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arxiv: 1906.09644 · v1 · pith:RTO7QFZQnew · submitted 2019-06-23 · 🧮 math.MG · math.FA

Gromov--Hausdorff Distance to Simplexes

D.S.Grigor'ev , A.O.Ivanov , A.A.Tuzhilin This is my paper

Pith reviewed 2026-05-25 17:54 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords Gromov-Hausdorff distancesimplexesmetric spacespartitionsminimal spanning treesbounded metric spaces
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The pith

Gromov-Hausdorff distances from any bounded metric space to simplexes are controlled by partition geometry.

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

The paper extends earlier results on Gromov-Hausdorff distances to simplexes—metric spaces in which every pair of distinct points has the same positive distance—from compact spaces to all bounded metric spaces. It shows that the same partition geometry used in the compact setting determines these distances once boundedness replaces compactness, and that the finite case reduces to lengths of minimal spanning trees. The work also shortens some proofs from the compact case. A reader would care because many natural metric spaces are bounded without being compact, so the formulas become usable in a larger setting.

Core claim

The geometric properties of partitions determine the Gromov-Hausdorff distance from a bounded metric space to a simplex, generalizing the compact-space formulas and yielding minimal spanning tree lengths in the finite case.

What carries the argument

Geometry of partitions of the metric space, which enters the distance formulas to simplexes and reduces to minimal spanning tree lengths when the space is finite.

If this is right

  • The distance formulas hold for every bounded metric space, not only compact ones.
  • In the finite case the distance equals the length of a minimal spanning tree.
  • Some proofs originally given for compact spaces become shorter.
  • Partition geometry remains the central tool without any compactness requirement.

Where Pith is reading between the lines

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

  • The same partition approach might be tested on specific bounded non-compact spaces such as infinite discrete sets with uniform distance.
  • One could ask whether the formulas remain valid when the target simplex is allowed to vary continuously with the source space.

Load-bearing premise

The partition geometry that controls the distances for compact spaces continues to control them when the space is merely bounded.

What would settle it

A bounded metric space for which the Gromov-Hausdorff distance to a simplex differs from the value computed from its partitions.

Figures

Figures reproduced from arXiv: 1906.09644 by A.A.Tuzhilin, A.O.Ivanov, D.S.Grigor'ev.

Figure 1
Figure 1. Figure 1: The graph of the function g(λ) = 2dGH(λ∆, X) for αm(X) = 0 and various values of dm(X) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph of the function g(λ) = 2dGH(λ∆, X), the point A lies below the strip S. • if α − m(X) + dm(X) ≤ λ ≤ αm(X) + d + m(X), then max λ − αm(X), dm(X) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graph of the function g(λ) = 2dGH(λ∆, X), the point A belongs to the strip S. Let B be the intersection point of the graphs of the functions diam X −λ and λ−αm(X). Again, the vertical dashed lines partition the figure into three parts, and the marginal parts contain the graphs of exact values of the function g. In the middle part the bold line bounds a 5-gone, that [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 4
Figure 4. Figure 4: g diam X - λ λ - αm --(X) d (X) m O diam X λ diam X αm αm (X) --(X) dm + (X) λ - αm (X) A S T B [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph of the function g(λ) = 2dGH(λ∆, X), the point B lies above the strip S. In this case, the function g can be calculated exactly for all λ. Corollary 2.20. Let X be an arbitrary bounded metric space, m = #λ∆ ≤ #X. Suppose that 2d + m(X) ≤ diam X − αm(X), then 2dGH(λ∆, X) = max diam X − λ, λ − αm(X) [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The corresponding calculations essentially use geometry of partitions of these spaces. In the finite case, it gives the lengths of minimal spanning trees. A similar theory for compact metric spaces was worked out previously. In the present paper we generalize those results to any bounded metric space, and also, we simplify some proofs.

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 generalizes prior results on explicit formulas for the Gromov-Hausdorff distance from a metric space X to simplexes (metric spaces with all nonzero distances equal) from the compact case to arbitrary bounded metric spaces. The formulas are expressed in terms of geometric invariants of partitions of X; in the finite case these reduce to lengths of minimal spanning trees. Some proofs from the compact setting are simplified.

