some of these points by proving several results dealing with explicit lower bounds for the Gromov–Hausdorff distance which involve different standard metric invariants. We also study a modified version of the Gromov–Hausdorff distance which is motivated by practical applications and both prove a structural theorem for it and study its topological equivalence to the usual notion. This structural theorem provides a decomposition of the modified Gromov–Hausdorff distance as the supremum over a family of pseudometrics, each of which involves the comparison of certain discrete analogues of curvature. This modified version relates the standard Gromov–Hausdorff distance to the work of Boutin and Kemper, and Olver.
Keywords
Gromov–Hausdorff distance Metric geometry Curvature sets1 Introduction
The Gromov–Hausdorff distance is a useful tool for studying topological properties of families of metric spaces. According to Berger [2], Gromov first introduced the notion of Gromov–Hausdorff distance in his ICM 1979 address in Helsinki on synthetic Riemannian geometry. The goal of the program he put forward was the study of all (Riemannian) metric structures: to give some structure to this space and to study completeness, possible convergences, compact families, and related concepts. Gromov made use of the Gromov–Hausdorff distance as a tool for attacking the proof of his theorem on groups of polynomial growth [12].
The Gromov–Hausdorff distance has received attention in the applied literature, where the motivation for its use originated in the area of object matching under invariances [24, 25]. The idea is to regard objects as metric spaces in a manner such that the choice of metric with which these objects are endowed dictates the type of invariance that is desired. A standard example is that of comparing objects in Euclidean space under invariance to rigid isometries: in that case objects are given the (restriction of the) Euclidean metric.
It is known that solving for the GH distance between finite metric spaces leads to NPhard problems [20]. Applied researchers have tackled the numerical computation of the GH distance using ad hoc optimization techniques [6, 24, 25] and not many inroads have been made into producing lower bounds for the GH distance, see [10, 20, 21]. See [23] for properties of the related Gromov–Wasserstein distance in the context of metric spaces endowed with probability measures.
In this paper we identify a number of new lower bounds for the Gromov–Hausdorff distance, all of which can be computed in polynomial time. We also study a certain modified version of the Gromov–Hausdorff distance which is in turn related to a family of isometry invariants of metric spaces that provides full classification of compact metric spaces up to isometry. We believe that the material in this paper will provide more understanding about the use of the Gromov–Hausdorff distance in applications, as well as about its relationship with preexisting work.
1.1 Organization of the Paper
In Sect. 1.2 we set up basic terminology. In Sect. 2 we first introduce the definition of several isometry invariants of compact metric spaces and discuss their ability to discriminate between certain metric spaces. In Sect. 3 we recall the main properties of the standard Gromov–Hausdorff distance and the topology it generates. Then, in Sect. 3.1, we state Theorem 3.4: this theorem establishes a hierarchy of lower bounds for the GH distance between two given compact metric spaces, which involves, in a precise sense, the comparison of all the invariants defined in Sect. 2. In Sect. 3.1 we also look into the numerical implementation and computational complexity of the lower bounds stated in Theorem 3.4.
In Sect. 4, we explain how, with a small change to expression (2), we obtain another possible distance between compact metric spaces, which we call the modified Gromov–Hausdorff distance. The definition of this distance is motivated by computational considerations [6, 24, 25], and it leads to solving two independent or decoupled matching problems. In that section we give several examples, and by an explicit construction, we also prove that this new definition gives us a distance which is different from the standard GH distance. In Theorem 4.1 we prove that this modified Gromov–Hausdorff does provide a legitimate distance on the collection of all compact metric spaces, and in Theorem 4.2 we prove that both the standard GH distance and the modified GH distance are topologically equivalent within Gromov–Hausdorff precompact classes of compact metric spaces. The modified GH distance turns out to be a lower bound for the standard GH distance.
In Sect. 5 we discuss another family of isometry invariant of metric spaces, called curvature sets, which were first considered by Gromov in [13]. We discuss how these invariants absorb useful information from compact metric spaces, in a manner that suggests that they may be of interest in practical applications. In addition, we also show how curvature sets are intimately related to the constructions of Boutin and Kemper [4], and Olver [26]. Theorem 5.1 provides a decomposition of the modified GH distance as the supremum over a family of pseudometrics on the collection of all compact metric spaces, where each of these pseudometrics involves the comparison of curvature sets of the intervening spaces.
Finally, in Sect. 6 we give some remarks about possible extensions.
With the goal of providing a reference for some aspects of the Gromov–Hausdorff distance that are not covered elsewhere, we provide proofs for all our results, and in order to maximize readability, we give all proofs of our mathematical statements at the end of the section where they are stated.
1.2 Background and Notation
Recall that a metric space is a pair (X,d _{ X }) where X is a set and d _{ X }:X×X→ℝ^{+} with the properties (I) d _{ X }(x,x′)=d _{ X }(x′,x) for all x,x′∈X; (II) d _{ X }(x,x″)≤d _{ X }(x,x′)+d _{ X }(x′,x″) for all x,x′,x″∈X; and (III) d _{ X }(x,x′)=0 if and only if x=x′.
By \({{\mathcal{B}}}({X})\) we denote all Borel sets of X. Recall that a set S⊂X is called an εnet of X if for all x∈X there exists s∈S with d _{ X }(x,s)≤ε. If λ≥0 and (X,d _{ X }) is any metric space, then λ⋅X will denote the metric space (X,λ⋅d _{ X }). We denote by \(\mathcal{G}\) the collection of all compact metric spaces. Given (X,d _{ X }) and (Y,d _{ Y }) in \(\mathcal{G}\), a map φ:X→Y is called an isometry if d _{ X }(x,x′)=d _{ Y }(φ(x),φ(x′)) for x,x′∈X and φ is surjective. When this happens, one says that X and Y are isometric. Given a fixed set ℑ, we say that a function \(\iota:\mathcal{G}\rightarrow\mathfrak{I}\) is an isometry invariant of metric spaces, if ι(X)=ι(Y) whenever X and Y are isometric.
