Explicit Geodesics in Gromov-Hausdorff Space

We provide an alternative, constructive proof that the collection $\mathcal{M}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on $\mathcal{M}$. We also provide several interesting examples of geodesics on $\mathcal{M}$, including a geodesic between $\mathbb{S}^0$ and $\mathbb{S}^n$ for any $n\geq 1$.


Geodesics on Gromov-Hausdorff space
The collection of compact metric spaces, denoted M throughout this paper, is a valid pseudometric space when endowed with the Gromov-Hausdorff distance [3,4]. We will denote this space by (M, d GH ). Furthermore, the space (M/∼, d GH ), where we define is a metric space [3]. It is known that (M/∼, d GH ) is separable, complete [8], and geodesic [6]. Specifically, the authors of [6] use a compactness result to argue that for any pair of points in M/∼, there exists a midpoint in M/∼, which implies that (M/∼, d GH ) is a geodesic space [3,Theorem 2.4.16]. However, this proof is not constructive. The goal of our paper is to provide a constructive proof through the explicit description of a certain class of geodesics on (M/∼, d GH ), which we call straight-line geodesics. The key ingredient in our construction is a proof showing that there exists an optimal correspondence between any two compact metric spaces. While obvious when considering finite metric spaces, establishing this result for general compact metric spaces requires some work. Our result is inspired by a similar result proved by Sturm about geodesics on the space of metric measure spaces [9]. We use our result to construct (1) an explicit geodesic between S 0 and S n , for any n ∈ N, and (2) explicit, infinite families of deviant (i.e., non-straightline) and branching geodesics between the one-point discrete space and the n-point discrete space, for any n ≥ 2.
As a consequence of this definition, for any geodesic γ such that γ(0) = x and γ(1) = x , one has L(γ) = d X (x, x ). The metric space (X, d X ) is called a geodesic space if for any x, x ∈ X, there exists a geodesic γ connecting x and x .
Next, given a metric space (X, d X ) and two nonempty subsets A, B ⊆ X, the Hausdorff distance between A and B is defined as where ϕ and ψ are both isometric embeddings [3] where the second-to-last equality follows from the triangle inequality and the ob- It is known that (M/∼, d GH ) is complete [8]. Details about the topology generated by the Gromov-Hausdorff distance can be found in [3]. One important fact is that it allows the existence of many compact sets in (M/∼, d GH ), in the sense below. Recall that for a compact metric space X, for ε > 0, the ε-covering number cov X (ε) is defined to be the minimum number of ε-balls required to cover X. Theorem 1.1 (Gromov's precompactness theorem, [8]). Given a bounded function N : (0, ∞) → N and D > 0, let C(N, D) ⊆ (M/∼, d GH ) be the collection of all [X] such that diam(X) < D and cov X (ε) ≤ N (ε) for each ε > 0. Then C(N, D) is precompact.
In our constructions, we will use an equivalent formulation of the Gromov-Hausdorff distance following [3,Chapter 7]. Given (X, d X ), (Y, d Y ) ∈ M, we say that a relation R ⊆ X × Y is a correspondence if for any x ∈ X, there exists y ∈ Y such that (x, y) ∈ R, and for any y ∈ Y , there exists x ∈ X such that (x, y) ∈ R. The set of all such correspondences will be denoted R(X, Y ). In the case Y = X, a particularly useful correspondence is the diagonal correspondence The Gromov-Hausdorff distance d GH : M × M → R + can be formulated as dis(R).

