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arxiv: 2409.02248 · v3 · submitted 2024-09-03 · 🧮 math.MG

Some novel constructions of optimal Gromov-Hausdorff-optimal correspondences between spheres

Sa\'ul Rodr\'iguez Mart\'in This is my paper

Pith reviewed 2026-05-23 21:11 UTC · model grok-4.3

classification 🧮 math.MG
keywords Gromov-Hausdorff distancespheresoptimal correspondencesgeodesic metricsmetric geometryS^3S^4circle
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The pith

Explicit correspondences establish the Gromov-Hausdorff distance between the 3-sphere and 4-sphere as half arccos of negative one fourth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit correspondences between spheres equipped with their geodesic metrics that achieve the infimum in the Gromov-Hausdorff distance definition. It first supplies alternative proofs for the distances between the circle and every higher-dimensional sphere. It then uses related constructions to fix the distance between the three-sphere and four-sphere at one half times arccos of negative one fourth, confirming the n=3 case of an existing conjecture. A reader cares because these values give concrete numbers for how spheres of different dimensions can be matched metrically.

Core claim

By constructing explicit optimal correspondences, the paper shows that the Gromov-Hausdorff distance between S^3 and S^4 equals one half arccos of negative one fourth and supplies alternative proofs that the distance between S^1 and S^n equals one half arccos of one over n plus one for each n.

What carries the argument

Explicit constructions of correspondences between spheres that realize the infimum defining the Gromov-Hausdorff distance.

If this is right

  • The n=3 case of the Lim-Mémoli-Smith conjecture is settled.
  • Alternative proofs now exist for the distances between the circle and all higher spheres.
  • The same style of explicit maps works for both the circle cases and the three-to-four sphere case.
  • These distances supply exact benchmark values for metric comparisons among spheres.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constructions may extend to compute distances between other pairs of spheres such as S^4 and S^5.
  • The explicit maps could serve as test cases for numerical algorithms that approximate Gromov-Hausdorff distances on manifolds.
  • The pattern of optimal distortion might reveal a general formula across all pairs of spheres.

Load-bearing premise

The correspondences constructed in the paper achieve the infimum that defines the Gromov-Hausdorff distance.

What would settle it

Either a proof of a strictly larger lower bound or the exhibition of any correspondence whose distortion is strictly smaller than one half arccos of negative one fourth.

Figures

Figures reproduced from arXiv: 2409.02248 by Sa\'ul Rodr\'iguez Mart\'in.