For a Riemannian manifold (X,g _{ X }) we denote by vol _{ X }(⋅) the Riemannian volume measure on X; its total volume by Vol(X)=vol _{ X }(X); and its geodesic distance function by d _{ X }.
For k∈ℕ let Π _{ k } denote the set of all permutation matrices of size k×k. It will be useful to consider the following notation: D _{ X } is the map that assigns each finite subset \(\mathbb{X}\) of the metric space (X,d _{ X }) with its distance matrix, that is, \(\mathbf{D}_{X}(\mathbb{X}) =({\!}({d_{X}(x,x')})\!)_{x,x'\in\mathbb{X}}\).
Recall that a subset A of a topological space Z is precompact whenever its closure \(\overline{A}\) is a compact subset of Z.
2 Isometry Invariants of Metric Spaces
In Theorem 3.4 we prove the quantitative stability of several isometry invariants of metric spaces that we now define.
Definition 2.1

Diameter: diam(X):=max_{ x,x′} d _{ X }(x,x′).

Separation: sep(X):=inf_{ x≠x′} d _{ X }(x,x′).

Circumradius: rad(X):=min_{ x }max_{ x′} d _{ X }(x,x′).

Eccentricity Function: ecc _{ X }:X→ℝ^{+} given by x↦max_{ x′} d _{ X }(x,x′).

Distance set: \(\boldsymbol{\mathcal{D}}_{X}:=\{d_{X}(x,x'),\,x,x'\in X\}\).

Local distance sets: \(\boldsymbol{\mathcal {L}}_{X}:X\rightarrow {{\mathcal{B}}}({\mathbb{R}^{+}})\) given by x↦{d _{ X }(x,x′), x′∈X}.
In the applied literature, local distance sets have been considered by Grigorescu and Petkov [11], eccentricities by Hilaga et al. [17] and Hamza and Krim [1], global distance sets by Osada et al. [27] and Boutin and Kemper [4].
Example 2.1
(Two nonisometric metric spaces with the same distance set)
Example 2.2
(Two nonisometric spaces with matching eccentricities)
Remark 2.1
Note that by the preceding examples, the lower bounds given by (11) and (14) in Theorem 3.4 are independent.
Remark 2.2

\(\boldsymbol{\mathcal{L}}_{X}(x) \subseteq[0,\mathbf {ecc}_{X}(x)]\) for all x∈X. There is equality for connected metric spaces.

rad(X)=min_{ p } ecc _{ X }(p)≤ecc _{ X }(x)≤max_{ p } ecc _{ X }(p)=diam(X), for all x∈X.
Example 2.3
One has rad(T(a,b,c))=min(max(a,b),max(b,c),max(c,a)).
3 The Gromov–Hausdorff Distance and Lower Bounds
In this section we recall the main properties of the GH distance and then in Sect. 3.1 we state and prove a theorem about the quantitative stability of the invariants introduced in Definition 2.1.
Definition 3.1
The Hausdorff distance is indeed a metric on the collection of closed subsets of a compact metric space (Z,d) [7, Proposition 7.3.3].
Definition 3.2
(Chap. 3 of [13])
The Gromov–Hausdorff distance \(d_{\mathcal{G}\mathcal{H}}({X},{Y})\) between compact metric spaces (X,d _{ X }) and (Y,d _{ Y }) is defined to be the infimal ε>0 s.t. there exists a metric d on X⊔Y with \(d_{_{X\times X}}=d_{X}\) and \(d_{_{Y\times Y}} = d_{Y}\) for which the Hausdorff distance between X and Y (as subsets of (X⊔Y,d)) is less than ε. From now on let \(\mathcal{G}\) denote the collection of all compact metric spaces.
Definition 3.3
(The Gromov–Hausdorff topology)
One says that \(\{(X_{n},d_{X_{n}})\}_{n\in \mathbb{N}}\subset\mathcal{G}\) Gromov–Hausdorff converges to \(X\in\mathcal{G}\) if and only if \(d_{\mathcal{G}\mathcal{H}}({X_{n}},{X})\rightarrow0\) as n↑∞.
Theorem 3.1
(Chap. 10 [29])
The space \((\mathcal{G},d_{\mathcal{G}\mathcal{H}})\) is separable and complete.
Definition 3.4
(Covering number)
For each ρ≥0 let cov _{ X }(ρ) denote the minimal number of open balls of radius ρ with which one can cover the compact metric space X.
The topology generated by the GH distance (see Definition 3.3) is rather coarse and this allows the existence of rich families of precompact sets.
Theorem 3.2
(Gromov’s precompactness theorem, [7])
For a bounded function N:ℝ^{+}→ℕ and D>0 let \(\mathcal {F}(N;D)\subset\mathcal{G}\) denote the collection of all compact metric spaces X with diam(X)≤D and s.t. cov _{ X }(ρ)≤N(ρ) for each ρ>0. Then, \(\mathcal{F}(N;D)\) is precompact for the Gromov–Hausdorff topology.
