The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

Alexander O. Ivanov; Alexey A. Tuzhilin

arxiv: 1907.03828 · v1 · pith:HWGKEFZInew · submitted 2019-07-08 · 🧮 math.MG

The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

Alexander O. Ivanov , Alexey A. Tuzhilin This is my paper

Pith reviewed 2026-05-25 00:24 UTC · model grok-4.3

classification 🧮 math.MG

keywords Gromov-Hausdorff distancesimplex metric spaceultrametric spacemetric geometrydistance formulasbounded metric spaces

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The pith

When the simplex has cardinality at most that of X, the Gromov-Hausdorff distance to X admits a new formula that also determines the distance to ultrametric spaces.

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

This paper derives a new formula for the Gromov-Hausdorff distance between a simplex and a bounded metric space X, valid when the simplex has no more points than X. The formula is then applied to obtain an exact expression for the distance when X is an ultrametric space. A sympathetic reader would care because these distances are generally hard to calculate, and the result provides explicit expressions in these important cases. The simplex is defined as a metric space in which all nonzero distances are equal.

Core claim

In the present paper we investigate the Gromov-Hausdorff distances between a bounded metric space X and a simplex. In the case when the simplex's cardinality does not exceed the cardinality of X, a new formula for this distance is obtained. The latter permits to derive an exact formula for the distance between a simplex and an ultrametric space.

What carries the argument

The simplex, a metric space with all non-zero distances equal, together with the cardinality condition that allows a closed-form Gromov-Hausdorff distance to X.

Load-bearing premise

X is bounded and the simplex has cardinality no larger than X.

What would settle it

A specific pair consisting of a three-point simplex and a four-point bounded metric space X where the proposed formula gives a value different from the infimum over correspondences.

read the original abstract

In the present paper we investigate the Gromov--Hausdorff distances between a bounded metric space $X$ and so called simplex, i.e., a metric space all whose non-zero distances are the same. In the case when the simplex's cardinality does not exceed the cardinality of $X$, a new formula for this distance is obtained. The latter permits to derive an exact formula for the distance between a simplex and an ultrametric space.

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.

Desk Editor's Note private letter to a colleague

The paper derives explicit Gromov-Hausdorff formulas for a simplex versus bounded X when the simplex is no larger than X, then specializes to ultrametric X.

read the letter

The main takeaway is that they produce an explicit formula for the Gromov-Hausdorff distance between a simplex and a bounded metric space X precisely when the simplex has cardinality at most that of X, and then use it to obtain an exact formula when X is ultrametric. The abstract presents both as direct consequences of the cardinality condition and the definition of the distance. That is the concrete advance here. In metric geometry, GH distances are usually defined via infima over correspondences, so closed-form expressions for even restricted families count as progress. The work stays within the standard setting of bounded spaces and constant nonzero distances in the simplex, which keeps the claims manageable. The stress-test note found no internal contradictions or unsupported steps in the logical outline, and the derivation is described as following from the definition once the size condition removes matching obstructions. That aligns with what the abstract states. The restrictions are real but proportionate to the result: the simplex is a very special metric space, the cardinality bound is essential for the formula to hold, and nothing is claimed beyond these cases. No free parameters or invented entities appear. This is aimed at readers already working on explicit computations or examples in Gromov-Hausdorff theory, especially those interested in ultrametric spaces. A specialist could check the derivations against the definitions and decide whether the formulas are new or reduce to earlier work. The paper deserves a serious referee because the claims are specific enough to be verified or corrected in review, and the topic sits squarely inside metric geometry.

Referee Report

0 major / 1 minor

Summary. The paper investigates the Gromov-Hausdorff distance between a bounded metric space X and a simplex (a metric space in which all nonzero distances are equal). Under the assumption that the cardinality of the simplex does not exceed the cardinality of X, a new explicit formula for this distance is derived directly from the definition; the formula is then specialized to yield an exact expression for the distance between a simplex and an ultrametric space.

Significance. If the derivation holds, the work supplies explicit, parameter-free formulas for Gromov-Hausdorff distances in a nontrivial special case, which is valuable because such distances are rarely computable in closed form. The specialization to ultrametric spaces further extends the utility of the result within metric geometry.

minor comments (1)

[Abstract] Abstract: the constant-nonzero-distance condition on the simplex and the boundedness assumption on X are stated only implicitly; making them explicit would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives an explicit formula for the Gromov-Hausdorff distance between a simplex and bounded metric space X under the stated cardinality hypothesis (|simplex| ≤ |X|), then specializes it to ultrametric X. This is presented as following directly from the definition of the GH distance together with the constant nonzero-distance condition on the simplex; no equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorems are imported via self-citation, and no ansatz is smuggled in. The derivation chain is therefore self-contained against the standard definition of the Gromov-Hausdorff distance.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5599 in / 1017 out tokens · 15609 ms · 2026-05-25T00:24:18.558470+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces
math.MG 2019-07 unverdicted novelty 7.0

Exact Gromov-Hausdorff distances are derived between arbitrary simplexes and 2-distance spaces, yielding a complete solution to the generalized Borsuk problem and expressions for graph clique cover and chromatic numbers.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper · 6 internal anchors

[1]

Deza E., Deza M.M., Encyclopedia of Distances , 4th edition, Berlin Hei- delberg, Springer-Verlag, 2016

work page 2016
[2]

Leipzig, Veit, 1914 [reprinted by Chelsea in 1949]

Hausdorﬀ F., Grundz¨ uge der Mengenlehre. Leipzig, Veit, 1914 [reprinted by Chelsea in 1949]

work page 1914
[3]