SAMIR CHOWDHURY AND FACUNDO MÉMOLI
In particular, a correspondence is optimal if the infimum is achieved. We will denote by R opt (X, Y ) the set of all closed optimal correspondences. We have Our main result is the explicit construction of straight-line geodesics: where for each (x, y), (x , y ) ∈ R and t ∈ (0, 1), Not all geodesics between compact metric spaces are of the form given by Theorem 1.2. Furthermore, branching of geodesics may happen in Gromov-Hausdorff space. We explore the deviance and branching phenomena in Section 1.1. In Section 1.2 we construct explicit geodesics between S 0 and S n . The proofs of Proposition 1.1 and Theorem 1.2 are given in Section 2.
1.1. Deviant and branching geodesics. The following lemma will be useful in the sequel: Then, in fact, Proof of Lemma 1.3. Suppose the inequality is strict. Suppose also that s ≤ t.
Then by the triangle inequality, we obtain This is a contradiction. Similarly we get a contradiction for the case t < s. This proves the lemma.
1.1.1. Deviant geodesics. For any n ∈ N, let ∆ n denote the n-point discrete space, often called the n-point unit simplex. Fix n ∈ N, n ≥ 2. We will construct an infinite family of deviant geodesics between ∆ 1 and ∆ n , named as such because they deviate from the straight-line geodesics given by Theorem 1.2. As a preliminary step, we describe the straight-line geodesic between ∆ 1 and ∆ n of the form given by Theorem 1.2. Let {p} and {x 1 , . . . , x n } denote the underlying sets of ∆ 1 and ∆ n . There is a unique correspondence R := {(p, x 1 ), . . . , (p, x n )} between these two sets. According to the setup in Theorem 1.2, the straight-line geodesic between ∆ 1 and ∆ n is then given by the metric spaces (R, d γ R (t) ), for t ∈ (0, 1). Here This corresponds to the all-t matrix with 0s on the diagonal. Finally, we note that the unique correspondence R necessarily has distortion 1. Thus d GH (∆ 1 , ∆ n ) = 1 2 . Now we give the parameters for the construction of a certain family of deviant geodesics between ∆ 1 and ∆ n . For any α ∈ (0, 1] and t ∈ [0, 1], define Next let m be a positive integer such that 1 ≤ m ≤ n, and fix a set X n+m :=

This is a block matrix
We first claim that δ t is the distance matrix of a pseudometric space. Symmetry is clear. We now check the triangle inequality. In the cases 1 ≤ i, j, k ≤ n and n + 1 ≤ i, j, k ≤ n + m, the points x i , x j , x k form the vertices of an equilateral triangle with side length t. Suppose 1 ≤ i, j ≤ n and n + 1 ≤ k ≤ n + m. Then the triple x i , x j , x k forms an isosceles triangle with equal longest sides of length t, and a possibly shorter side of length or just a third equal side with length t in the remaining cases. The case 1 ≤ i ≤ n, n + 1 ≤ j, k ≤ n + m is similar. This verifies the triangle inequality. Also note that δ t is the distance matrix of a bona fide metric space for t ∈ (0, 1). For t = 1, we identify the points x i and x i−n , for n + 1 ≤ i ≤ n + m, to obtain ∆ n , and for t = 0, we identify all points together to obtain ∆ 1 . This allows us to define geodesics between ∆ 1 and ∆ n as follows. Let α denote the vector (α 1 , . . . , α m ). We define a map γ α : [0, 1] → M by writing where we can take quotients at the endpoints as described above. We now verify that these curves are indeed geodesics. There are three cases: . By using the diagonal correspondence , we check case-by-case that dis( ) ≤ |t − s|. Thus for any It follows by Lemma 1.3 that γ α is a geodesic between ∆ 1 and ∆ n . Furthermore, since α ∈ (0, 1] m was arbitrary, this holds for any such α. Thus we have an infinite family of geodesics A priori, some of these geodesics may intersect at points other than the endpoints. By this we mean that there may exist t ∈ (0, 1) and α = β ∈ (0, 1] m such that [γ α (t)] = [γ β (t)] in M/ ∼. This is related to the branching phenomena that we describe in the next section. For now, we give an infinite subfamily of geodesics that do not intersect each other anywhere except at the endpoints. Recall that the separation of a finite metric space (X, d X ) is the smallest positive distance in X, 52 SAMIR CHOWDHURY AND FACUNDO MÉMOLI which we denote by sep(X). If sep(X) < sep(Y ) for two finite metric spaces X and Y , then d GH (X, Y ) > 0.
Note that one could choose the diameter of ∆ n to be arbitrarily small and still obtain deviant geodesics via the construction above.

