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REVIEW 3 major objections 4 minor 25 references

On spaces of Euclidean triangles and triangulated Euclidean surfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read An asymmetric log-ratio metric on unit-area Euclidean triangles is complete and Finsler, with geodesics governed by one dominant coordinate.

desk verdict Single-triangle results are correct and citable; the triangulated-surface geodesic proof has a real gap (fix via affine interpolation), plus a factor-2 derivative typo in Section 3. read the letter →

arxiv 2504.17328 v2 pith:YBRQWOSG submitted 2025-04-24 math.GT

Ismail Saglam (MPI) , Ken'Ichi Ohshika (MPI) , