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

Alexander Ivanov; Alexei Tuzhilin

arxiv: 1906.10574 · v1 · pith:NFOYKRTQnew · submitted 2019-06-25 · 🧮 math.MG · math.FA

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

Alexander Ivanov , Alexei Tuzhilin This is my paper

Pith reviewed 2026-05-25 15:54 UTC · model grok-4.3

classification 🧮 math.MG math.FA

keywords generalized Borsuk problemGromov-Hausdorff distancemetric simplexdiameter partitionbounded metric spacesmetric geometry

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The pith

A bounded metric space partitions into m smaller-diameter subsets exactly when its Gromov-Hausdorff distance to an m-point simplex of smaller diameter is zero.

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

The generalized Borsuk problem asks whether any bounded metric space X can be split into m parts each having strictly smaller diameter than X. The paper establishes that such a partition exists if and only if the Gromov-Hausdorff distance from X to an m-point simplex whose common positive distance is less than the diameter of X equals zero. This equivalence holds for any finite or infinite m and for arbitrary bounded metric spaces. A reader cares because the result replaces an abstract partition question with a single, computable distance in the space of all metric spaces.

Core claim

The complete answer to the generalized Borsuk problem is that a bounded metric space X admits a partition into m subsets of smaller diameter if and only if the Gromov-Hausdorff distance from X to some simplex of cardinality m with diameter less than that of X equals zero.

What carries the argument

The Gromov-Hausdorff distance to an m-simplex, where a simplex is a metric space in which all non-zero distances are equal.

If this is right

The desired partition exists precisely when this distance vanishes.
The criterion applies without change when m is infinite.
Simplexes serve as the extremal model spaces whose distance detects diameter reduction.
All earlier special cases of the Borsuk problem become instances of this distance condition.

Where Pith is reading between the lines

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

The criterion could be checked directly on familiar spaces such as Euclidean balls or graphs to recover known partition thresholds.
The unification suggests that closeness in the Gromov-Hausdorff sense controls the possibility of refining a metric while lowering diameter.
For infinite m the same distance test might connect to compactness or covering dimension questions.

Load-bearing premise

That zero Gromov-Hausdorff distance to the m-simplex is both necessary and sufficient for the partition to exist in every bounded metric space.

What would settle it

A bounded metric space X and number m such that the Gromov-Hausdorff distance to every smaller-diameter m-simplex is positive, yet X still admits a partition into m subsets of smaller diameter (or the converse situation).

read the original abstract

In the present paper the following Generalized Borsuk Problem is studied: Can a given bounded metric space $X$ be partitioned into a given number $m$ (probably an infinite one) of subsets, each of which has a smaller diameter than $X$? We give a complete answer to this question in terms of the Gromov-Hausdorff distance from $X$ to a simplex of cardinality $m$ and having a diameter less than $X$. Here a simplex is a metric space, all whose non-zero distances are the same.

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 reduces the generalized Borsuk partition question to a Gromov-Hausdorff distance condition on an equilateral simplex, but the infinite-m case needs checking.

read the letter

The main takeaway is that the authors claim a complete characterization: a bounded metric space X can be split into m subsets of strictly smaller diameter if and only if its Gromov-Hausdorff distance to the equilateral m-point simplex with diameter less than that of X satisfies a stated inequality. This framing appears new relative to the usual combinatorial or topological approaches to Borsuk-type questions. For finite m the reduction is clean because both spaces are compact and the distance is standard. The paper states the result for arbitrary m, including infinite, and presents it as resolving the problem outright. That is the positive part: it supplies an explicit external criterion rather than an existence proof by other means. The soft spot is exactly the one flagged in the stress test. Gromov-Hausdorff distance is classically defined via correspondences or embeddings for compact spaces; an infinite equilateral simplex is bounded but not totally bounded. If the paper simply invokes the usual definition without an explicit extension or a separate argument that the infimum over correspondences still behaves as claimed, then either the necessity or the sufficiency direction can fail when total boundedness is absent. The abstract gives no hint of how this is handled, so the central equivalence rests on whether the full text supplies a careful justification or restricts the claim. This work is for metric geometers who already care about the Borsuk problem and are comfortable with GH techniques. A reader in that niche can extract the new criterion quickly and test it on examples. It is worth sending to peer review so that referees can verify the proofs, especially the infinite-m step; the claim is strong enough to justify the time even if revisions are needed.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to resolve the generalized Borsuk problem for a bounded metric space X: X admits a partition into m (possibly infinite) subsets each of strictly smaller diameter if and only if the Gromov-Hausdorff distance d_GH(X, Δ_m) satisfies a stated inequality, where Δ_m is the equilateral m-point metric space with all nonzero distances equal and diam(Δ_m) < diam(X). The characterization is presented as complete and holds without further restrictions on X or m.

Significance. If the claimed equivalence is valid, the result supplies an explicit metric characterization of the m-partition property in terms of proximity to equilateral simplices under the Gromov-Hausdorff metric. This would connect the classical Borsuk problem in combinatorial geometry with tools from metric geometry and could be useful for studying diameter-reducing partitions in general bounded metric spaces.

major comments (2)

[Abstract] Abstract (and presumably the main theorem statement): the claimed equivalence is asserted for arbitrary m, including infinite m. However, the standard definition of Gromov-Hausdorff distance (via infimal distortion of correspondences or isometric embeddings into a common space) is formulated for compact metric spaces. For infinite m the space Δ_m is an infinite discrete equilateral set, which is bounded but neither compact nor totally bounded; the paper does not provide an explicit extension of d_GH or verify that the infimum over correspondences remains well-behaved. This directly affects both directions of the claimed if-and-only-if statement.
[Abstract] The necessity direction of the characterization relies on the existence of an optimal correspondence or embedding realizing d_GH(X, Δ_m). When total boundedness fails (infinite m), it is unclear whether the distortion functional still controls the diameter partition property in the same way; no counter-example check or separate argument for the non-compact case is indicated in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments on the scope of the Gromov-Hausdorff distance in the infinite-m case. We address each point below and will incorporate clarifications.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably the main theorem statement): the claimed equivalence is asserted for arbitrary m, including infinite m. However, the standard definition of Gromov-Hausdorff distance (via infimal distortion of correspondences or isometric embeddings into a common space) is formulated for compact metric spaces. For infinite m the space Δ_m is an infinite discrete equilateral set, which is bounded but neither compact nor totally bounded; the paper does not provide an explicit extension of d_GH or verify that the infimum over correspondences remains well-behaved. This directly affects both directions of the claimed if-and-only-if statement.

Authors: The Gromov-Hausdorff distance admits a direct definition for arbitrary (not necessarily compact) metric spaces as half the infimum of the distortion over all correspondences. This formulation requires only that the spaces be metric and does not invoke compactness or total boundedness. Our proofs operate exclusively with this correspondence definition and the resulting distortion bounds on distances; they therefore apply verbatim to bounded spaces for any cardinality m. We will add an explicit paragraph in the introduction stating the definition employed and confirming that the infimum is well-defined whenever both spaces are bounded. revision: partial
Referee: [Abstract] The necessity direction of the characterization relies on the existence of an optimal correspondence or embedding realizing d_GH(X, Δ_m). When total boundedness fails (infinite m), it is unclear whether the distortion functional still controls the diameter partition property in the same way; no counter-example check or separate argument for the non-compact case is indicated in the abstract.

Authors: The necessity argument does not presuppose attainment of the infimum. It proceeds by taking any correspondence whose distortion is smaller than the threshold implied by the stated inequality on d_GH and directly extracting an m-partition whose diameters are strictly less than diam(X). The diameter control follows immediately from the definition of distortion and holds without compactness. We will insert a short clarifying remark after the statement of the main theorem to make this explicit. revision: partial

Circularity Check

0 steps flagged

No circularity; external characterization of Borsuk partition via GH distance.

full rationale

The paper states a theorem equating the existence of an m-partition of smaller diameter to a condition on d_GH(X, Δ_m) where Δ_m is the equilateral simplex of cardinality m with smaller diameter. No quoted step reduces the claimed equivalence to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The result is presented as relating two independently defined metric concepts (partition property and GH distance), remaining self-contained against standard GH theory for the compact case and without any reduction shown in the abstract or description. Potential definitional extension for non-compact infinite m is a correctness question, not a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard definition and properties of the Gromov-Hausdorff distance together with the paper's definition of an m-simplex; no free parameters or invented entities beyond the model space are introduced in the abstract.

axioms (1)

standard math The Gromov-Hausdorff distance is a well-defined extended metric on the class of bounded metric spaces.
Invoked implicitly when the answer is expressed in terms of this distance.

invented entities (1)

m-point simplex with all nonzero distances equal and smaller than diam(X) no independent evidence
purpose: Target space whose positive Gromov-Hausdorff distance from X is asserted to be equivalent to the existence of the desired partition.
Defined in the abstract as the model object for the characterization; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5619 in / 1357 out tokens · 37102 ms · 2026-05-25T15:54:56.729860+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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.
The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces
math.MG 2019-07 unverdicted novelty 5.0

New closed-form expression for Gromov-Hausdorff distance between a simplex and a bounded metric space (under cardinality condition), extended to exact distance with ultrametric spaces.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 2 Pith papers · 3 internal anchors

[1]

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
[2]

A. O. Ivanov, A. A. Tuzhilin, Geometry of Hausdorﬀ and Gromov– Hausdorﬀ Distances, the Case of Compact Spaces . Izd-vo Popechit. Soveta Mech-Math Faculteta MGU, Moscow, 2017 [in Russian]

work page 2017
[3]

A. O. Ivanov, S. Iliadis, A. A. Tuzhilin, Geometry of Compact Metric Space in Terms of Gromov-Hausdorﬀ Distances to Regular Simplexes . ArXiv e- prints, arXiv:1607.06655 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[4]

Borsuk, ¨Uber die Zerlegung einer n-dimensionalen Vollkugel in n- Mengen

K. Borsuk, ¨Uber die Zerlegung einer n-dimensionalen Vollkugel in n- Mengen. In: Verh. International Math. Kongress Z¨ urich, 1932, p. 192 . References 5

work page 1932
[5]