1.1.2.
Branching. The structure of d GH permits branching geodesics, as illustrated in Figure 1. We use the notation (a) + for any a ∈ R to denote max(0, a). As above, fix n ∈ N, n ≥ 2, and consider the straight-line geodesic between ∆ 1 and ∆ n described at the beginning of Section 1.1.1. Throughout this section, we denote this geodesic by γ : [0, 1] → M. We will construct an infinite family of geodesics which branch off from γ. For convenience, we will overload notation and write, for each t ∈ [0, 1], the distance matrix of γ(t) as γ(t). Recall from above that γ(t) is a symmetric n × n matrix with the following form:  Let (a i ) i∈N be any sequence such that 0 < a 1 < a 2 < · · · < 1. For each t ∈ [0, 1], define the (n + 1) × (n + 1) matrix γ (a1) (t) to be the symmetric matrix with the following upper triangular form: : 1 ≤ i < n, j = n + 1 (t − a 1 ) + : i = n, j = n + 1 0 : For t > a 1 , we have d GH γ(t), γ (a1) (t) > 0 because any correspondence between γ(t), γ (a1) (t) has distortion at least t − a 1 . Thus γ (a1) branches off from γ at a 1 .
The construction of γ (a1) (t) above is a special case of a one-point metric extension. Such a construction involves appending an extra row and column to the distance matrix of the starting space; explicit conditions for the entries of the new row and column are stated in [7, Lemma 5.1.22]. In particular, γ (a1) (t) above satisfies these conditions. Procedurally, the γ (a1) (t) construction can be generalized as follows. Let (•) denote any finite subsequence of (a i ) i∈N . We also allow (•) to be the empty subsequence. Let a j denote the terminal element in this subsequence. Then for any a k , k > j, we can construct γ (•,a k ) as follows: (1) Take the rightmost column of γ (•) (t), replace the only 0 by (t−a k ) + , append a 0 at the bottom. (2) Append this column on the right to a copy of γ (•) (t).
(3) Append the transpose of another copy of this column to the bottom of the newly constructed matrix to make it symmetric.
The objects produced by this construction satisfy the one-point metric extension conditions [7, Lemma 5.1.22] and hence are distance matrices of pseudometric spaces. By taking the appropriate quotients, we obtain valid distance matrices. Symmetry is satisfied by definition, and the triangle inequality is satisfied because any triple of points forms an isosceles triangle with longest sides equal. We write Γ (•) (t) to denote the matrix obtained from γ (•) (t) after taking quotients. As an example, we obtain the following matrices after taking quotients for γ (a1) (t) above, for 0 ≤ t ≤ a 1 (below left) and for a 1 < t ≤ 1 (below right): Now let (a ij ) k j=1 be any finite subsequence of (a i ) i∈N . For notational convenience, we write (b i ) i instead of (a ij ) k j=1 . Γ (bi)i is a curve in M; we need to check that it is moreover a geodesic.
Let s ≤ t ∈ [0, 1]. Then Γ (bi)i (s) and Γ (bi)i (t) are square matrices with n + p and n + q columns, respectively, for nonnegative integers p and q. It is possible that the matrix grows in size between s and t, so we have q ≥ p. Denote the underlying point set by {x 1 , x 2 , . . . , x n+p , . . . , x n+q }. Then define Here B is possibly empty. Note that R is a correspondence between Γ (bi)i (s) and Γ (bi)i (t), and by direct calculation we have dis(R) ≤ |t − s|. Hence we have d GH Γ (bi)i (s), Γ (bi)i (t) ≤ 1 2 · |t − s| = |t − s| · d GH (∆ 1 , ∆ n ). An application of Lemma 1.3 now shows that Γ (bi)i is a geodesic.

