Gromov-Hausdorff Geometry of Metric Trees

A.A. Tuzhilin; A.O. Ivanov; I.N. Mikhailov

REVIEW 2 major objections 4 minor 1 cited by

Gromov-Hausdorff Geometry of Metric Trees

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For subsets of an arbitrary metric tree, the Gromov-Hausdorff distance to the whole tree equals the Hausdorff distance under a condition on outer boundary points; every subset of the real line satisfies it.

desk verdict A genuinely new equality result for Gromov–Hausdorff distances on tree clouds, with one advertised corollary that needs repair and one load-bearing unpublished citation. read the letter →

arxiv 2412.18888 v1 pith:ZG7X22HI submitted 2024-12-25 math.MG

A.O. Ivanov , I.N. Mikhailov , A.A. Tuzhilin This is my paper · ORCID

classification math.MG MSC 53C2354E3505C05

keywords Gromov-HausdorffdistanceHausdorffmetrictreesultrametrizationsubsetsofthereallinecanonicalgeodesiccloudsinfinite

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

This paper asks when the Gromov–Hausdorff distance between a metric tree and one of its subsets equals the much easier Hausdorff distance. Its main theorem, Theorem 8.2, says this happens for an arbitrary, possibly infinite metric tree $T$ and a non-empty subset $X$ whenever the points of $T\setminus X$ are all 'internal' (each lies on some path with both endpoints in $X$), or, failing that, when the Hausdorff distance $d_H(X,T)$ is strictly larger than the oriented distance from the non-internal points to $X$. Since every subset of the real line with finite Hausdorff distance to $\mathbb{R}$ has only internal points, a corollary is the exact equality $d_{GH}(X,\mathbb{R})=d_H(X,\mathbb{R})$ for all such subsets. The authors use the equality to build explicit shortest geodesics in the Gromov–Hausdorff class connecting such subsets to their ambient tree.

What carries the argument

The mechanism is ultrametrization $U(X)$: the quotient of a metric space $X$ by the pseudometric $|xy|_u=\inf\{\max_i |x_{i-1}x_i|\}$ taken over dotted lines $x=x_0,\dots,x_n=y$. This quotient collapses every path-connected space to a single point, and the paper's lower bound comes from the inequality $d_{GH}(X,Y)\ge d_{GH}(U(X),U(Y))$; for a tree $T$, $U(T)=\Delta_1$, so $d_{GH}(X,T)\ge \frac12\operatorname{diam} U(X)$. The tree hypothesis is then used to prove the reverse half, $d_H(X,T)=\frac12\operatorname{diam} U(X)$, by showing that roughly half the length of any sufficiently short dotted line between two points of $X$ lands inside the tree's interior region. The second piece of machinery is the canonical Hausdorff geodesic $C_t=B_t(A)\cap B_{d-t}(B)$, which is already a shortest geodesic in the space of closed subsets; once $d_{GH}=d_H$, Theorem 9.1 turns this curve into a shortest geodesic in the Gromov–Hausdorff cloud.

What would settle it

The most direct check is to search for any two metric spaces $X,Y$ with $d_{GH}(X,Y)<d_{GH}(U(X),U(Y))$; such a pair contradicts Theorem 4.1, the lower-bound lemma the whole proof leans on. For the theorem itself, one can look for a metric tree $T$ and a subset $X$ satisfying the boundary condition ($\partial_X T=\emptyset$, or $d_H(X,T)>\overrightarrow{d}_H(\partial_X T,X)$) for which $d_{GH}(X,T)<d_H(X,T)$; even one such pair would refute Theorem 8.2.