Example 3.1
(Simplices)
Consider the family {Δ _{ n }, n∈ℕ} of compact metric spaces. One has diam(Δ _{ n })=1 for all n∈ℕ but clearly there exists no function N:ℝ^{+}→ℕ as in Theorem 3.2. We see in Example 4.1 below that \(d_{\mathcal{G}\mathcal{H}}({\varDelta_{n}},{\varDelta_{m}})=\frac{1}{2}\) for all n≠m; hence {Δ _{ n }}_{ n∈ℕ} cannot have a converging subsequence.
Remark 3.1
(Precompact families of Riemannian manifolds)
Theorem 3.3 item 5 below provides an alternative expression for the GH distance.
Definition 3.5
(Correspondence)

∀a∈A, there exists b∈B s.t. (a,b)∈R.

∀b∈B, there exists a∈X s.t. (a,b)∈R.
Example 3.2
Theorem 3.3
[7]
 (1)Let (X,d _{ X }), (Y,d _{ Y }) and (Z,d _{ Z }) be metric spaces then$$d_{\mathcal{G}\mathcal{H}}({X},{Y})\leq d_{\mathcal{G}\mathcal{H}}({X},{Z}) + d_{\mathcal{G}\mathcal{H}}({Y},{Z}).$$
 (2)
Assume that (X,d _{ X }) and (Y,d _{ Y }) are compact metric spaces. Then \(d_{\mathcal{G}\mathcal{H}}({X},{Y})=0\) if and only if (X,d _{ X }) and (Y,d _{ Y }) are isometric.
 (3)Let \(\mathbb{X}\) be a subset of the compact metric space (X,d _{ X }). Then$$d_{\mathcal{G}\mathcal{H}}\bigl ({(X,d_X)},{(\mathbb{X},{d_X}_{_{\mathbb{X}\times\mathbb {X}}})}\bigr )\leq d_{\mathcal{H}}^{X} ({\mathbb{X}},{X} ).$$
 (4)For compact metric spaces (X,d _{ X }) and (Y,d _{ Y }):$$d_{\mathcal{G}\mathcal{H}}({X},{Y}) \leq \frac{1}{2}\max\bigl({\mathbf {diam}} ({X} ),{\mathbf {diam}} ({Y} )\bigr).$$(6)
 (5)For compact metric spaces (X,d _{ X }) and (Y,d _{ Y }),$$ d_{\mathcal{G}\mathcal{H}}({X},{Y})=\frac{1}{2}\inf_{R\in {\mathcal{R}}({X},{Y})}\sup_{\tiny \begin{array}{ccc}x_1,x_2\in X\\ y_1,y_2\in Y\\ s.t.\,(x_i,y_i)\in R\\\end{array}} \bigld_X(x_1,x_2)d_Y(y_1,y_2)\bigr.$$(7)
Remark 3.2
Note that items 1 and 2 of the theorem encode the symmetry of the GH distance: let Z=X; then \(d_{\mathcal{G}\mathcal{H}}({X},{Y})\leq d_{\mathcal{G}\mathcal{H}}({X},{X})+d_{\mathcal{G}\mathcal{H}}({Y},{Z})=d_{\mathcal{G}\mathcal{H}}({Y},{X})\) for all \(X,Y\in\mathcal{G}\). Hence, by exchanging the roles of X and Y, one sees that \(d_{\mathcal{G}\mathcal{H}}({X},{Y})=d_{\mathcal{G}\mathcal{H}}({Y},{X})\).
Remark 3.3
Note that (2) asserts that the infimum over all correspondences \(R\in {\mathcal{R}}({X},{Y})\) in (7) can be restricted to all those correspondences with the form described in Example 3.2.
Example 3.3
(Distance between homothetic spaces)
Example 3.4
Fix \((X,d_{X})\in\mathcal{G}\). Consider the sequence \(\{(X,\frac{1}{n}\cdot d_{X})\}_{n\in \mathbb{N}}\subset\mathcal{G}\). Then, this sequence Gromov–Hausdorff converges to the metric space consisting of a single point.
Remark 3.4
(Gromov–Hausdorff distance and the BQAP)
 (1)
δ _{ ij }∈{0,1} for all i,j;
 (2)
∑_{ i } δ _{ ij }≥1 for all j;
 (3)
∑_{ j } δ _{ ij }≥1 for all i;
Actually, we prove next that, when n=m, min_{ δ∈D } L(δ) reduces to a BQAP. It is known that, as an instance of binary integer quadratic programming, the BQAP is an NPhard problem [28]. Indeed, it is clear that for any δ∈D there exist P∈Π _{ n } (n×n permutations matrices) such that δ _{ ij }≥P _{ ij } for all 1≤i,j≤n. Then, since Γ _{ ikjl } is nonnegative for all 1≤i,j,k,l≤n, it follows that L(δ)≥L(P). Therefore the minimal value of L(δ) is attained at some δ∈Π _{ n }.
3.1 Gromov–Hausdorff Stability of Metric Invariants
Theorem 3.4 below makes precise a sense in which the metric invariants of Sect. 2 are organized into a hierarchy of lower bounds for the GH distance.
Theorem 3.4
Remark 3.5
Similar hierarchies of lower bounds are possible in contexts when one assumes that more structure is given to the spaces. One concrete example of this is the case of metric measure spaces: compact metric spaces enriched with probability measures, where instead of the GH distance one constructs a mass transportation variant called the Gromov–Wasserstein distance [23]. In the more extreme case when one assumes that the spaces are restricted to a subclass of \(\mathcal{G}\) given by the collection of all compact Riemannian manifolds without boundary, then another similar hierarchy is possible, where now the GH distance is supplanted by a certain spectral version of the Gromov–Wasserstein distance, and the intervening lower bounds involve invariants that absorb spectral information of the underlying spaces [22].
