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REVIEW 2 major objections 3 minor 27 references

Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper proves that the Gromov–Hausdorff distance between a finite-dimensional normed space and a subset at finite Hausdorff distance is at least the Hausdorff distance divided by twice the Jung constant, and equals it for sup-norm spaces

desk verdict The weak bound in Theorem 3.1 is a genuine, well-proved new result, but the stronger Theorem 6.1 and the headline sup-norm equality rest on a step that is internally contradictory; the paper needs serious repair before it can be trusted. read the letter →

arxiv 2607.18447 v1 pith:MXDOYQYI submitted 2026-07-20 math.MG

Henry Adams , Semeon A. Bogatyi , Florian Frick , Daniil A. Ilyukhin , Alexander O. Ivanov , Ivan N. Mikhailov , Alexey A. Tuzhilin , Anton A. Vikhrov This is my paper · ORCID
classification math.MG MSC 46B2054E35
keywords Gromov–HausdorffdistanceHausdorffJungconstantrelativenormedspacesintersectionpropertysup-normnervelemma
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 establishes a lower bound on the Gromov–Hausdorff distance between a finite-dimensional normed space V and any subset X at finite Hausdorff distance: d_GH(X,V) ≥ (1/(2J_s(V))) d_H(X,V), where J_s(V) is the relative Jung constant. When V satisfies a geometric condition called the intersection property, the bound improves to use the absolute Jung constant J(V): d_GH(X,V) ≥ (1/(2J(V))) d_H(X,V). For finite-dimensional spaces with the sup-norm, J(V)=1/2 and the intersection property holds, so the Gromov–Hausdorff distance equals the Hausdorff distance exactly. This matters because lower bounds on the Gromov–Hausdorff distance are notoriously difficult, and the result shows that for a large class of spaces the two distances are proportional.

What carries the argument

The Jung constant J(V) is the supremum over unit-diameter subsets of the circumradius; the relative Jung constant J_s(V) restricts circumcenters to lie in the convex hull of the subset. The intersection property says that, far away from the origin, the intersection of a finite family of balls of bounded radius with the complement of a large ball is contractible whenever non-empty. The proof's engine is a diagram of nerve complexes of ball coverings, where the Jung constant controls the conversion between Vietoris–Rips and Čech scales, and the intersection property guarantees the nerve lemma applies after adding the 'complement of a large ball' sets; a contradiction in top homology then force

What would settle it

Take V = ℓ∞^2 and let X be a finite ε-net of the unit square for some ε > 0; if d_GH(X,V) turns out to be strictly less than d_H(X,V), Theorem 1.1 would be false. More generally, exhibit any finite-dimensional normed space satisfying the intersection property and a subset X for which d_GH(X,V) < d_H(X,V)/(2J(V)).

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Extended reading notes

Core claim

The central claim is that the Hausdorff distance controls the Gromov–Hausdorff distance from below, up to a constant determined by the Jung constant. For any finite-dimensional normed space V and subset X with d_H(X,V) < ∞, the paper proves d_GH(X,V) ≥ d_H(X,V)/(2J_s(V)); for spaces with the intersection property, the relative Jung constant J_s(V) can be replaced by the usual Jung constant J(V). As a corollary, in a finite-dimensional sup-norm space, d_GH(X,V) = d_H(X,V) for every such X. The proof builds a correspondence between V and X, extends it to a triangulation, and then uses a nerve-lemma/homology argument involving Čech and Vietoris–Rips complexes, with the intersection property ens

Load-bearing premise

The proof of the strong lower bound assumes that a point y chosen outside X can be treated as a fixed point of the correspondence maps h and g after translating coordinates, a step that is asserted without being justified.

Editorial extensions

If this is right

  • For any finite-dimensional sup-norm space (e.g., ℓ∞^n), the Gromov–Hausdorff distance from a subset to the whole space is exactly its Hausdorff distance.
  • For an n-dimensional space with the max-norm and n ≥ 3, the bound gives d_GH(X,V) ≥ (n/(2(n-1))) d_H(X,V), improving on the trivial inequality.
  • The equivalence of finiteness of Hausdorff distance, Gromov–Hausdorff distance, and being an ε-net (Corollary 3.5) means the coarse category of subsets of a finite-dimensional normed space is governed by ε-nets.
  • For any 2-dimensional normed space, or any space with a cylindrical norm, the strong lower bound holds with the absolute Jung constant.
  • The result extends the known equality d_GH = d_H from graph subsets to a class of normed spaces, suggesting a wider phenomenon.

