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Quasi-isometric modification of Gromov-Hausdorff distance

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Pith's one-line read The paper defines a quasi-isometric analog of Gromov–Hausdorff distance, proves it is metrizable, and shows that any two spaces with finite quasi-isometric distortion can be connected by a continuous curve of controlled finite length.

desk verdict New coarse distance with a nice hierarchy, but the abstract's 'any two spaces' claim is wrong. read the letter →

arxiv 2602.04826 v3 pith:Z2V5YL37 submitted 2026-02-04 math.MG

Alexei Naianzin This is my paper · ORCID
classification math.MG MSC 53C2351F3054E35
keywords quasi-isometryGromov–Hausdorffdistancemetricspacescorrespondencesquasi-isometricdistortioncoarsestructurepath-connectedspaceconvergence
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 sets out to build a distance on all metric spaces that measures how far two spaces are from being quasi-isometric, in the spirit of Gromov–Hausdorff distance but without requiring compactness or boundedness. It introduces two tools: the quasi-isometric distance dhat and the quasi-isometric distortion of a correspondence, leading to a genuine metric D on equivalence classes. The main structural result is that convergence in this distance sits between Gromov–Hausdorff and pointed Gromov–Hausdorff convergence, and many geometric properties pass to limits. The capstone is Theorem 5.5: given a correspondence with finite distortion r, the interpolated spaces R_t form a continuous finite-length path in (M,D), with length at most e^{2r}−e^r. The abstract's unconditional 'any two metric spaces' is stronger than the theorem's hypothesis, which requires finite r—that is, quasi-isometric spaces.

What carries the argument

The load-bearing object is the quasi-isometric distortion of a correspondence, defined as the infimum of r>0 such that for all paired points, distances can differ only by exponential factors: (1/e^r)|y,y'|−e^r+1 ≤ |x,x'| ≤ e^r|y,y'|+e^{2r}−e^r. This single quantity replaces the additive distortion of classical Gromov–Hausdorff theory with a multiplicative-exponential one, making it invariant under quasi-isometry rather than isometry. The interpolation curve R_t then uses a convex combination of the two metrics, and the exponential bounds are exactly what make the D-distance between nearby R_t small and the total length finite.

What would settle it

Take X={p} and Y=R with the usual metric. The only correspondence is R={p}×R; the lower bound in q-disR reads |y−y'|/e^r −e^r+1 ≤ 0 for all y,y'∈R, which forces |y−y'|≤e^r(e^r−1) for all y,y' and is impossible for unbounded y,y'. Hence q-disR=∞, so D is infinite and no finite-length path connects them—contradicting the abstract's literal 'any two metric spaces' reading.

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Extended reading notes

Core claim

The central discovery is that a single number attached to a correspondence—the quasi-isometric distortion q-disR—controls both the distance D(X,Y) (as the infimum over all correspondences) and the existence of a continuous deformation between X and Y. The map t↦R_t, where R_t inherits the metric (1−t)|xx'|+t|yy'|, is a continuous path in the space M of equivalence classes of separable metric spaces, and its D-length is bounded by e^{2r}−e^r whenever q-disR≤r. This makes the quasi-isometric distance topologically and coarsely equivalent to a true metric D, and implies that the coarse structure of M is monogenic. The theorem should be read as applying to quasi-isometric spaces; the abstract's

Load-bearing premise

The theorem's path-connectedness requires a correspondence R between X and Y with finite quasi-isometric distortion (q-disR ≤ r for some finite r), which holds exactly when X and Y are quasi-isometric; without it—as for a point versus the real line—the distance is infinite and the finite-length path is not obtained.

Editorial extensions

If this is right

  • Gromov–Hausdorff convergence implies quasi-isometric convergence implies pointed Gromov–Hausdorff convergence, so the new distance is a strictly weaker way to converge that still remembers coarse structure.
  • Limit spaces inherit total boundedness, finite or infinite diameter, separability, properness (when complete), the length-space property, geodesicity, δ-hyperbolicity, and the CAT(κ) condition.
  • The metric D induces the same topology and coarse structure as dhat, with explicit two-way estimates: dhat=r gives D≤ln(1+2r), and D=r gives dhat≤e^{2r}−e^r.
  • Any two quasi-isometric spaces can be connected by a continuous curve in M of finite D-length, so the equivalence-class space is linearly connected and its coarse structure is monogenic.
  • For finite-dimensional normed spaces, the new distance coincides with the logarithmic bi-Lipschitz distance between norms, making the natural interpolation between ℓ_p spaces a geodesic.