The Structure of Superspace

Edwards D., “The Structure of Superspace”, // In: Studies in Topology , ed. by Stavrakas N.M. and Allen K.R., New York, London, San Francisc o, Academic Press, Inc., 1975. References 10

work page 1975
[4]

Groups of Polynomial growth and Expanding Maps

Gromov M., “Groups of Polynomial growth and Expanding Maps” // in: Publications Mathematiques I.H.E.S., 53, 1981

work page 1981
[5]

Gromov--Hausdorff Distance to Simplexes

Grigor’ev D.S., Ivanov A.O., Tuzhilin A.A., “Gromov–Hausdorﬀ distanc e to simplexes” // ArXiv e-prints, arXiv:1906.09644, 2019; Chebyshevskii Sbornik, to appear

work page internal anchor Pith review Pith/arXiv arXiv 1906
[6]

Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

Ivanov A.O., Tuzhilin A.A., “Solution to Generalized Borsuk Problem in Terms of the Gromov–Hausdorﬀ Distances to Simplexes” // ArXiv e- prints, arXiv:1906.10574, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1906
[7]

Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes

Iliadis S., Ivanov A., Tuzhilin A., “Geometry of Compact Metric Space in Terms of Gromov-Hausdorﬀ Distances to Regular Simplexes” // Ar Xiv e-prints, arXiv:1607.06655, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[8]

Burago, Yu

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

work page 2001
[9]

Isometry Group of Gromov--Hausdorff Space

Ivanov A. O., Tuzhilin A. A., “Isometry group of Gromov–Hausdor ﬀ space” // Matematicki Vesnik, Vol. 71 (1–2), 123–154, 2019; arXiv:1806.02100, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2019
[10]

Gromov--Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings

Ivanov A.O., Tuzhilin A.A., “Gromov–Hausdorﬀ Distance, Irreduc ible Cor- respondences, Steiner Problem, and Minimal Fillings” // ArXiv e-print s, arXiv: 1604.06116 , 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[11]

Geometry of Gromov–Hausdorﬀ metric sp ace

Ivanov A., Tuzhilin A., “Geometry of Gromov–Hausdorﬀ metric sp ace” // Bulletin de l’Academie Internationale CONCORDE. no. 3, pp. 47–57, 20 17

work page
[12]

Calculation of Minimum Spanning Tree Edges Lengths using Gromov--Hausdorff Distance

Tuzhilin A.A., “Calculation of Minimum Spanning Tree Edges Lengths using Gromov–Hausdorﬀ Distance” // ArXiv e-prints, arXiv:1605.01566, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[1] [1]

Deza E., Deza M.M., Encyclopedia of Distances , 4th edition, Berlin Hei- delberg, Springer-Verlag, 2016

work page 2016

[2] [2]

Leipzig, Veit, 1914 [reprinted by Chelsea in 1949]

Hausdorﬀ F., Grundz¨ uge der Mengenlehre. Leipzig, Veit, 1914 [reprinted by Chelsea in 1949]

work page 1914

[3] [3]

The Structure of Superspace

Edwards D., “The Structure of Superspace”, // In: Studies in Topology , ed. by Stavrakas N.M. and Allen K.R., New York, London, San Francisc o, Academic Press, Inc., 1975. References 10

work page 1975

[4] [4]

Groups of Polynomial growth and Expanding Maps

Gromov M., “Groups of Polynomial growth and Expanding Maps” // in: Publications Mathematiques I.H.E.S., 53, 1981

work page 1981

[5] [5]

Gromov--Hausdorff Distance to Simplexes

Grigor’ev D.S., Ivanov A.O., Tuzhilin A.A., “Gromov–Hausdorﬀ distanc e to simplexes” // ArXiv e-prints, arXiv:1906.09644, 2019; Chebyshevskii Sbornik, to appear

work page internal anchor Pith review Pith/arXiv arXiv 1906

[6] [6]

Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

Ivanov A.O., Tuzhilin A.A., “Solution to Generalized Borsuk Problem in Terms of the Gromov–Hausdorﬀ Distances to Simplexes” // ArXiv e- prints, arXiv:1906.10574, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1906

[7] [7]

Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes

Iliadis S., Ivanov A., Tuzhilin A., “Geometry of Compact Metric Space in Terms of Gromov-Hausdorﬀ Distances to Regular Simplexes” // Ar Xiv e-prints, arXiv:1607.06655, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[8] [8]

Burago, Yu

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

work page 2001

[9] [9]

Isometry Group of Gromov--Hausdorff Space

Ivanov A. O., Tuzhilin A. A., “Isometry group of Gromov–Hausdor ﬀ space” // Matematicki Vesnik, Vol. 71 (1–2), 123–154, 2019; arXiv:1806.02100, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2019

[10] [10]

Gromov--Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings

Ivanov A.O., Tuzhilin A.A., “Gromov–Hausdorﬀ Distance, Irreduc ible Cor- respondences, Steiner Problem, and Minimal Fillings” // ArXiv e-print s, arXiv: 1604.06116 , 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[11] [11]

Geometry of Gromov–Hausdorﬀ metric sp ace

Ivanov A., Tuzhilin A., “Geometry of Gromov–Hausdorﬀ metric sp ace” // Bulletin de l’Academie Internationale CONCORDE. no. 3, pp. 47–57, 20 17

work page

[12] [12]

Calculation of Minimum Spanning Tree Edges Lengths using Gromov--Hausdorff Distance

Tuzhilin A.A., “Calculation of Minimum Spanning Tree Edges Lengths using Gromov–Hausdorﬀ Distance” // ArXiv e-prints, arXiv:1605.01566, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016