Reading between the lines

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

  • If the proof's normalization step can be justified, the same technique may yield lower bounds for other classes of metric spaces where a Jung-type constant and a nerve-lemma argument are available.
  • The equality for sup-norm spaces suggests that exact Gromov–Hausdorff distance computations to dense subsets might be reduced to computing Hausdorff distance, which is far easier to estimate numerically.
  • The dependence on the intersection property hints that the Gromov–Hausdorff distance to a subset is controlled by the failure of ball-intersections to be contractible at large scales; spaces where this failure is bounded may admit similar proportional bounds.
  • A testable extension: for other polytopal norms such as ℓ1^n, compute the Jung constant and search for subsets that achieve the lower bound d_GH = d_H/(2J), which would establish sharpness of the inequality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the Gromov–Hausdorff distance between a finite-dimensional normed space V and a subset X with finite Hausdorff distance. Theorem 3.1 gives a lower bound d_GH(X,V) ≥ d_H(X,V)/(2J_s(V)) in terms of the relative Jung constant. The authors then introduce an 'intersection property' and prove a stronger lower bound d_GH(X,V) ≥ d_H(X,V)/(2J(V)) in Theorem 6.1, from which they derive Theorem 1.1: for sup-norm spaces, d_GH(X,V) = d_H(X,V). The proof of Theorem 3.1 uses a triangulation, a surjectivity lemma, and Gulevich's theorem, and appears solid. The proof of Theorem 6.1 is toplogical, using Čech and Vietoris–Rips complexes, nerve lemmas, and the intersection property.

Significance. If correct, the paper would make a real contribution: it gives a quantitative lower bound for the Gromov–Hausdorff distance in terms of the Hausdorff distance and the Jung constant, and it would establish the striking equality d_GH = d_H for sup-norm spaces. The weak estimate (Theorem 3.1) is convincingly proved and appears to be a publishable result in its own right. However, the proof of the strong estimate (Theorem 6.1) contains two load-bearing errors, so the central claims of the paper are not established as written.

major comments (2)
  1. [Section 6, paragraph after diagram (1)] The proof chooses y ∈ V \ ⋃_{x∈X} B(x;(r+ε)J), so d(y,X) ≥ (r+ε)J > 0 and in particular y ∉ X. The next sentence asserts that after translating copies of V we may assume y = h(y) = g(h(y)). This is impossible: h maps V into X and g maps X into V, so h(y) ∈ X; the equality h(y) = y would imply y ∈ X, contradicting d(y,X) > 0. An isometric copy of X cannot be translated to identify h(y) and the g-preimage of y with y unless the correspondence already has a special property. All subsequent estimates of the form |y h(z)| ≥ |y z| − r and |y g(z)| ≥ |y z| − r, used to locate the centers w in U2 and U3, rely on this identification. Thus the normalization step is invalid and the lower-bound estimates are unproved.
  2. [Section 6, simpliciality check for the extended map g] In checking that the second new simplicial map g remains simplicial, the proof takes a simplex Δ with W_k = U_2 and chooses z ∈ (∩_{i<k} B(x_i;(r+ε)J)) ∩ U_2. It then writes {g(x_0),...,g(x_{k-1}),g(z)} ∈ VR(...), but g is defined only on X, and the nerve condition gives z ∈ V, not z ∈ X. For example, in V = R with X = 2Z, a ball centered at an even integer can meet U_2 in an open interval containing no even integer, so g(z) is undefined. The subsequent estimate |y w| ≥ |y g(z)| − |g(z)w|, which forces w ∈ U_3, depends on g(z). Replacing z by a nearby point of X would introduce an error of order d_H(X,V), which need not be small compared with the 6r gap between U_2 and U_3. Therefore the extended map g in diagram (3) is not proved simplicial, and the contiguity/homology contradiction at the end of Section 6 collapses.
minor comments (3)
  1. [Section 6, definition of U_3] The intersection property is stated for complements V \ \overline{B}(0;t') with t' > t, but U_3 is defined with threshold t. This quantifier mismatch should be clarified, though it is likely fixable by shifting the thresholds.
  2. [Section 6, g-simpliciality paragraph] In the check that g remains simplicial, the complexes denoted VR(X;...) and Č(X;...) should presumably be VR(V;...) and Č(V;...), since g takes values in V. The current notation appears to be a typo.
  3. [Lemma 5.2] The display in Lemma 5.2 is hard to parse: the sequence of spaces and the labels 'h', 'g', 'ι_ν' do not clearly indicate the order of the induced simplicial maps. This should be typeset more explicitly.