Remark 3.6
Notice that for all n,m∈ℕ, \(F_{\mathbb{S}^{n},\mathbb{S}^{m}}(x,y)=0\) for all \(x\in \mathbb{S}^{n}\) and \(y\in \mathbb{S}^{m}\). Hence, all lower bounds in Theorem 3.4 are unable to discriminate between spheres of different dimension, see, however, Example 5.3.
Remark 3.7
(About the complexity associated to computing the lower bounds)
Notice that in the case both X and Y are finite, and given F _{ X,Y }, lower bound (8) above can be computed by solving X⋅Y bottleneck assignment problems [9, Chap. 6], each of which can be solved using the thresholding algorithm of [8, Sect. 5], with running time θ _{ N }:=O(N ^{2.5}logN), where N=max(X,Y). This lower bound is structurally the same as Lawler’s lower bound [28] in the context of the QAP. Now, the computation of F _{ X,Y } incurs cost N ^{2}⋅θ _{ N } as well and hence the total cost of computing (8) is 2⋅N ^{2}⋅θ _{ N }.
Similarly, the computation of \(\mathcal{A}(X,Y)\) incurs a running time θ _{ N }+N ^{2}⋅c _{ N } where c _{ N } is the cost of computing \(d_{\mathcal{H}}^{\mathbb{R}^{+}}({\boldsymbol {\mathcal {L}}_{X}(x_{0})},{\boldsymbol {\mathcal {L}}_{Y}(y_{0})})\) for a given pair (x _{0},y _{0})∈X×Y, which can be bounded by N ^{2}.
Finally, computing the RHS of (11) incurs cost θ _{ N }+2⋅N ^{2}.
We now turn our attention to the proof of Theorem 3.4. The proofs of the following three lemmas are given at the end of this section.
Lemma 3.1
Lemma 3.2
Lemma 3.3
Proof of Theorem 3.4
The validity of (12) follows directly from Lemma 3.3.
Note that as we saw in Remark 2.2, for any compact metric space X, min_{ x∈X } ecc _{ X }(x)=rad(X) and max_{ x∈X } ecc _{ X }(x)=diam(X), applying Lemma 3.2, one readily obtains (13).
For (14) notice that \(\boldsymbol {\mathcal {L}}_{X}(X)=\boldsymbol {\mathcal {D}}_{X}\), \(\boldsymbol {\mathcal {L}}_{Y}(Y)=\boldsymbol {\mathcal {D}}_{Y}\), and apply Lemma 3.3. The proof of (15) follows directly from Lemma 3.2 and the observation that \(\max\{\boldsymbol {\mathcal{D}}_{X}\}={\mathbf {diam}}({X})\). □
Proof of Lemma 3.1
Let ε>0 and \(R\in\mathcal{R}(A,B)\) be s.t. d(a,b)<ε for all (a,b)∈R. Since R is a correspondence between A and B it follows that inf_{ b∈B } d(a,b)<ε for all a∈A and inf_{ a∈A } d(a,b)<ε for all b∈B. Recalling (5) it follows that \(d_{\mathcal{H}}^{Z}(A,B)\leq\varepsilon\).
Assume now that \(d_{\mathcal{H}}^{Z}(A,B)<\varepsilon\). Then, for each a∈A there exist b∈B s.t. d(a,b)<ε. Then, we may define ϕ:A→B s.t. d(a,ϕ(a))<ε for all a∈A. Similarly, define ψ:B→A s.t. d(ψ(b),b)<ε for all b∈B. Consider \(R(\phi,\psi)\in {\mathcal{R}}({A},{B})\) as in Example 3.2. By construction d(a,b)<ε for all (a,b)∈R and hence we are done. □
Proof of Lemma 3.2
Assume that \(\varepsilon> d_{\mathcal{H}}^{\mathbb{R}} ({A},{B} )\). Then, for any a∈A there exists b∈B with a−b<ε. In particular, ε+b>a≥infA, and hence ε+b>infA for all b∈B. It follows that ε+infB>infA and similarly, ε+infA>infB. Thus, ε>infA−infB. The inequality for the difference of suprema is similar. □
Proof of Lemma 3.3
Finally, note that t−s<ε for all (t,s)∈S _{ R }. The conclusion now follows from Lemma 3.1. □
4 The Modified Gromov–Hausdorff Distance
We now consider a variant of the Gromov–Hausdorff distance, which we refer to as the modified Gromov–Hausdorff distance. The definition of this new distance is motivated by computational considerations [6, 24, 25].
This leads to the following definition:
Definition 4.1
(Modified Gromov–Hausdorff distance)
For brevity we will sometimes refer to the modified Gromov–Hausdorff distance by \(\widehat {\mbox {GH}}\) .
The definition of \(\widehat {\mbox {GH}}\) above expresses the fact that this distance can be computed by solving two decoupled or independent submatching problems: (I) finding the best map from X to Y, and (II) finding the best map in the opposite direction. This type of problem admits a binary integer programming formulation similar to the one in Remark 3.4 and is therefore still NPhard, but, for global optimization strategies such as those of [6, 25], having two decoupled problems is an important property that reduces the overall size of the optimization problem that one needs to solve in practice.
4.1 Properties of the Modified Gromov–Hausdorff Distance
We now prove that \(\widehat {\mbox {GH}}\) does indeed define a legitimate distance on collection the isometry classes of \(\mathcal{G}\). In addition, in this section we prove that these two distance are in general not equal, establish their topological equivalence, and also present several examples.
Theorem 4.1
 (1)
For all \(X,Y\in\mathcal{G}\), \(d_{\mathcal{G}\mathcal{H}}({X},{Y})\geq \widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})\).
