New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line

Ivan N. Mikhailov

REVIEW 2 major objections 4 minor 13 references

New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that unbounded subsets of the real line generate new geodesic lines in the Gromov–Hausdorff class, while scaling the integer lattice by $\lambda>1$ stays discontinuous.

desk verdict Two new results in Gromov–Hausdorff geometry: a clean, correct lower bound for the scaling of Z^n that blocks the naive contractibility proof, and a geodesic construction whose proof has a patchable ordering gap. read the letter →

arxiv 2504.14629 v1 pith:Q5ARZXTQ submitted 2025-04-20 math.MG

Ivan N. Mikhailov This is my paper · ORCID

classification math.MG MSC 53C2351F30

keywords Gromov–Hausdorffdistancegeodesiclinecloudunboundedsubsetoftherealintegerlatticescalingdiscontinuityℓ^1productcorrespondencedistortion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper works in the Gromov–Hausdorff metric, which measures how far two metric spaces are from being isometric, and asks whether natural one-parameter families of spaces are geodesics. Its first result is constructive: whenever $A$ is an unbounded subset of the real line and $X$ is a bounded metric space, the family $A\times_{\ell^1}(tX)$, $t\ge 0$, moves through the Gromov–Hausdorff class at constant speed, so it is a geodesic. Its second result is a counterexample: for every $\lambda>1$ and $n\in\mathbb{N}$, the distance between the integer lattice $\mathbb{Z}^n$ and its scaled copy $\lambda\mathbb{Z}^n$ is at least $1/2$, so the scaling curve $t\mathbb{Z}^n$ is not continuous, let alone geodesic. These two facts together show that the cloud of $\mathbb{R}^n$—the class of metric spaces at finite Gromov–Hausdorff distance from $\mathbb{R}^n$—cannot be contracted by the obvious homothety, leaving the question of whether such clouds are contractible genuinely open.

What carries the argument

The load-bearing object is the correspondence and its distortion, because Proposition 1 reduces the Gromov–Hausdorff distance to half the infimum distortion over all correspondences between two spaces. For the geodesic theorem the construction uses the $\ell^1$ product $A\times_{\ell^1} X$ with metric $d((a,x),(a',x'))=|a-a'|+|x-x'|$; the proof selects $2n+1$ points $p_1<\cdots<p_{2n+1}$ in the unbounded set $A$ whose consecutive gaps are huge compared with the distortion and diameters involved, forcing any correspondence to have distortion at least $(\operatorname{diam} X-\operatorname{diam} Y)$. For the lattice counterexample, the key tool is the asymptotic count of lattice points in a ball, $N(t)=\operatorname{Vol} B_1(0)\,t^n(1+o(1))$, which rules out bijective correspondences between $\mathbb{Z}^n$ and $\lambda\mathbb{Z}^n$ because the volumes scale by $\lambda^n>1$.

What would settle it

For $A=\{-1,-2,-3,\ldots\}$, compute $d_{GH}(A\times_{\ell^1}X,\,A\times_{\ell^1}(2X))$ for a two-point metric space $X$; Theorem 4 predicts exactly $\operatorname{diam} X/2$, so any smaller value refutes the theorem as stated. For Theorem 5, an explicit correspondence between $\mathbb{Z}$ and $2\mathbb{Z}$ with distortion less than $1$ would refute the bound $d_{GH}(\mathbb{Z},2\mathbb{Z})\ge 1/2$.

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Extended reading notes

Core claim

The paper's central claim is that the Gromov–Hausdorff class contains many geodesic lines built from an unbounded spine in the real line. Theorem 4 states that if $A\subset\mathbb{R}$ is unbounded, $A'$ is any metric space at finite Gromov–Hausdorff distance from $A$, and $X,Y$ are bounded, then $d_{GH}(A'\times_{\ell^1}X,\,B\times_{\ell^1}Y)\ge (\operatorname{diam} X-\operatorname{diam} Y)/2$; Corollary 2 converts this into the exact equality $d_{GH}(A\times_{\ell^1}t_1X,\,A\times_{\ell^1}t_2X)=|t_1-t_2|\operatorname{diam} X/2$, so the curve is a geodesic. The paper's second central result, Theorem 5, is that $d_{GH}(\mathbb{Z}^n,\lambda\mathbb{Z}^n)\ge 1/2$ for every $\lambda>1$ and $n\in\mathbb{N}$, proved by showing that a bijective correspondence between the two lattices would violate the asymptotic lattice-point count in balls. Consequently the curve $t\mathbb{Z}^n$ is discontinuous in the Gromov–Hausdorff class, and multiplication by $\lambda$ is discontinuous on the cloud of $\mathbb{R}^n$.