Significance. If the generalization is valid, the work extends the range of spaces for which GH distances to simplexes admit explicit partition-based expressions, which may be useful in metric geometry and data analysis on non-compact but bounded spaces. The claimed proof simplifications are a secondary positive feature.

major comments (2)
  1. [abstract and generalization section] The central generalization (abstract and §1) asserts that the partition-controlled formulas extend from compact to merely bounded spaces. However, the skeptic's concern is load-bearing: boundedness alone does not guarantee that infima over partitions are attained or that sequential-compactness arguments used in the compact case survive. The manuscript must exhibit, in the relevant proof section, an explicit replacement argument (e.g., a lemma showing that the relevant infima are realized or that ε-net constructions can be replaced by something weaker) that does not invoke total boundedness.
  2. [proof of the main generalization theorem] If the new proofs rely on any limiting procedure that extracts convergent subsequences of partitions or nets, the argument must be checked against the counter-example of a bounded but non-totally-bounded space (e.g., an infinite discrete space with distances bounded by 1). No such verification appears to be supplied.
minor comments (2)
  1. Notation for the simplex and for the partition functionals should be introduced once and used consistently; several symbols appear to be redefined in passing.
  2. The finite-case reduction to MST lengths is stated but the precise correspondence (which partition functional equals which MST quantity) could be stated as a numbered lemma for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying points where the generalization to bounded spaces requires clearer justification. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the arguments.

read point-by-point responses

  1. Referee: [abstract and generalization section] The central generalization (abstract and §1) asserts that the partition-controlled formulas extend from compact to merely bounded spaces. However, the skeptic's concern is load-bearing: boundedness alone does not guarantee that infima over partitions are attained or that sequential-compactness arguments used in the compact case survive. The manuscript must exhibit, in the relevant proof section, an explicit replacement argument (e.g., a lemma showing that the relevant infima are realized or that ε-net constructions can be replaced by something weaker) that does not invoke total boundedness.

    Authors: The proofs of the main results in §3 and §4 are direct and rely only on the uniform bound on diameters to control the relevant suprema and infima over partitions; no attainment of infima or sequential compactness is invoked. The arguments proceed by explicit comparison of partitions without extracting convergent subsequences. We agree that an explicit remark or short lemma clarifying this independence from total boundedness would strengthen the exposition and will add it to the proof section in the revised version. revision: yes

  2. Referee: [proof of the main generalization theorem] If the new proofs rely on any limiting procedure that extracts convergent subsequences of partitions or nets, the argument must be checked against the counter-example of a bounded but non-totally-bounded space (e.g., an infinite discrete space with distances bounded by 1). No such verification appears to be supplied.

    Authors: The simplified proofs do not employ limiting procedures or subsequence extraction; they consist of direct estimates and constructions that remain valid when the space is an infinite discrete metric space with diameter bound 1 (where the partition invariants reduce to the same combinatorial quantities as in the finite case). We will insert a brief verification paragraph addressing this class of examples to make the independence from compactness explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization from compact to bounded spaces uses independent extension of partition geometry

full rationale

The paper explicitly references prior results on compact metric spaces and extends them to bounded ones via partitions and MST lengths in the finite case. No equations or claims reduce a derived quantity to a fitted input or self-citation by construction; the central formulas are presented as new calculations that hold under boundedness alone. Self-citation of the compact case is load-bearing only for the starting point, not for the generalization itself, which is claimed to simplify proofs without invoking compactness-dependent limits. This matches the default non-circular outcome for an honest extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior compact-space theory plus standard definitions of Gromov-Hausdorff distance and metric-space partitions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of metric spaces and the definition of Gromov-Hausdorff distance
    Invoked throughout the distance calculations.
  • standard math Existence and properties of minimal spanning trees in finite metric spaces
    Used to interpret the finite-case formulas.

pith-pipeline@v0.9.0 · 5617 in / 1177 out tokens · 24739 ms · 2026-05-25T17:54:04.709931+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

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

  1. 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.

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

    math.MG 2019-06 unverdicted novelty 7.0

    The generalized Borsuk problem is solved by the criterion that a bounded metric space X admits an m-partition into smaller-diameter sets if and only if its Gromov-Hausdorff distance to an m-point simplex of smaller di...

  3. 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

12 extracted references · 12 canonical work pages · cited by 3 Pith papers · 4 internal anchors

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