Circularity Check

2 steps flagged · score 2.0 of 10

Central lower-bound theorems are not circular: Jung constants enter as fixed external inputs and the intersection property for the sup-norm case is proved in-paper. The flagged load-bearing problem is a proof-validity gap (unsatisfiable y=h(y) normalization; g applied to points outside its domain), recorded as 'other' flags rather than as circularity. Minor self-citations ([18],[2],[25]) are prese

  1. other [Section 6, proof of Theorem 6.1, paragraph after diagram (1)]
    "the same procedure, being applied to those copies of V that contain the images of g and h, enables us to assume that y = h(y) = g(h(y))"

    Flagged per reviewing rule as missing support / unsatisfiable premise, not as a circular reduction. y was chosen with d(y,X) >= (r+eps)J > 0, hence y is outside X, while h : V -> X forces h(y) in X; so y = h(y) is impossible. The subsequent margin estimates |y h(z)| >= |y z| - r, used to conclude w in U_2 and w in U_3, depend on this premise, so the proof of Theorem 6.1 is invalid as written. This is a correctness/validity gap: the argument does not presuppose its own conclusion, and no equation reduces to an input by construction.

  2. other [Section 6, proof of Theorem 6.1, paragraph beginning 'Let us check that the second new map g also remains simplicial']
    "If W_k = U_2, then there exists z in (intersection of B(x_i; (r+eps)J) for i<k) intersect U_2, and we have {g(x_0), ..., g(x_{k-1}), g(z)} in VR(X; r+2(r+eps)J)"

    Flagged per reviewing rule as omitted justification / domain error, not as a circular reduction. The nerve condition only gives z in V (it may lie in U_2 outside X), while g is defined only on X; hence g(z) is undefined. Example: X = 2Z in R, a ball intersection inside U_2 is an open interval containing no point of X. The conclusion w in U_3 and the contiguity/homology contradiction rest on this invalid step. This is a proof gap, not a self-referential derivation; the theorem's inputs (J(V), intersection property) are unaffected.

full rationale

Derivation-chain audit. Theorem 3.1 (weak estimate d_GH >= d_H/(2 J_s)): the proof affinely extends a low-distortion selection f: V -> X, applies Bourgin/Webster surjectivity ([8, Prop. 4.1]), and bounds d_H(V,X) <= J_s(V) diam(f*(Delta)) via Gulevich's external identity G(V) = J_s(V) ([15], Theorem 2.6). J_s(V) is an input whose values are quoted from [21], [6], [12], [22], [13]; it is never produced by the theorem. Theorem 6.1 (strong estimate under the intersection property): the nerve diagram is assembled from the Chazal-de Silva-Oudot contiguity lemma ([10], Lemma 5.2) and the definitional Jung bound (Lemma 5.3: VR(X;s) subset of Cech C(X; J s)); the Jung constant appears only as a fixed geometric invariant of V, and the intersection property is an introduced assumption verified in-paper for Euclidean and max-norm spaces (Lemmas 4.1, 4.2). Theorem 1.1 is the clean substitution J(sup-norm) = 1/2 ([21]) into Theorem 6.1 plus the standard upper bound d_GH <= d_H; no step renames a known result or fits a parameter to the target. Self-citation audit: [18] (co-author Ilyukhin) supplies Lemmas 4.3-4.4 (2D and cylindrical norms satisfy the intersection property) used only in Corollary 6.2, not in Theorem 1.1; [2] (Adams-Frick) is a published technique source for Lemma 5.2 and the contiguity argument; [25]/[19]/[1]/[3] are contextual. None makes the central derivation reduce to the authors' own prior conclusion: the cited lemmas concern ball intersections or nerve contiguity and are stated without the target GH inequality, so they are independent support. Correctness flags (recorded as 'other' steps, not circularity): the normalization 'y = h(y) = g(h(y))' is unsatisfiable since y not in X but h(y) in X, and g is applied to a point z certified only in V not in X. These are validity gaps in the proof of Theorem 6.1; a correctness reviewer should weigh them, but they are not instances of a derivation equivalent to its inputs by construction. Verdict: no significant circularity; score 2 reflects the minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