Load-bearing premise

The proof of the geodesic theorem needs the unbounded set to contain points going off to $+\infty$, since it places a long chain of far-apart points in increasing order; the theorem only assumes the set is unbounded, so one-sided unbounded sets such as the negative integers are not covered by the proof as written.

Editorial extensions

If this is right

For every unbounded $A\subset\mathbb{R}$ and bounded $X$, the curve $t\mapsto A\times_{\ell^1}(tX)$ is a geodesic in the Gromov–Hausdorff class with speed $\operatorname{diam} X/2$.
The lower bound $d_{GH}(A'\times_{\ell^1}X,\,B\times_{\ell^1}Y)\ge (\operatorname{diam} X-\operatorname{diam} Y)/2$ forces apart any two such spaces whose bounded factors have different diameters.
For every $\lambda>1$ and $n\in\mathbb{N}$, $d_{GH}(\mathbb{Z}^n,\lambda\mathbb{Z}^n)\ge 1/2$, so the multiplication map on the cloud of $\mathbb{R}^n$ is not continuous.
The curve $t\mathbb{Z}^n$ is therefore not a geodesic, and the standard homothety argument does not prove contractibility of the cloud of $\mathbb{R}^n$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same one-dimensional lower-bound argument should work whenever the unbounded spine is a subset of a line inside any normed space and the bounded factor is joined by the $\ell^1$ product, so such geodesics are not special to $\mathbb{R}$.
The volume-counting contradiction suggests that any uniformly discrete subset of $\mathbb{R}^n$ with minimum separation $1$ lies at distance at least $1/2$ from its $\lambda$-scaling, making the discontinuity generic rather than special to $\mathbb{Z}^n$.
Whether one-sided unbounded sets such as the negative integers satisfy the stated geodesic equality is not decided by the proof, because the point-selection step presupposes arbitrarily large positive elements.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper for Theorem 5: for every λ>1 and n, d_GH(Z^n, λZ^n) ≥ 1/2. The proof is a nice counting argument—a finite-distortion bijective correspondence would force the lattice ball count in λZ^n to dominate that in Z^n, but the volume asymptotics say the opposite, so the correspondence cannot be bijective; non-bijective gives distortion at least 1. That is clean, uses a standard lattice-point theorem, and it has a real consequence: the natural scaling map on the cloud of R^n is discontinuous, so the standard route to contractibility fails. That result is the real contribution.

The other main result, Theorem 4, constructs geodesics: for unbounded A⊂R and bounded X, A×_{ℓ1}(tX) is a geodesic line in the GH class. The lower bound d_GH(A'×X, B×Y) ≥ (diam X−diam Y)/2 is the engine, and the proof is a long distortion chain that essentially works.

The soft spots are all in the presentation of Theorem 4. After Lemma 3, the proof says 'without loss of generality' the pairs {b_{i0}, b_{i1}} lie on a line in ascending index order. That is not automatic from Lemma 3 as written; Lemma 3 gives betweenness for any i

There are also small presentational issues: t in the proof of Theorem 4 is never defined (it is an arbitrary value ≤ diam X), and the English needs editing. None of that touches the math.

Verdict: the paper deserves a serious referee. Theorem 5 is solidly new; Theorem 4 needs a rewritten proof with the ordering lemma explicit. I would send it out, asking for revisions rather than rejecting.

Referee Report

2 major / 4 minor

Summary. The paper studies geodesic lines in the Gromov-Hausdorff class, focusing on the cloud of the real line. Theorem 4 claims that for any unbounded A subset of R, any A' at finite Gromov-Hausdorff distance from A, any B subset of R, and any bounded metric spaces X,Y, the lower bound d_GH(A' x_{l1} X, B x_{l1} Y) >= (diam X - diam Y)/2 holds. From this, Corollary 2 derives that A x_{l1} (tX) is a geodesic line for bounded X and unbounded A. Theorem 5 proves that for every lambda > 1 and every n, d_GH(Z^n, lambda Z^n) >= 1/2, and the authors conclude that the scaling curve t Z^n is discontinuous and hence not a geodesic, and that multiplication by a fixed lambda is not continuous on the cloud of R^n.