 (2)\(\widehat {d}_{\mathcal{G}\mathcal{H}}({},{})\) is a strict metric on the isometry classes of spaces in \(\mathcal{G}\):

For \(X,Y\in\mathcal{G}\), \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})=0\) if and only if X and Y are isometric.

For \(X,Y,Z\in\mathcal{G}\), \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})\leq \widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Z})+\widehat {d}_{\mathcal{G}\mathcal{H}}({Z},{Y})\).

Theorem 4.2
Let \(\mathcal{F}\) be a GHprecompact family of compact metric spaces. Then, for any ε>0 there exists \(\delta=\delta(\mathcal {F},\varepsilon)>0\) s.t. whenever \(X,Y\in\mathcal{F}\) satisfy \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})<\delta\), then \(d_{\mathcal{G}\mathcal{H}}({X},{Y})<\varepsilon\).
Example 4.1
(GH and \(\widehat {\mbox {GH}}\) distances between simplices)
 (a)
dis(ϕ)≥1 for all ϕ:Δ _{ n }→Δ _{ m }, and
 (b)
\(\operatorname{infdis}(\varDelta_{m}\rightarrow\varDelta_{n})=0\).
Example 4.2
(Explicit formula for the modified Gromov–Hausdorff distance between metric spaces with three points)
Fix a _{1},a _{2},a _{3}>0 and \(a_{1}', a_{2}', a_{3}'>0\) that verify all triangle inequalities and denote T=T(a _{1},a _{2},a _{3}) and \(\mathrm {T}' =\mathrm {T}(a_{1}',a_{2}',a_{3}')\). Let {x _{1},x _{2},x _{3}} and {y _{1},y _{2},y _{3}} be the underlying sets of T and T′, respectively. We also introduce the convention that a _{1}=d _{T}(x _{2},x _{3}), a _{2}=d _{T}(x _{1},x _{3}), a _{3}=d _{T}(x _{1},x _{2}), and \(a_{1}' = d_{\mathrm {T}'}(y_{2},y_{3})\), \(a_{2}' = d_{\mathrm {T}'}(y_{1},y_{3})\), \(a_{3}' = d_{\mathrm {T}'}(y_{1},y_{2})\).
 (A)
Can be a bijection;
 (B)
Can map two points in T to one point in T′, and the remaining point in T to a different point in T′; or
 (C)
Can map all three points to one point.
In case (C), the distortion of any map is max_{ i } a _{ i }=diam(T).
It is of interest to ascertain whether the GH and \(\widehat {\mbox {GH}}\) distances are in some sense comparable (recall Theorem 4.1 item 1). A first question is whether GH and \(\widehat {\mbox {GH}}\) could be equal in general.
Remark 4.1
(The GH and \(\widehat {\mbox {GH}}\) distances are not equal in general)
Indeed, using the notation of Example 4.2, δ(X _{ α }↔Y _{ β })=max(α−β,α−1)=α−1>1, diam(X _{ α })=α, diam(Y _{ β })=β, γ(X _{ α }→Y _{ β })=γ(Y _{ β }→Y _{ α })=1. Thus, \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X_{\alpha}},{Y_{\beta}})=\frac{1}{2}\) by (18).
Alternatively, it is easy to construct maps ϕ:X _{ α }→Y _{ β } and ψ:Y _{ β }→X _{ α } with \(\max (\mathrm {dis} ({\phi} ),\mathrm {dis} ({\psi} ))=\frac{1}{2}\), see Fig. 2.
4.2 Proofs of Theorems 4.1 and 4.2
Lemma 4.1
Proof
Proof of Theorem 4.1
Item 1 is true by the definition of \(\widehat {\mbox {GH}}\) and (2).
In order to prove item 2 we need to prove symmetry, the triangle inequality, and the fact that \(d_{\mathcal{G}\mathcal{H}}({X},{Y})=0\) if and only if X and Y are isometric. Symmetry is clear, and the triangle inequality can be proved as follows: let \(X,Y,Z\in\mathcal{G}\) and δ _{1},δ _{2}>0 be s.t. \(\delta_{1}>\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Z})\) and \(\delta_{2}>\widehat {d}_{\mathcal{G}\mathcal{H}}({Y},{Z})\). Further, let ϕ _{1}:X→Z, ϕ _{2}:Y→Z, ψ _{1}:Z→X, ψ _{2}:Z→Y s.t. max(dis(ϕ _{1}),dis(ψ _{1}))<2δ _{1} and max(dis(ϕ _{2}),dis(ψ _{2}))<2δ _{2}. Let ϕ:X→Y be given by ψ _{2}∘ϕ _{1} and ψ:Y→X by ψ _{1}∘ϕ _{2}. From Lemma 4.1 one then sees that max(dis(ϕ),dis(ψ))<2(δ _{1}+δ _{2}), and hence \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})<\delta_{1}+\delta_{2}\), from which the triangle inequality follows.