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Extended reading notes

Core claim

The central discovery is a rigorous equality theorem: for an arbitrary metric tree $T$ and a non-empty subset $X\subset T$, if either $\partial_X T=\emptyset$ or $d_H(X,T)>\overrightarrow{d}_H(\partial_X T,X)$, then $d_{GH}(X,T)=d_H(X,T)$. Here $\partial_X T$ is the set of points of $T\setminus X$ that lie on no path between two points of $X$; these are the exposed ends of the tree relative to $X$. The proof passes through the ultrametrization $U(X)$, the quotient of $X$ by the pseudometric of dotted-line ultrametric length: because $U(T)$ is a single point, the lower bound $d_{GH}(X,T)\ge d_{GH}(U(X),U(T))=\frac12\operatorname{diam} U(X)$ applies, and the tree structure is used to prove $d_H(X,T)=\frac12\operatorname{diam} U(X)$ under the theorem's hypotheses. Applying the theorem to the line $\mathbb{R}$, regarded as an infinite metric tree, yields $d_{GH}(X,\mathbb{R})=d_H(X,\mathbb{R})$ for every subset $X$; with the canonical Hausdorff geodesic construction, this gives shortest curves in the Gromov–Hausdorff class and, for the integer lattice in $\ell_\infty^n$, the exact value $\frac12$.

Load-bearing premise

The load-bearing premise is that collapsing any two metric spaces to their ultrametrization quotients cannot increase their Gromov–Hausdorff distance; the paper cites this for arbitrary spaces from an unpublished companion, and if that bound fails for unbounded spaces the lower-bound half of the equality has no support.

Editorial extensions

If this is right

For every subset $X\subset\mathbb{R}$ with finite Hausdorff distance to $\mathbb{R}$, the exact Gromov–Hausdorff distance is known and equals $d_H(X,\mathbb{R})$; no search over correspondences is needed.
Any such subset $X$ is connected to $\mathbb{R}$ by an explicit shortest geodesic in the Gromov–Hausdorff class, namely the canonical Hausdorff curve $t\mapsto B_t(X)\cap B_{d-t}(\mathbb{R})$.
The same geodesic conclusion holds for closed subsets of arbitrary metric trees with finite vertex degrees satisfying the boundary condition, and for integer lattices $\mathbb{Z}^n$ in $\mathbb{R}^n$ with the $\ell_\infty$ norm, where the common distance is $1/2$.
The finite-tree theorem from the literature becomes a special case; infinite trees are covered, so the result applies to unbounded clouds such as the cloud of the real line.
For any metric space $X$, the lower bound $d_{GH}(X,Y)\ge \frac12\operatorname{diam}U(X)$ holds for every path-connected $Y$, so the ultrametrization diameter is a universal obstruction to being close to a connected space.

Reading between the lines

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

The authors do not pursue it, but the same ultrametrization lower bound should give equality $d_{GH}(X,T)=d_H(X,T)$ for other geodesic targets $T$ that collapse to a point under $U$ and have a unique-path structure generalized from trees; testing this on $\mathbb{R}^n$ with more general subsets would separate the role of trees from the role of path-connectedness.
Because the real-line equality makes the Hausdorff distance a two-sided estimate for the Gromov–Hausdorff distance, software that already computes Hausdorff distances between one-dimensional sets can be used directly for Gromov–Hausdorff queries in one dimension.
The lattice example suggests a testable pattern: other self-similar lattices, such as hexagonal or root lattices, may sit at exactly computable positive Gromov–Hausdorff distances from their ambient spaces, with the ultrametrization of the lattice controlling the value.

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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

The main content is good and new. The paper proves that for a subset X of an arbitrary metric tree T satisfying either no boundary points or a strict separation condition, dGH(X,T) = dH(X,T). It also settles the real-line case, dGH(X,R) = dH(X,R), for all subsets. That is a clean, useful result, and the canonical Hausdorff geodesic becoming a geodesic in the GH-cloud is a nice application. The proofs are mostly readable and the ultrametrization technique is used appropriately.

The soft spot that matters is Corollary 9.4. The claim that finite vertex degree implies properness is false. A tree made of segments with lengths 2^{-i} glued into [0,1) has all degrees at most 2, but the closed ball of radius 1 is the whole space, which is not compact. So the advertised canonical geodesic statement does not follow for arbitrary finite-degree infinite trees. The fix is straightforward: assume T is proper or locally compact, then Theorem 9.1 applies. This does not touch Theorems 8.2 or 9.2, so the main contribution stands.