SAMIR CHOWDHURY AND FACUNDO MÉMOLI
The finite subsequence (b i ) i of (a i ) i∈N was arbitrary. Thus we have an infinite family of geodesics which branch off from γ. Since the increasing sequence (a i ) i∈N ∈ (0, 1) N was arbitrary, the branching could occur at arbitrarily many points along γ. 1.2. An explicit geodesic from S 0 to S n . Let n ∈ N. Consider the spheres S 0 and S n equipped with the canonical geodesic metric, such that each sphere has diameter π. We will now construct an explicit geodesic between S 0 and S n .
Proof of Proposition 1.2. Let U and L denote the closed upper hemisphere and open lower hemisphere of S n , respectively. Then we have U L = S n . Moreover, let s, s ∈ U be two points realizing the diameter of S n via an arc in U (note that such an arc exists for each S n when n ≥ 1). Also let {p, q} denote the two points of S 0 . Now we construct a correspondence between S 0 and S n : Then we have It follows that d GH (S 0 , S n ) ≤ π 2 . Next we wish to show the reverse inequality. Let T be an arbitrary correspondence between S 0 and S n , and write P := {x ∈ S n : (p, x) ∈ T } , Q := {x ∈ S n : (q, x) ∈ T } .
Then, P = ∅, Q = ∅, and S n = P ∪ Q, by the definition of correspondences. Now recall the Lusternik-Schnirelmann theorem ([1, p. 117], also see [5, p. 33]): for every family of n + 1 closed sets covering S n , one of the sets contains a pair of antipodal points. Applying this result by taking n copies of P and one copy of Q as the cover of S n , we get that at least one of the sets P and Q contains a pair of antipodal points. Without loss of generality, suppose P contains a pair of antipodal points (a, a ). Let (a n ), (a n ) be sequences in P such that for each n ∈ N, we have a n ∈ B(a, 1 n ) and a n ∈ B(a , 1 n ). By the triangle inequality, one has that |d S n (a n , a n ) − d S n (a, a )| ≤ d S n (a n , a) + d S n (a n , a ) < 2 n . Also note that |d S n (a, a ) − d S 0 (p, p)| = π. Then we obtain |d S n (a n , a n ) − d S 0 (p, p)| = |d S n (a n , a n ) − d S n (a, a ) + d S n (a, a ) − d S 0 (p, p)| > π − 2 n . By letting n → ∞, it follows that dis(T ) ≥ π. Since T was an arbitrary correspondence, we obtain d GH (S 0 , S n ) ≥ π 2 . Thus we obtain d GH (S 0 , S n ) = π 2 . It now follows that the correspondence R defined in the proof of Proposition 1.2 is an optimal correspondence between S 0 and S n . In particular, the definition of R suggests that one may define a geodesic from S n to S 0 by "shrinking" U and L to the north and south poles, respectively. Proposition 1.3. Let U and L denote the closed upper hemisphere and open lower hemisphere of S n , respectively. Also let {p, q} denote the two points of S 0 , and let µ ∈ U , λ ∈ L denote the north and south poles of S n , respectively. For each t ∈ (0, 1), define Then γ is a geodesic from S 0 to S n .
Proof of Proposition 1.3. First let t ∈ (0, 1), and define a correspondence between S 0 and X t by Then we have dis(R t ) = (1−t)π, and so d GH Similarly we obtain d GH (γ(t), S n ) ≤ tπ 2 . We wish to show that these inequalities are actually equalities. Without loss of generality, suppose that d GH (S 0 , γ(t)) < (1−t)π 2 . Then we obtain This is a contradiction, by Proposition 1.2. Thus for each t ∈ (0, 1), we have d GH (S 0 , γ(t)) = (1−t)π 2 and d GH (S n , γ(t)) = tπ 2 . Next let s ∈ (0, 1). We wish to show d GH (γ(s), γ(t)) = |t − s| · d GH (S 0 , S n ). We have two cases: (1) s ≤ t, and (2) s > t. Both cases are similar, so we just show the first case. Before proceeding, notice that since s ≤ t, we have U t ⊆ U s and L t ⊆ L s , and so X t ⊆ X s . We will define some notation for convenience. For each x ∈ U s let c µ x denote the shortest geodesic segment connecting x to the north pole µ. Next, for each x ∈ L s let c λ x denote the shortest geodesic segment connecting x to the south pole λ. Also write bd(U t ) and bd(L t ) to denote the boundaries of U t and L t . Now define a map ϕ U : U s → U t by: Also define a map ϕ L : L s → L t by: Finally define ϕ : X s → X t as follows: Observe that for any x ∈ X s we have: Proof of Theorem 1.2. Suppose we can find a curve γ : [0, 1] → M such that γ(0) = (X, d X ) and γ(1) = (Y, d Y ), and for all s, t ∈ [0, 1], .
) for all s, t ∈ [0, 1], and we will be done. So we will show the existence of such a curve γ. Let R ∈ R opt (X, Y ), i.e., let R be a correspondence between X and Y such that dis(R) = 2 d GH (X, Y ). Such a correspondence always exists by Proposition 1.1.

Discussion
While we provide an explicit construction of straight-line geodesics, it is natural to ask: can we characterize other classes of geodesics in (M/ ∼, d GH )? In Section 1.1.1, we constructed infinite families of deviant (i.e., non-unique) geodesics between ∆ 1 and ∆ n . In Section 1.1.2, we provided a parametric construction by which the straight-line geodesic between ∆ 1 and ∆ n could be made to branch off into arbitrarily many nodes at arbitrarily many locations.
As stated at the end of Section 1.1, the existence of branching and deviant geodesics shows the negative result that (M/∼, d GH ) cannot have curvature bounded from above or below. In light of this result, it is interesting to point out the work of Sturm showing that the space of metric measure spaces [9] has nonnegative curvature when equipped with an L 2 -Gromov-Wasserstein metric.