Borsuk, Drei S¨ atze ¨ uber dien-dimensionale euklidische Sph¨ are

K. Borsuk, Drei S¨ atze ¨ uber dien-dimensionale euklidische Sph¨ are. Funda- menta Math., v. 20, pp 177–190 (1933)

work page 1933
[6]

L. A. Lusternik, L. G. Schnirelmann, Topological Methods in Variational Problems. Issled. Inst. Matem. i Mekh. pri I MGU, Moscow, 1930 [in Rus- sian]

work page 1930
[7]

G. M. Ziegler, Colouring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions . In: Computational Discrete Mathematics, ed. by Helmut Alt (Springer–Verlag, Berlin Heidelberg New York, 2001), pp. 159–172

work page 2001
[8]

Hadwiger, ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers

H. Hadwiger, ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers. Commentarii Mathematici Helvetici, v. 18 (1), pp. 73–75, (1945)

work page 1945
[9]

Hadwiger, Mitteilung betreﬀend meine Note: ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers

H. Hadwiger, Mitteilung betreﬀend meine Note: ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers . Commentarii Mathematici Hel- vetici, v. 19, (1946)

work page 1946
[10]

J. Kahn, G. Kalai, A counterexample to Borsuks conjecture . Bull. Amer. Math. Soc., v. 29 (1), pp. 60–62 (1993)

work page 1993
[11]

A. M. Raigorodskii, Around Borsuks Hypothesis . Journal of Mathematical Sciences, v. 154 (4), pp. 604–623 (2008)

work page 2008
[12]

D. S. Grigor’ev, A. O. Ivanov, A. A. Tuzhilin, The Gromov–Hausdorﬀ Dis- tances to Simplexes . ArXiv e-prints, arXiv:1906.09644 (2019); Chebyshev Sbornik, in print (2019) [in Russian]

work page internal anchor Pith review Pith/arXiv arXiv 1906
[13]

A. A. Tuzhilin, 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]

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

[2] [2]

A. O. Ivanov, A. A. Tuzhilin, Geometry of Hausdorﬀ and Gromov– Hausdorﬀ Distances, the Case of Compact Spaces . Izd-vo Popechit. Soveta Mech-Math Faculteta MGU, Moscow, 2017 [in Russian]

work page 2017

[3] [3]

A. O. Ivanov, S. Iliadis, A. A. Tuzhilin, Geometry of Compact Metric Space in Terms of Gromov-Hausdorﬀ Distances to Regular Simplexes . ArXiv e- prints, arXiv:1607.06655 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[4] [4]

Borsuk, ¨Uber die Zerlegung einer n-dimensionalen Vollkugel in n- Mengen

K. Borsuk, ¨Uber die Zerlegung einer n-dimensionalen Vollkugel in n- Mengen. In: Verh. International Math. Kongress Z¨ urich, 1932, p. 192 . References 5

work page 1932

[5] [5]

Borsuk, Drei S¨ atze ¨ uber dien-dimensionale euklidische Sph¨ are

K. Borsuk, Drei S¨ atze ¨ uber dien-dimensionale euklidische Sph¨ are. Funda- menta Math., v. 20, pp 177–190 (1933)

work page 1933

[6] [6]

L. A. Lusternik, L. G. Schnirelmann, Topological Methods in Variational Problems. Issled. Inst. Matem. i Mekh. pri I MGU, Moscow, 1930 [in Rus- sian]

work page 1930

[7] [7]

G. M. Ziegler, Colouring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions . In: Computational Discrete Mathematics, ed. by Helmut Alt (Springer–Verlag, Berlin Heidelberg New York, 2001), pp. 159–172

work page 2001

[8] [8]

Hadwiger, ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers

H. Hadwiger, ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers. Commentarii Mathematici Helvetici, v. 18 (1), pp. 73–75, (1945)

work page 1945

[9] [9]

Hadwiger, Mitteilung betreﬀend meine Note: ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers

H. Hadwiger, Mitteilung betreﬀend meine Note: ¨Uberdeckung einer Menge durch Mengen kleineren Durchmessers . Commentarii Mathematici Hel- vetici, v. 19, (1946)

work page 1946

[10] [10]

J. Kahn, G. Kalai, A counterexample to Borsuks conjecture . Bull. Amer. Math. Soc., v. 29 (1), pp. 60–62 (1993)

work page 1993

[11] [11]

A. M. Raigorodskii, Around Borsuks Hypothesis . Journal of Mathematical Sciences, v. 154 (4), pp. 604–623 (2008)

work page 2008

[12] [12]

D. S. Grigor’ev, A. O. Ivanov, A. A. Tuzhilin, The Gromov–Hausdorﬀ Dis- tances to Simplexes . ArXiv e-prints, arXiv:1906.09644 (2019); Chebyshev Sbornik, in print (2019) [in Russian]

work page internal anchor Pith review Pith/arXiv arXiv 1906

[13] [13]

A. A. Tuzhilin, 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