That \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})=0\) when X and Y are isometric follows from item 1 and the similar claim for the standard GH distance (Theorem 3.3). Assume now that \(X,Y\in\mathcal{G}\) are s.t. \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})=0\). Then, this implies the existence of a sequence {ϕ _{ n }}_{ n∈ℕ} of maps ϕ _{ n }:X→Y with dis(ϕ _{ n })→0 as n↑∞. From now on the proof follows standard steps which we only sketch, see [7, Sect. 7.3]. Since X is compact, there is a countable dense S⊂X which we henceforth fix. By a diagonal procedure one can choose a subsequence {n _{ k }}_{ k }⊂ℕ s.t. for every x∈S, \(\{\phi_{n_{k}}(x)\}_{k}\) converges in Y. Define a map ϕ:S→Y as the pointwise limit of \(\{\phi_{n_{k}}\}_{k}\): \(\phi(x)=\lim_{k}\phi_{n_{k}}(x)\) for x∈S. Since \(\mathrm {dis} ({\phi_{n_{k}}} )\rightarrow0\) as k↑∞, one has \(d_{X}(x,x')=\lim_{k}d_{Y}(\phi_{n_{k}}(x),\phi_{n_{k}}(x'))=d_{Y}(\phi (x),\phi (x'))\) for all x,x′∈S. Thus, ϕ:S→X is distance preserving, and since S is dense, it can be extended to a distance preserving map from X to Y. Similarly, there exists ψ:Y→X distance preserving, and hence ψ∘ϕ is distance preserving from X into itself, and since X is compact, ψ∘ϕ must be surjective. It follows that ψ must be surjective and therefore an isometry. □
Proof of Theorem 4.2
5 Curvature Sets and a Structural Theorem for the Modified Gromov–Hausdorff Distance
We now establish a connection between the Gromov–Hausdorff distance and the work of Boutin and Kemper [4, 5] and Olver [26].
Boutin and Kemper have studied the characterization of certain metric spaces by their distribution of distances and distribution of triangles. Roughly, to a finite metric space \((\mathbb {X},d_{\mathbb {X}})\) one attaches \(D_{2}(\mathbb{X})=\{d_{\mathbb{X}}(x,x');\,x,x'\in \mathbb{X}\}\) and \(D_{3}(\mathbb{X})=\{T(x,x',x'');\,x,x',x''\in \mathbb {X}\}\), where T(x,x′,x″) is the three point pseudometric space with metric given by restriction of \(d_{\mathbb{X}}\) to {x,x′,x″}. Notice that one can regard \(D_{2}(\mathbb{X})\) as the set of all 2point pseudometric spaces arising from \(\mathbb{X}\). The ensuing question is whether these metric invariants are able to characterize finite metric spaces in a certain restricted class up to isometry.
One of the motivations that Boutin and Kemper cite is
Open Problem 1
Suppose A and B are sets with n elements each (n≥3). A metric d is given on A s.t. d(x,y)∈{0,1,2} for all x,y∈A. A similar metric is given on B. Now suppose that the n−1 element subsets of A and B can be labeled A _{1},…,A _{ n } and B _{1},…,B _{ n } in a way such that each A _{ i } is isometric to B _{ i }. Does this force A to be isometric to B?
In this paper we point out that Gromov [13] has made use of similar constructions in his considerations, where for a compact metric space (X,d _{ X }) and k∈ℕ, he defines K _{ k }(X), the kth curvature set of X, as the collection of all the kpoints pseudometric spaces arising from X by restriction of the metric d _{ X }:
Definition 5.1
(Curvature sets, [13])
Remark 5.1
In a completely different language, the pioneering work of Olver on joint invariants [26] has established that smooth planar curves X and Y are rigidly isometric if and only if K _{4}(X)=K _{4}(Y). In a similar manner, Olver proved that two smooth surfaces X and Y embedded in ℝ^{3} are rigidly isometric if and only if K _{7}(X)=K _{7}(Y). Curves and surfaces are regarded as metric spaces once endowed with the restriction of the Euclidean metric. Olver’s motivation for considering these joint invariants comes from the desire to avoid directly estimating curvatures of, say curves, from discrete data sampled from the curve—an inherently noisy process.
The works of Boutin and Kemper, and Olver therefore suggest that one defines a distance between certain classes of objects based on quantifying the dissimilarity between their corresponding curvature sets. We show next that this idea is actually realized by the \(\widehat {\mbox {GH}}\) distance.
5.1 The Structural Theorem
We prove the following structural theorem for the \(\widehat {\mbox {GH}}\) distance which decomposes the computation of \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})\) into a direct comparison of the curvature sets of X and Y of successively higher order. This theorem implies in particular that \(\widehat {\mbox {GH}}\) metrizes the topology (20) defined by Gromov.
Theorem 5.1
(Structural theorem)
In the statement, \(d_{\mathcal{H}}^{{\mathbf {Sym}}_{k}^{+}}\) is the Hausdorff distance in \({\mathbf {Sym}}_{k}^{+}\): the set of all symmetric matrices with nonnegative entries and zero diagonal, which we view as a metric space with metric \(d_{{\mathbf {Sym}}_{k}^{+}}(A,B):=\max_{ij}a_{ij}b_{ij}\), for A=((a _{ ij })) and B=((b _{ ij })) in \({\mathbf {Sym}}_{k}^{+}\).
As an application of Theorem 5.1 and an explicit computation of \(\mathbf {K}_{3}({\mathbb{S}^{1}})\) and \(\mathbf {K}_{3}({\mathbb{S}^{2}})\), in Example 5.3 we lower bound the GH distance between \(\mathbb{S}^{1}\) and \(\mathbb{S}^{2}\) by \(\frac{\pi}{12}\).
5.2 Remarks About Curvature Sets
Remark 5.2
Note that \(\boldsymbol {\mathcal {D}}_{X}\simeq \mathbf {K}_{2}({X})\) where the isomorphism notion ≃ is \({\mathbf {Sym}}_{2}^{+}\simeq \mathbb{R}_{+}\).
Example 5.1

K _{1}(T)={0}.