No parameters are fitted: the Jung constants are pre-existing quantities from the literature. The central weak theorem rests on standard external theorems (Bourgin, Gulevich). The strong theorem rests on the newly introduced intersection property (domain assumption), on the co-author preprint [18] for two of its instances, and on an asserted normalization step that is internally contradictory; the latter is the main unresolved burden.

assumptions (7)
  • standard math Bourgin–Webster lemma: a continuous map between finite-dimensional normed spaces with finite distortion and dim V ≥ dim W is surjective and forces dim V = dim W.
    Used in the proof of Theorem 3.1 (Lemma 3.4) to conclude the piecewise-affine extension f* is surjective; cited from [8].
  • standard math Gulevich's theorem: the measure of nonconvexity equals the relative Jung constant, G(V) = J_s(V).
    Used in Theorem 3.1 to bound dist(v, {f(y_i)}) by J_s(V)·diam; cited from [15].
  • standard math Klee–Garkavi theorem: J(V) = J_s(V) iff V is an inner-product space or dim V ≤ 2.
    Used in remarks and Corollary 3.6; cited from [22] and [13].
  • standard math Known Jung constant values: J(V) = 1/2 for sup-norm, J_s(V) = (n−1)/n for n-dimensional max-norm (n ≥ 3), J(V) = sqrt(n/(2(n+1))) for inner-product spaces.
    Input constants for Theorem 1.1 and Corollary 3.6; cited from [21], [6], and [12].
  • domain assumption The intersection property: for each ρ > 0 there is t > 0 such that V setminus overlineB(0; t′) meets any finite intersection of balls of radius ≤ ρ in an empty or contractible set.
    Hypothesis of Theorem 6.1. Proved for Euclidean and max-norm spaces in Lemmas 4.1–4.2; claimed for dim-2 and cylindrical norms in Lemmas 4.3–4.4 via the unpublished co-author preprint [18]; counterexamples are given in Remark 4.5. The classification is left open (Problem 4.6).
  • standard math Leray's nerve lemma and the Chazal–de Silva–Oudot covering lemma (Lemma 5.2) for Vietoris–Rips and Čech complexes.
    Used in the contradiction diagram of Theorem 6.1; Lemma 5.1 from [17] and Lemma 5.2 from [10]/[2].
  • ad hoc to paper WLOG re-normalization of the correspondence so that y = h(y) = g(h(y)).
    Asserted in Section 6 after diagram (1). Unsatisfiable as stated because y ∉ X (distance ≥ (r+ε)J > 0 from X) while h(y) ∈ X. This is the review's central objection.
invented entities (1)
  • Intersection property (new condition on normed spaces) independent evidence
    purpose: Hypothesis of Theorem 6.1 enabling the replacement of the relative Jung constant J_s(V) by the absolute Jung constant J(V) in the GH lower bound.
    The property has falsifiable content: it is verified for Euclidean and max-norm spaces (Lemmas 4.1–4.2), claimed for dim-2 and cylindrical norms via [18], and explicitly refuted for icosahedral, dodecahedral, octahedral, and tube-like unit balls (Remark 4.5, Figures 1–2). Its classification is posed as an open problem (Problem 4.6).

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Pith. "Pith review of Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces." pith.science (2026). https://pith.science/paper/MXDOYQYI

@misc{pith2026260718447,
  author       = {Pith},
  title        = {Pith review of: Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXDOYQYI}},
  note         = {Machine review of arXiv:2607.18447}
}
abstract

For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov--Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov--Hausdorff distance.

Figures

Figures reproduced from arXiv: 2607.18447 by the authors.

Figure 1
Figure 1. The norms with unit ball the icosahedron, dodecahedron, or octahedron do not [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. A norm which does not satisfy the intersection property, with [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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