Reading between the lines

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

  • The finite-length path theorem only bites when q-disR<∞, i.e., when X and Y are quasi-isometric; for non-quasi-isometric pairs such as a point and the real line, D is infinite and the abstract's 'any two metric spaces' statement would need a different construction.
  • The correspondence-interpolation idea suggests a general recipe: any criterion that bounds distortion of correspondences yields a controlled deformation of the underlying spaces, potentially transferable to other equivalence relations such as rough isometries or measure-preserving maps.
  • The exponential bounds are not optimal, as the paper itself notes through the finite-dimensional norm example; tightening them could turn finite-length paths into actual geodesics in more cases.
  • Because D is a genuine metric, the completion of the space of quasi-isometric equivalence classes under D may be a useful object for studying coarse-geometric boundary points, analogous to Gromov boundary constructions.
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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

3 major / 5 minor

Summary. The paper proposes a quasi-isometric analogue of the Gromov–Hausdorff distance. It defines a distance d_hat(X,Y) as the infimum of r such that X and Y are (1+r,r,r)-quasi-isometric, and a second distance D(X,Y) obtained by minimizing the quasi-isometric distortion q-disR over correspondences R. The main results are: Gromov–Hausdorff convergence implies quasi-isometric convergence and quasi-isometric convergence implies pointed Gromov–Hausdorff convergence (Cor. 3.6, Prop. 3.7); preservation of several metric properties under limits (Cor. 3.8); mutual bounds showing that d_hat and D induce the same topology and coarse structure (Prop. 5.4); and a deformation theorem (Thm. 5.5) constructing a continuous path R_t between X and Y, of length at most e^{2r}-e^r, whenever a correspondence R has q-disR≤r. The paper also discusses metrization, a monogenic coarse structure, and an example in finite-dimensional l^p spaces.

Significance. If the main estimates are correct, the construction of D is a useful contribution to the coarse/quasi-isometric geometry of metric spaces: it gives a unified, quantitative way to compare spaces up to quasi-isometry, with explicit inequalities. The proof of the triangle inequality for D (Prop. 5.3) and the mutual bounds in Prop. 5.4 make the paper essentially self-contained, and the path-deformation theorem is a nice device. However, the paper's headline claim in the abstract—that any two metric spaces can be connected by a curve of finite length—is false as stated, and Example 10 contains a Banach–Mazur definitional mismatch. These issues are load-bearing for the presentation and need correction before the paper can be accepted.

major comments (3)
  1. [Abstract and Theorem 5.5] The abstract states that 'the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.' Theorem 5.5 requires, as a hypothesis, a correspondence R with q-disR≤r for a finite r. This hypothesis is not satisfied by all metric spaces. For X={p} and Y=R, the only correspondence is {p}×R, and the lower inequality in (5.1) forces 0 ≥ e^{-r}|y-y'|-e^r+1 for all y,y'∈R, which is impossible for unbounded y,y'. Thus D({p},R)=∞. In a generalized metric, the length of any curve between two points is at least their distance, so no finite-length curve can connect them; a continuous path would also force finite distance by compactness of [0,1]. The correct statement is that any two metric spaces admitting a finite-q-dis correspondence—equivalently, quasi-isometric spaces—can be connected. The abstract and the 'linearly connected' claim in
  2. [Proposition 5.4, first direction] The proof contains a displayed inference with reversed inequalities. After constructing an (A,B+C)-correspondence with A=1+r and B+C=2r, the text says 'Hence if d_hat(X,Y)=r, then e^D≤1+r, e^{2D}-e^D≤2r, whence D≤ln(1+2r).' From an (A,B+C)-correspondence, the condition for q-disR≤s is e^s≥A and e^{2s}-e^s≥B+C. Thus for A=1+r, B+C=2r, the correct necessary conditions are e^D≥1+r and e^{2D}-e^D≥2r, not the displayed upper bounds. The final inequality D≤ln(1+2r) is nevertheless true by taking e^D=1+2r, but the proof as written is not correct and should be rewritten.
  3. [Example 10] The example identifies D(X,Y) with the Banach–Mazur distance, but it defines D_BM(X,Y)=inf_T ln max{||T||,||T^{-1}||}, whereas the standard Banach–Mazur distance (and the formula quoted from [Tom89, Prop. 37.6]) is inf_T ln(||T||·||T^{-1}||). Scaling T makes the max-version equal to one half of the standard logarithmic BM distance: for any T, choosing c with ||cT||=||(cT)^{-1}||=sqrt(||T||||T^{-1}||) gives log max = (1/2)log(||T||||T^{-1}||). Therefore the asserted equality 'D equals Banach–Mazur distance' and the numerical geodesic computations in Example 10 are off by a factor of 1/2 unless the nonstandard convention is stated explicitly and the quoted [Tom89] formula is adjusted. The asymptotic-cone argument itself is plausible and would support D=inf_T log max{||T||,||T^{-1}||}.
minor comments (5)
  1. [Proposition 3.7 proof] The proof says 'there exist maps g_k : X_k → X' and 'choose p arbitrarily' and 'd_X(g_k(p_k),p)' but the target space should be Y, not X. Please replace X with Y (and d_X with d_Y) throughout that paragraph.
  2. [Section 5, terminology] D is called a 'metric', but by the paper's own definition in Section 3 a metric must be finite. Since D may be infinite (e.g., D({pt},R)=∞), it should be called a generalized metric, as was done for ρ in Section 4.
  3. [Example 5] The reference to 'Theorem 2.' is incomplete; it should presumably be Theorem 2.9. The proof sketch for d_GH(α_k Z,Z) ≥ 1/4 is very terse and would benefit from a few more details.
  4. [Corollary 5.6] The phrase 'any two metric spaces at distance at most r' should explicitly include the hypothesis of Theorem 5.5, i.e., the existence of a finite-q-dis correspondence. As written it could be read as claiming the false global statement.
  5. [Section 4.1] M is defined as the set of equivalence classes of separable metric spaces, while the abstract speaks of the class of all metric spaces. Please align the set-theoretic scope of these statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the metric D, the mutual bounds with the quasi-isometric distance, and the finite-length path construction are all derived from explicit definitions and independent external results; the abstract's unconditional path-connectedness claim is an overclaim, not a circular reduction.