A second issue, more of a referee inconvenience: Theorem 4.1, the ultrametrization lower bound for arbitrary metric spaces, is cited to the authors' unpublished paper [7]. The stress-test note suggests the bound may be derivable by an elementary argument, but the paper itself does not give one. A referee who cannot verify [7] will have to trust an unproved load-bearing lemma. I would ask the authors to include a proof of Theorem 4.1 in full generality or replace the citation with a published version.

The citation pattern is otherwise fine: the paper clearly credits the finite-tree predecessor [1] and the earlier ultrametrization work. I do not see any circularity. The main theorems are new statements and the external lower bound is a background lemma, not the target claim.

For whom: metric geometers working on GH distances, especially tree clouds. The paper deserves a serious referee; it likely needs minor to moderate revision, mainly fixing Corollary 9.4 and shoring up the reliance on [7].

Referee Report

2 major / 4 minor

Summary. The paper studies the Gromov–Hausdorff (GH) distance between a subset X of a metric tree T and T itself. It introduces the ultrametrization U(X) and uses the lower bound dGH(X,T) ≥ dGH(U(X),U(T)) to show that, under a condition on the 'end points' of X in T (either no end points, or the Hausdorff distance from X to T exceeds the oriented Hausdorff distance from the end set to X), one has dGH(X,T)=dH(X,T). This is applied to subsets of the real line, giving dGH(X,R)=dH(X,R). The equality is then used to show that the canonical Hausdorff geodesic between X and the ambient tree is a shortest geodesic in the GH cloud, under a properness assumption. The paper also proves analogous results for R^n with the ℓ∞ norm and its integer lattice.

Significance. The equality theorems provide one of the rare cases where the Gromov–Hausdorff distance can be computed exactly, and the ultrametrization method is elegant and potentially useful. The proof of Theorem 8.2 is coherent and the condition in its statement is used correctly. The real-line result (Theorem 9.2) is a clean corollary. The geodesic construction in the GH class is a valuable contribution. However, the advertised extension to all finite-degree metric trees in Corollary 9.4 relies on a false properness implication and must be corrected. The paper also depends on an unpublished self-cited theorem for a load-bearing inequality, which should be proved in the paper or replaced by a published reference.

major comments (2)

[Corollary 9.4] The proof asserts 'Since the degree of each vertex T is finite, then T is proper.' This implication is false. For example, let T be the metric tree homeomorphic to [0,1) with vertices at 1-2^{-i} (i=0,1,...) and edge lengths 2^{-(i+1)}; every vertex has degree at most 2, but the closed ball of radius 1 about 0 is the whole space, which is not compact because the vertex sequence is Cauchy with no limit in T. Hence T is not proper, and Theorem 9.1 cannot be invoked. To obtain the advertised geodesic statement, the hypothesis must be strengthened to properness (or local compactness plus completeness) of T; the current Corollary 9.4 is unproved as stated.
[Theorem 4.1] The lower bound dGH(X,Y) ≥ dGH(U(X),U(Y)) is stated for arbitrary metric spaces and attributed to the unpublished manuscript [7]; earlier references [4] and [6] cover only finite and bounded spaces. This theorem is used essentially in Theorems 8.1, 8.2, and 9.5. Since [7] is not publicly available, the paper should either include a proof of the arbitrary case or cite a published source.

minor comments (4)

[Abstract/Introduction] The abstract heading is given in Russian ('Аннотация'); the paper should be uniformly in English.
[Theorem 9.2 proof] The proof would benefit from a short justification that dH(X,R)<∞ implies every point of R\X is internal w.r.t. X; as written the claim is plausible but not immediate.
[Theorem 9.1 proof] In the displayed equality, the summation index runs over 'n−1' terms but the subdivision has n intervals; the indexing should be adjusted for consistency.
[References] References [7] and [8] are self-citations listed as 'to appear' and 'in print'; since they support load-bearing results, the authors should update them with full publication details or include the necessary proofs in this paper.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional circularity: the main equality is proved from the ultrametrization lower bound plus a direct diameter argument. The only circularity-adjacent issue is that the arbitrary-space form of the load-bearing lower bound is cited to the authors' own unpublished manuscript [7].