 Since \({\mathbf {Sym}}_{3}^{+}\simeq \mathbb{R}_{+}^{3}\), we see that$$\mathbf {K}_{3}({\mathrm {T}})\simeq\Biggl\{ \left ( \begin{array}{c}0 \\0 \\0 \end{array} \right )\Biggr\}\cup \bigcup _{P\in\varPi_3} \Biggl\{P \left ( \begin{array}{c}a_1 \\a_2 \\a_3 \end{array} \right )\Biggr\} \cup \bigcup_i\bigcup _{P\in\varPi_3} \Biggl\{P \left ( \begin{array}{c}a_i \\a_i \\0 \end{array} \right )\Biggr\}.$$
Example 5.2
(Computation of \(\mathbf {K}_{3}({\mathbb{S}^{1}})\) and \(\mathbf {K}_{3}({\mathbb{S}^{2}})\))
Since \({\mathbf {Sym}}_{3}^{+}\simeq \mathbb{R}_{+}^{3}\), we see that \(\mathbf {K}_{3}({\mathbb{S}^{1}})\) is isomorphic to the (hollow) regular tetrahedron in \(\mathbb{R}^{3}_{+}\) with vertexes (0,0,0), (0,π,π), (π,0,π), and (π,π,0).
Example 5.3
(Lower bound for the Gromov–Hausdorff distance between \(\mathbb{S}^{1}\) and \(\mathbb{S}^{2}\))
We claim that \(\widehat {d}_{\mathcal{G}\mathcal{H}}({\mathbb{S}^{1}},{\mathbb{S}^{2}})\geq\frac{\pi}{12}\).
Compare with Remark 3.6, which tells us that none of the isometry invariants playing a role in the bounds of Theorem 3.4 is able to distinguish between spheres of different dimension.
Example 5.4
(A counterexample)
For any ℓ∈ℕ there exist two nonisometric metric spaces X and Y such that K _{ k }(X)=K _{ k }(Y) for all k≤ℓ. Indeed, pick n,m∈ℕ with n>m≥ℓ and let X=Δ _{ n } and Y=Δ _{ m }. Since their cardinality is different, X and Y are not isometric; but any subset \(\mathbb{X}\) of X with at most k≤ℓ points embeds isometrically into Y and viceversa. Then, given any k≤ℓ, K _{ k }(Δ _{ n })=K _{ k }(Δ _{ m }).
Nonetheless, note that since by Theorem 4.1 \(\widehat {\mbox {GH}}\) is a distance on \(\mathcal{G}\), then Theorem 5.1 implies that, for given compact metric spaces X and Y, the totality of all curvature sets are able to discriminate whether X and Y are isometric.
Remark 5.3
One can interpret (21) as providing a decomposition of \(\widehat {d}_{\mathcal{G}\mathcal{H}}({X},{Y})\) into different terms indexed by k∈ℕ which provide increasingly more information about the similarity of X and Y. Note in particular that since \(d_{\mathcal{H}}^{{\mathbf {Sym}}_{2}^{+}} ({\mathbf {K}_{2}({X})},{\mathbf {K}_{2}({X})} )=d_{\mathcal{H}}^{\mathbb{R}^{+}} ({\boldsymbol {\mathcal {D}}_{X}},{\boldsymbol {\mathcal {D}}_{Y}} )\), the first term in this decomposition is given by a comparison of the distance sets of X and Y, cf. Remark 5.2. This observation provides an alternative way of proving that \(d_{\mathcal{G}\mathcal{H}}({X},{Y})\geq\frac{1}{2}d_{\mathcal{H}}^{\mathbb{R}^{+}} ({\boldsymbol {\mathcal {D}}_{X}},{\boldsymbol {\mathcal {D}}_{Y}} )\), compare with Theorem 3.4.
5.3 The proof of Theorem 5.1
Lemma 5.1
Let X be a compact metric space and let {x _{1},…,x _{ n }}⊂X be an εnet for X. For each i=1,…,r let V _{ i } be the Voronoi cell corresponding to x _{ i }, i.e. V _{ i }={x  d _{ X }(x,x _{ i })<min_{ j≠i } d _{ X }(x,x _{ j })}. Then, for each i, V _{ i }⊆B(x _{ i },ε).
Proof of Lemma 5.1
Let z∈V _{ i } for some i∈{1,2,…,n}. Assume that z∉B(x _{ i },ε), that is, d _{ X }(z,x _{ i })≥ε. By hypothesis there exists j∈{1,2,…,n} with z∈B(x _{ j },ε), which implies d _{ X }(z,x _{ j })<ε≤d _{ X }(z,x _{ i }), a contradiction. □
Proof of Theorem 5.1
Assume now that for all k∈ℕ, \(d_{\mathcal{H}}^{{\mathbf {Sym}}_{k}^{+}} ({\mathbf {K}_{k}({X})},{\mathbf {K}_{k}({Y})} )<2\eta\). Fix ε>0 and let \(\mathbb{X}_{\varepsilon}=\{x_{1},\ldots,x_{n}\}\subset X\) be an ε/2net for X. Then, there exists M∈K _{ n }(Y) and \(\mathbb{Y}_{\varepsilon}'=\{y_{1}',\ldots,y_{n}'\}\in Y\) s.t. \(M=\mathbf{D}_{Y}(\mathbb{Y}_{\varepsilon}')\) and \(d_{{\mathbf {Sym}}_{n}^{+}}(\mathbf {D}_{X}(\mathbb{X}_{\varepsilon}),M)<2\eta\). Let \(\phi_{\varepsilon}:\mathbb{X}_{\varepsilon}\rightarrow Y\) be given by \(x_{i}\mapsto y_{i}'\) for i=1,2,…,n. Then, by construction, dis(ϕ _{ ε })<2η. Similarly, construct \(\mathbb{Y}_{\varepsilon}\), an ε/2net for Y and \(\psi_{\varepsilon}:\mathbb{Y}_{\varepsilon}\rightarrow X\) s.t. dis(ψ _{ ε })<2η.