full rationale

The derivation chain is self-contained. q-disR is defined in (5.1); D is defined as its infimum in (5.2); Proposition 5.3 proves the triangle inequality by composing correspondences and estimating q-dis of the composition; Proposition 5.4 proves the two-sided bounds between D and d_hat by explicitly constructing selectors and graphs; Theorem 5.5 constructs the path t ↦ R_t and bounds its D-length using the q-dis inequality directly. None of these steps fits a fitted parameter to a target or re-labels an input as a prediction. The only author-affiliated reference, [Aru23], appears in the introduction as general motivation and is not load-bearing. The external references (Roe, BBI, Drutu–Kapovich, Tomczak-Jaegermann, Chittenden–Frink) are standard, independent, and not used as a self-citation chain. The abstract's claim that 'any two metric spaces' can be connected by a finite-length curve is stronger than Theorem 5.5, which requires a correspondence with finite q-disR; for X={pt} and Y=R, q-disR=∞ and D=∞, so the unconditional statement is false. That is a correctness or scope error, but not circularity: the theorem's hypothesis is not secretly equal to its conclusion, and the proof does not assume the result it purports to establish.

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

The paper introduces no fitted parameters or invented entities. Its central claims rest on standard mathematical definitions (metric spaces, correspondences, quasi-isometries, coarse structures) and on the paper-specific exponential parametrization of q-dis. The main domain assumption is that the spaces under discussion admit a finite-q-dis correspondence for the path theorem.

assumptions (4)
  • standard math Standard definitions and facts about Gromov–Hausdorff distance, correspondences, ε-isometries, and coarse structures (from [BBI+01], [Roe03]).
    Used throughout Sections 2 and 4 as background; no alternative to these definitions is considered.
  • domain assumption M is the set of equivalence classes of separable metric spaces under d_hat=0; the paper assumes this class forms a set.
    Section 4 defines M on separable spaces to avoid set-theoretic issues; the path theorem is stated for arbitrary metric spaces but the corollaries use M.
  • domain assumption For any two quasi-isometric spaces, there exists a correspondence with finite q-disR (constructed from the q-i embeddings).
    Theorem 5.5 assumes a correspondence with q-disR≤r exists; the proof of Prop 5.4 shows q-i spaces admit one, but the abstract extends this to all metric spaces, which is false.
  • domain assumption Asymptotic cone machinery: an (A,B)-quasi-isometry between normed spaces induces a bi-Lipschitz homeomorphism between asymptotic cones with Lipschitz constants ≤ A (Lemma 10.83 of [DK18]).
    Used in Example 10 to relate D to the Banach–Mazur distance; this is an external result and the paper's application of it contains a factor-of-2 mismatch.

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Pith. "Pith review of Quasi-isometric modification of Gromov-Hausdorff distance." pith.science (2026). https://pith.science/paper/Z2V5YL37

@misc{pith2026260204826,
  author       = {Pith},
  title        = {Pith review of: Quasi-isometric modification of Gromov-Hausdorff distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2V5YL37}},
  note         = {Machine review of arXiv:2602.04826}
}
read the original abstract

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.

Figures

Figures reproduced from arXiv: 2602.04826 by the authors.

Figure 1
Figure 1. Quasi-isometric but not ε-isometric spaces Proof. We check by direct computation. Let f : X → Y, f′ : Y → Z, then dZ(f ′ fx1, f′ fx2) ≤A ′ dY (fx1, fx2) + B ′ ≤ A ′AdX(x1, x2) + A ′B + B ′ ≤ A ′AdX(x1, x2) + A ′B + AB′ ; dZ(f ′ fx1, f′ fx2) ≥ 1 A′ dY (fx1, fx2) − B ′ ≥ 1 A′A dX(x1, x2) − B A′ − B ′ ≥ 1 A′A dX(x1, x2) − BA′ − AB′ ; dX(gg′ f ′ fx, x) ≤dX(gg′ f ′ fx, gfx) + dX(gfx, x) ≤ AC′ + B + C ≤ AC′ + A ′C + B + B… view at source ↗

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Works this paper leans on

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