self citation load bearing [Theorem 4.1 and its use in Theorem 8.2, Sections 4 and 8]
"The following inequality was first formulated in the work [4] for finite metric spaces and subsequently generalized to the case of bounded [6] and arbitrary [7] metric spaces. Theorem 4.1 ([4],[6],[7]). For any non-empty metric spaces X and Y, we have the following inequality: dGH (X, Y ) ≥ dGH ( U (X), U(Y ) )."

The proof of Theorem 8.2 derives its lower bound dGH(X,T) ≥ diam U(X)/2 directly from Theorem 4.1, and Theorem 8.1 uses the same inequality. The needed full generality for arbitrary metric spaces, including unbounded trees and their subsets, is attributed to [7], an unpublished manuscript by coauthor I.N. Mikhailov, rather than proved in the present paper or supported by a non-overlapping reference. This makes an important premise partly self-citational. However, this is not a definitional circularity: Theorem 4.1 is a separate lower-bound statement that does not assert the target equality dGH(X,T)=dH(X,T), and Theorem 8.2 independently proves the matching upper bound by a dotted-line diameter argument.

full rationale

The central derivation is not circular in the sense of assuming its conclusion. Theorem 8.2 obtains dGH(X,T) ≥ diam U(X)/2 from the ultrametrization inequality and then proves diam U(X) = 2dH(X,T) directly from the tree structure and the definition of the internal-point set; the equality is not baked into an input. Theorem 9.2 is a direct application of Theorem 8.2 to R, and Theorem 9.1 assumes dH = dGH as a hypothesis before proving shortness of the canonical Hausdorff geodesic, so it does not smuggle the equality back in. The main caveat is that the arbitrary-metric-space version of Theorem 4.1, which is load-bearing, is cited to the authors' own unpublished work [7]; this is a self-citation burden and prevents a score of 0, but it does not reduce the paper's main results to a fit or to a self-citation chain. The false assertion in Corollary 9.4 that finite vertex degree implies properness is a genuine mathematical correctness issue, but it is not a circularity and therefore does not raise the circularity score.

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

The paper introduces no fitted parameters or new entities. It relies on external theorems, some from the authors' own prior and unpublished work, and on one false assumption in Corollary 9.4.

assumptions (4)

domain assumption For arbitrary metric spaces X,Y, dGH(X,Y) ≥ dGH(U(X),U(Y)) (Theorem 4.1)
Cited to [7] (Mikhailov, to appear), an author's unpublished work. The finite and bounded cases are in [4] and [6]. Load-bearing for Theorem 8.2.
domain assumption A subset of R^n has finite GH distance to R^n iff it is an ε-network (Theorem 3.1 from [9])
Used in Theorem 9.2 to handle infinite distances; from the authors' own prior work.
standard math Every metric tree T has U(T) = Δ1
A metric tree is path-connected by definition; Note 4.4.
ad hoc to paper A metric tree with finite vertex degrees is proper
Asserted in Corollary 9.4 proof; false for trees with edge lengths tending to 0, e.g., a ray with lengths 1/2^k is degree 2 but not proper.

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Cite this review

Pith. "Pith review of Gromov-Hausdorff Geometry of Metric Trees." pith.science (2026). https://pith.science/paper/ZG7X22HI

@misc{pith2026241218888,
  author       = {Pith},
  title        = {Pith review of: Gromov-Hausdorff Geometry of Metric Trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZG7X22HI}},
  note         = {Machine review of arXiv:2412.18888}
}

read the original abstract

In this paper, we study metric trees, without any finiteness restrictions. For subsets of such trees, a condition that guarantees that the Hausdorff and Gromov--Hausdorff distances from the subset to the entire metric tree are the same is obtained. This result allows to construct a new class of shortest geodesics (in the proper class of all metric spaces) connecting such subset of a metric tree with the tree itself. In particular, the technique elaborated is demonstrated on subsets of the real line.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces
math.MG 2026-07 reject novelty 7.0 of 10

The paper gives a Gromov–Hausdorff lower bound via the relative Jung constant (sound) but its stronger Jung-constant version relies on an impossible normalization step.