Figure 1
Figure 1. Figure 1: A low dimensional depiction of the maps F ′ (left) and F ′′ (right). map in higher dimension, ϕn : S n+1 → S n , has distortion ηn (defined in Proposition 2.4), and ηn > ζn for n ≥ 2, so it cannot be used prove dGH(S n+1 , S n ) = 1 2 ζn. Our map F is defined as F ′ for points near the north pole (so that its distortion is not π), F ′′ for points in S 3 (so that F is surjective) and an interpolation betwee… view at source ↗
Figure 2
Figure 2. Figure 2: The Voronoi cells V 2n i and Wi in the case n = 1 Note that V 2n i is obtained by taking a cone (see Equation (5)) from V 2n−1 i with respect to the point (0, . . . , 0, 1) ∈ R 2n+1, so by Equation (6) and Proposition 2.4c) we have diam(V 2n i ) = diam(V 2n−1 i ) = arccos  −2n 2n + 2 = arccos  −n n + 1 Also, letting q1, . . . , q2n+1 be the vertices of a regular 2n+1-gon inscribed in S 1 , we define th… view at source ↗
Figure 3
Figure 3. Figure 3: Some subsets of S 2n+1 we will use in our construction. Now, note that the metric correspondence R2n from Section 3 was the union of the graphs of two odd maps, one map f : S 2n → {q0, . . . , q2n} ⊆ S 1 and one map g : S 1 → {p0, . . . , p2n} ⊆ S 2n (the domains of f, g are actually only dense subsets of S 2n , S 1 respectively). Then, the restriction of our map Φ to S 2n will be just the map f: we let Φ(… view at source ↗
Figure 4
Figure 4. Figure 4: The map Φ : D → S 1 . This figure depicts the restriction of Φ to S 2n = S 2 and also its restriction to two geodesics (colored blue) between points of S 2n and N, the north pole of S 3 . We utilize two copies of N for more clarity. Note that all points in the geodesic segment [p ′ , N] with α > π 2n+1 = π 3 are mapped to −q0 · e iπ 2n+1 = q2. In this section, we identify S 1 with {z ∈ C; |z| = 1}, and for… view at source ↗
Figure 5
Figure 5. Figure 5: The image of Φ is half of S 1 (case n = 2) Note that graph of the restriction of Φ to S 2n is S2n j=0 V 2n j × {bj} ⊆ S 2n × S 1 , which is contained in the relation R2n with distortion 2πn 2n+1 that we used in Section 3. So for any (p, 0),(p ′ , 0) ∈ D we have |dS 1 (Φ(p, 0), Φ(p ′ , 0)) − dS 2n+1 ((p, 0),(p ′ , 0))| ≤ 2πn 2n + 1 . (12) Also note that Φ maps most points of D to {q0, . . . , q2n}; letting … view at source ↗
Figure 6
Figure 6. Figure 6: A depiction of the sets A and B := D \ A. 4.2 Proof that Φ has distortion 2πn 2n+1 We want to prove that, for any two points (p, α) and (p ′ , α′ ) in D ⊆ H 2n+1 + , we have |dS 1 (Φ(p, α), Φ(p ′ , α′ )) − dS 2n+1 ((p, α),(p ′ , α′ ))| ≤ 2πn 2n + 1 . That means that neither of the following two inequalities can happen: dS 1 (Φ(p, α), Φ(p ′ , α′ )) − dS 2n+1 ((p, α),(p ′ , α′ )) > 2πn 2n + 1 . (13) dS 2n+1 … view at source ↗
Figure 7
Figure 7. Figure 7: The function f : [0, π/2] → [0, 1] [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: This figure represents the restriction of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: A case where the inequality fails for n ≥ 7. In order to proceed, suppose for the sake of reaching a contradiction that d(F(x), F(x ′ )) > d(x, x′ ) + ζ3. As in the previous case, we need to introduce some notation: let p, p′ ∈ 4σ(x) cannot be exactly −p ′ , but it can be arbitrarily close to it. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Graph of U3(κ) − L(π − 2, κ) − ζ3. Claim 5.5. We have κ ∈ [0.7, 1.4] and α + α ′ ∈ [π − 2, 2]. Proof. Clearly α + α ′ > π − 2, as we are in the case where α, α′ ≥ π 2 − 1. The facts that α + α ′ < 2 and κ ≥ 0.7 follow from Lemmas 5.3 and 5.4. In order to deduce that κ < 1.4 one can consider the following upper bounds for d(F(x), F(x ′ )) obtained from the triangle inequality: d(σ(x), σ(x ′ )) + d(F(x), σ(… view at source ↗
Figure 12
Figure 12. Figure 12: Points p, q, r in the cases where ∠prq is acute and obtuse. If cos(t) ≥ cos(a) cos(b) , then by the cosine rule the angle ∠prq at r is at most π 2 , so f(t) = d(p, r) = b. If cos(t) < cos(a) cos(b) , then ∠prq is obtuse, and the maximal distance from p to points of the segment qr is given by Lemma A.5. Note that as r moves further from q, the angle between 0⃗p and the plane containing the origin, q and r … view at source ↗
read the original abstract

In this article, as a first contribution, we provide alternative proofs of recent results by Harrison and Jeffs which determine the precise value of the Gromov-Hausdorff (GH) distance between the circle $\mathbb{S}^1$ and the $n$-dimensional sphere $\mathbb{S}^n$ (for any $n\in\mathbb{N}$) when endowed with their respective geodesic metrics. Additionally, we prove that the GH distance between $\mathbb{S}^3$ and $\mathbb{S}^4$ is equal to $\frac{1}{2}\arccos\left(\frac{-1}{4}\right)$, thus settling the case $n=3$ of a conjecture by Lim, M\'emoli and Smith.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript provides alternative proofs of results by Harrison and Jeffs on the exact Gromov-Hausdorff distance between S^1 and S^n (n any natural number) under geodesic metrics, and proves that d_GH(S^3, S^4) equals ½ arccos(−1/4), thereby settling the n=3 case of the conjecture of Lim, Mémoli and Smith.