6 Discussion
Several aspects remain to be explored, most interestingly perhaps the numerical estimation of \(\widehat {\mbox {GH}}\) using the structural theorem (Theorem 5.1). Strengthening the claim of Theorem 4.2 for specific subfamilies of \(\mathcal{G}\) also appears to be of interest.
Notes
Acknowledgements
Supported by ONR grant number N000140910783 and DARPA grant number HR00110510007.
References
 1.Ben Hamza, A., Krim, H.: Geodesic object representation and recognition. In: Lecture Notes in Computer Science, vol. 2886, pp. 378–387. Springer, Berlin (2003) Google Scholar
 2.Berger, M.: Encounter with a geometer. II. Not. Am. Math. Soc. 47(3), 326–340 (2000) MATHGoogle Scholar
 3.Bloom, G.S.: A counterexample to a theorem of S. Piccard. J. Comb. Theory, Ser. A 22(3), 378–379 (1977) MATHCrossRefGoogle Scholar
 4.Boutin, M., Kemper, G.: On reconstructing npoint configurations from the distribution of distances or areas. Adv. Appl. Math. 32(4), 709–735 (2004) MathSciNetMATHCrossRefGoogle Scholar
 5.Boutin, M., Kemper, G.: Lossless representation of graphs using distributions. arXiv eprints, October 2007 Google Scholar
 6.Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Efficient computation of isometryinvariant distances between surfaces. SIAM J. Sci. Comput. 28(5), 1812–1836 (2006) MathSciNetMATHCrossRefGoogle Scholar
 7.Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. AMS Graduate Studies in Math, vol. 33. Am. Math. Soc., Providence (2001) MATHGoogle Scholar
 8.Burkard, R., Cela, E.: Linear assignment problems and extensions. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, supplement, vol. A, pp. 75–149. Kluwer Academic, Dordrecht (1999) Google Scholar
 9.Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009) MATHCrossRefGoogle Scholar
 10.Chazal, F., CohenSteiner, D., Guibas, L., Mémoli, F., Oudot, S.: GromovHausdorff stable signatures for shapes using persistence. In: Proc. of SGP (2009) Google Scholar
 11.Grigorescu, C., Petkov, N.: Distance sets for shape filters and shape recognition. IEEE Trans. Image Process. 12(10), 1274–1286 (2003) MathSciNetCrossRefGoogle Scholar
 12.Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHÉS 53, 53–73 (1981) MathSciNetMATHGoogle Scholar
 13.Gromov, M.: Metric Structures for Riemannian and NonRiemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston (1999) MATHGoogle Scholar
 14.Grove, K.: Metric and topological measurements of manifolds. In: Proceedings of the International Congress of Mathematicians, Vols. I, II, Kyoto, 1990, pp. 511–519. Math. Soc. Japan, Tokyo (1991) Google Scholar
 15.Grove, K., Markvorsen, S.: Curvature, triameter, and beyond. Bull. Am. Math. Soc. 27(2), 261–265 (1992) MathSciNetMATHCrossRefGoogle Scholar
 16.Grove, K., Markvorsen, S.: New extremal problems for the Riemannian recognition program via Alexandrov geometry. J. Am. Math. Soc. 8(1), 1–28 (1995) MathSciNetMATHCrossRefGoogle Scholar
 17.Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3d shapes. In: SIGGRAPH ’01: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 203–212. ACM, New York (2001) CrossRefGoogle Scholar
 18.Kalton, N.J., Ostrovskii, M.I.: Distances between Banach spaces. Forum Math. 11(1), 17–48 (1999) MathSciNetMATHCrossRefGoogle Scholar
 19.Kokkendorff, S.L.: Characterizing the round sphere by mean distance. Differ. Geom. Appl. 26(6), 638–644 (2008) MathSciNetMATHCrossRefGoogle Scholar
 20.Mémoli, F.: On the use of GromovHausdorff distances for shape comparison. In: Proceedings of Point Based Graphics 2007, Prague, Czech Republic (2007) Google Scholar
 21.Mémoli, F.: Gromov–Hausdorff distances in Euclidean spaces. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, June 2008, pp. 1–8 (2008) CrossRefGoogle Scholar
 22.Mémoli, F.: A spectral notion of Gromov–Wasserstein distances and related methods. Appl. Comput. Math. 30, 363–401 (2011) MATHGoogle Scholar
 23.Mémoli, F.: Gromov–Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11(4), 417–487 (2011) MathSciNetMATHCrossRefGoogle Scholar
 24.Mémoli, F., Sapiro, G.: Comparing point clouds. In: SGP ’04: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 32–40. ACM, New York (2004) CrossRefGoogle Scholar
 25.Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5(3), 313–347 (2005) MathSciNetMATHCrossRefGoogle Scholar
 26.Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1(1), 3–68 (2001) MathSciNetMATHCrossRefGoogle Scholar
 27.Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Shape distributions. ACM Trans. Graph. 21(4), 807–832 (2002) CrossRefGoogle Scholar
 28.Pardalos, P.M., Wolkowicz, H. (eds.): Quadratic Assignment and Related Problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16. Am. Math. Soc., Providence (1994). Papers from the workshop held at Rutgers University, New Brunswick, New Jersey, May 20–21, 1993 MATHGoogle Scholar
 29.Petersen, P.: Riemannian Geometry. Springer, New York (1998) MATHGoogle Scholar
 30.Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. Am. Math. Soc., Providence (1996) MATHGoogle Scholar
 31.Ulam, S.M.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience, New York/London (1960) MATHGoogle Scholar