Reference graph

Works this paper leans on

9 extracted references · 7 canonical work pages · cited by 1 Pith paper

[7]

I. N. Mikhailov, Ultrametric spaces and clouds (to appear)

work page
[1]

Adams, S

H. Adams, S. Majhi, F. Manin, Z. Virk, N. Zava, Lower-bounding the Gromov–Hausdorﬀ distance in metric graphs , 2024, ArXiv e-prints, arXiv:2411.09182

work page arXiv 2024
[4]

G. E. Carlsson, F. Memoli, Characterization, stability and convergence of hierarchic al clustering methods, J. Mach. Learn., vol. 11, no. 47, 1425-1470, 2010

work page 2010
[6]

Sunhyuk Lim, Facundo Memoli, Zane Smith, The Gromov- Hausdorﬀ distance between spheres , 2022, ArXiv e-prints, arXiv:2105.00611v5

arXiv 2022
[2]

Burago, Yu

D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry , Graduate Studies in Mathematics 33 AMS, Providence, RI, 2001

work page 2001
[3]

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

arXiv 2021
[5]

Kuratowski, Quelques probl` emes concernant les espaces m´ etriques non-s ´ eparables, Fundamenta Mathematicae 1935, v

C. Kuratowski, Quelques probl` emes concernant les espaces m´ etriques non-s ´ eparables, Fundamenta Mathematicae 1935, v. 25, pp. 534–545

work page 1935
[8]

I. N. Mikhailov , New geodetic V class Gromov–Hausdorﬀ, lying in the cloud of th e real line, Chebyshevsky collection, 2025 (in print). References 10

work page 2025

Show all 9 references

[9]

I. N. Mikhailov, A. A. Tuzhilin, When the Gromov–Hausdorﬀ distance between ﬁnite- dimensional space and its subset is ﬁnite? , Commun. Math. Res., accepted to appear in volume 41, 2024

2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [7]

I. N. Mikhailov, Ultrametric spaces and clouds (to appear)

work page

[2] [1]

Adams, S

H. Adams, S. Majhi, F. Manin, Z. Virk, N. Zava, Lower-bounding the Gromov–Hausdorﬀ distance in metric graphs , 2024, ArXiv e-prints, arXiv:2411.09182

work page arXiv 2024

[3] [4]

G. E. Carlsson, F. Memoli, Characterization, stability and convergence of hierarchic al clustering methods, J. Mach. Learn., vol. 11, no. 47, 1425-1470, 2010

work page 2010

[4] [6]

Sunhyuk Lim, Facundo Memoli, Zane Smith, The Gromov- Hausdorﬀ distance between spheres , 2022, ArXiv e-prints, arXiv:2105.00611v5

arXiv 2022

[5] [2]

Burago, Yu

D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry , Graduate Studies in Mathematics 33 AMS, Providence, RI, 2001

work page 2001

[6] [3]

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

arXiv 2021

[7] [5]

Kuratowski, Quelques probl` emes concernant les espaces m´ etriques non-s ´ eparables, Fundamenta Mathematicae 1935, v

C. Kuratowski, Quelques probl` emes concernant les espaces m´ etriques non-s ´ eparables, Fundamenta Mathematicae 1935, v. 25, pp. 534–545

work page 1935

[8] [8]

I. N. Mikhailov , New geodetic V class Gromov–Hausdorﬀ, lying in the cloud of th e real line, Chebyshevsky collection, 2025 (in print). References 10

work page 2025