Significance. If the main results are correct, the paper makes a worthwhile contribution: it constructs new geodesic lines in the Gromov-Hausdorff class inside the cloud of the real line, and it identifies a concrete obstruction to the standard scaling argument for contractibility of clouds of R^n. Theorem 5 is particularly clean: the proof uses only the elementary fact that a non-bijective correspondence between Z^n and lambda Z^n has distortion at least 1, combined with the standard lattice-point asymptotic, and it contains no free parameters or circular reasoning. The lower-bound technique in Theorem 4, once the missing justifications are supplied, would be a useful tool for further work on clouds. The conceptual conclusion about discontinuity of the scaling ray is clearly stated and is a valid negative result about a natural approach, not an overclaim about contractibility itself.

major comments (2)

[Theorem 4, proof, first paragraph] The proof chooses points p_1 < p_2 < ... < p_{2n+1} in A with consecutive gaps larger than 100(t+c+w+diam Y). This is possible only if A is unbounded above. The theorem statement assumes only that A is an unbounded subset of R, and sets such as A = -N admit no increasing sequence with arbitrarily large consecutive gaps. A reflection argument (applying the theorem to -A and -A') would repair the proof, but as written the stated theorem is not proved for all unbounded A.
[Theorem 4, proof after Lemma 3] The step 'without loss of generality, we can assume that pairs of points {b_{ij}, b_{i j+1}} are located on a line in ascending order of indices i' is asserted without proof. The subsequent equality |b_{1 i_1} b_{2n+1 i_{2n+1}}| = sum_{k=1}^{2n} |b_{k i_k} b_{k+1 i_{k+1}}| and the alternating-index construction i_{2n+1} = i_1 both depend on the selected points being in monotone order. Lemma 3 provides a betweenness condition for every triple i<j<k, but the paper does not show how this condition forces the required global ordering of the pairs, especially for adjacent pairs. The proof needs an explicit argument (for example, an induction using a third pair to separate consecutive pairs) before this ordering can be used.

minor comments (4)

[Corollary 2, proof] The text states 'by Theorem 2 the equality d_GH(t_1 X, t_2 X) = |t_1 - t_2| diam X holds'; the factor 1/2 is missing. The subsequent displayed inequality and the final result are correct, so this is only a typographical slip.
[Theorem 4, proof, displayed chain] In the long inequality chain, the notation |a_{1 i_1} a_{2n+1 i_{2n+1}}| and |p_{1 i_1} p_{2n+1 i_{2n+1}}| is confusing because a_i and p_i are not indexed by the choice indices i_j; these expressions should be |a_1 a_{2n+1}| and |p_1 p_{2n+1}|, respectively.
[Throughout] There are numerous typographical errors that should be corrected: 'abitrary' in the abstract, 'gedesic', 'discontinous', 'Mo reover', 'dist ance', and the corrupted text '/emdash.cyr' in the proof of Theorem 5.
[Theorem 4, proof, final inequality] After deriving c + 2w/(2n+1) + 2n/(2n+1) diam Y >= 2n/(2n+1) t, the passage to the limit 'n arbitrarily large and t arbitrarily close to diam X' is correct, but it would be clearer to state explicitly that the term 2w/(2n+1) tends to 0 and that t can be chosen after n because t depends only on X.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the results are derived from external metric-geometry facts and the lattice-point theorem; no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 4's lower bound follows from choosing a finite chain p_1 < ... < p_{2n+1} in an unbounded A with large gaps, estimating distortions through correspondences S and R, and using triangle inequalities on the real line; the 'without loss of generality' ordering of pairs {b_{ij}, b_{i,j+1}} is intended as a consequence of Lemma 3, and even if that step is under-justified, it is a proof gap rather than a circularity because it is not an input to the theorem. Corollary 2 combines Theorem 4 with Theorem 2 (the geodesic formula for bounded scaling, from [2],[11]) and Lemma 2, so the geodesic conclusion does not assume itself. Theorem 5 uses the external lattice-point asymptotic Theorem 3 from [8] to compare balls in Z^n and lambda Z^n; the conclusion d_GH(Z^n, lambda Z^n) >= 1/2 is derived from the counting asymptotics and the elementary fact that a non-bijective correspondence between countable spaces has distortion at least 1. No fitted parameter is used as a prediction, no central premise is justified only by a self-citation, and the sole self-citation [10] appears only in the bibliography and is not load-bearing. The potential issue with one-sided unbounded sets such as -N concerns the hypotheses of Theorem 4, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the paper's estimates are parameter-free. The only non-standard background premise is the geometric fact about unbounded subsets of R, which is not stated precisely and is the weakest point of the proof.