Significance. If the optimality of the constructed correspondences is rigorously established, the work supplies concrete exact values for GH distances between spheres and independent proofs, strengthening the body of explicit computations in metric geometry.

major comments (2)
  1. [S^3–S^4 construction and lower-bound argument] The central claim that d_GH(S^3, S^4) = ½ arccos(−1/4) rests on the constructed correspondence R ⊂ S^3 × S^4 having distortion exactly equal to the claimed value and on a matching lower bound that holds for every correspondence. The manuscript must isolate the lower-bound argument (likely in the section treating the S^3–S^4 case) and verify that it does not tacitly rely on equivariance or radiality properties satisfied only by the particular R.
  2. [Abstract and the S^3–S^4 optimality proof] The abstract asserts that the constructed correspondences achieve the infimum defining the GH distance; however, without an explicit verification that the distortion of R equals the claimed constant and that no smaller distortion is possible, the exact equality does not follow. This verification is load-bearing and must be supplied with error bounds or exhaustive case analysis if the construction is finite.
minor comments (2)
  1. Clarify the notation for the distortion function and the precise definition of the correspondence R early in the text so that the upper-bound calculation can be followed without ambiguity.
  2. Add a short comparison table or statement contrasting the new proofs with those of Harrison–Jeffs to highlight the technical differences.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript to improve the isolation and independence of the lower-bound argument for the S^3–S^4 case.

read point-by-point responses

  1. Referee: [S^3–S^4 construction and lower-bound argument] The central claim that d_GH(S^3, S^4) = ½ arccos(−1/4) rests on the constructed correspondence R ⊂ S^3 × S^4 having distortion exactly equal to the claimed value and on a matching lower bound that holds for every correspondence. The manuscript must isolate the lower-bound argument (likely in the section treating the S^3–S^4 case) and verify that it does not tacitly rely on equivariance or radiality properties satisfied only by the particular R.

    Authors: We agree that the lower-bound argument should be presented in a self-contained manner. The existing proof derives the lower bound from the general definition of distortion and the geometry of the spheres without invoking equivariance or radiality of the specific R; however, to address the concern we will isolate this argument in a dedicated subsection of the S^3–S^4 section and add an explicit statement confirming that the bound applies to arbitrary correspondences. revision: yes

  2. Referee: [Abstract and the S^3–S^4 optimality proof] The abstract asserts that the constructed correspondences achieve the infimum defining the GH distance; however, without an explicit verification that the distortion of R equals the claimed constant and that no smaller distortion is possible, the exact equality does not follow. This verification is load-bearing and must be supplied with error bounds or exhaustive case analysis if the construction is finite.

    Authors: The body of the paper already contains both the explicit computation that the distortion of R equals ½ arccos(−1/4) and the matching lower bound that rules out smaller values. The construction is continuous rather than finite, so error bounds are not applicable. We will nevertheless revise the abstract to note that both the upper and lower bounds are established in the text, and we will add a short concluding remark in the S^3–S^4 section that summarizes the equality. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit constructions and proofs are self-contained

full rationale

The paper provides alternative proofs for the GH distance between S^1 and S^n and an explicit construction settling the S^3-S^4 case at ½ arccos(-1/4). These are standard mathematical arguments: a concrete correspondence yields an upper bound on d_GH by definition of the infimum, while any matching lower bound must be shown independently (e.g., via distortion inequalities that do not presuppose the form of the constructed map). No equations, fitted parameters, or self-citations reduce the claimed equality to its own inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definition of the Gromov-Hausdorff distance and the geodesic metrics on spheres; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Geodesic metric on spheres is the standard shortest-path distance
    Invoked when the spheres are endowed with their respective geodesic metrics.

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Reference graph

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