assumptions (4)

standard math Standard Gromov-Hausdorff distance properties, including Proposition 1 relating d_GH to correspondence distortion.
Used throughout; cited from Burago-Burago-Ivanov [2].
standard math Theorem 2: for a bounded metric space X, d_GH(t1 X, t2 X) = |t1 - t2| diam X / 2 and tX is a geodesic.
Used in Corollary 2 for the upper bound; cited from [2] and [11].
standard math Theorem 3: number of integer points in a Euclidean ball of radius t is Vol(B_1) t^n (1 + o(1)).
Used in Theorem 5 to compare lattice point counts in scaled balls; cited from Kang-Sobolev [8].
domain assumption An unbounded subset A of R contains arbitrarily long increasing chains with large consecutive gaps.
Theorem 4 proof relies on this to build the points p_i; it holds only when A is unbounded above, which the paper does not state explicitly.

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Pith. "Pith review of New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line." pith.science (2026). https://pith.science/paper/Q5ARZXTQ

@misc{pith2026250414629,
  author       = {Pith},
  title        = {Pith review of: New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5ARZXTQ}},
  note         = {Machine review of arXiv:2504.14629}
}

abstract

In the paper we prove that, for arbitrary unbounded subset $A\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\times_{\ell^1} (tX)$, $t\in[0,\,\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $\lambda > 1$, $n\in\mathbb{N}$, the following inequality holds: $d_{GH}\bigl(\mathbb{Z}^n,\,\lambda\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. We conclude that a curve $t\mathbb{Z}^n$, $t\in(0,\,\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all metric spaces lying on the finite Gromov--Hausdorff distance from $\mathbb{R}^n$ on some~$\lambda > 0$ is also discontinous with respect to the Gromov--Hausdorff distance.

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Reference graph

Works this paper leans on

13 extracted references · 6 canonical work pages

[1]

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[9]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

S. A. Bogaty, A. A. Tuzhilin, Gromov–Hausdorﬀ class: its completeness and cloud geometr y, ArXiv e-prints, arXiv:2110.06101, 2021

arXiv 2021

[2] [2]

Burago, Yu

D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry , Graduate Studies in Mathe- matics 33, AMS, 2001

2001

[3] [3]

Edwards, The structure of superspace , Studies in Topology, Academic Press, 1975

D. Edwards, The structure of superspace , Studies in Topology, Academic Press, 1975

work page 1975

[4] [4]

Gromov, Groups of polynomial growth and expanding maps , Publications Mathematiques I.H.E.S., 53 1981

M. Gromov, Groups of polynomial growth and expanding maps , Publications Mathematiques I.H.E.S., 53 1981

work page 1981

[5] [5]

Gromov, Structures m´ etriques pour les vari´ et´ es riemanniennes, Textes Math

M. Gromov, Structures m´ etriques pour les vari´ et´ es riemanniennes, Textes Math. 1 (1981)

1981

[6] [6]

G. E. Carlsson, F. Memoli, Characterization, stability and convergence of hierarchi cal clustering methods, J. Mach. Learn

work page

[7] [7]

Gromov, Metric structures for Riemannian and non-Riemannian space s, Birkh¨ user (1999)

M. Gromov, Metric structures for Riemannian and non-Riemannian space s, Birkh¨ user (1999)

1999

[8] [8]

Sobolev, Distribution of integer lattice points in a ball centred at a diophantine point , Mathematika 56(01):118 - 134, 2010

Hyunsuk Kang, Alexander V. Sobolev, Distribution of integer lattice points in a ball centred at a diophantine point , Mathematika 56(01):118 - 134, 2010

work page 2010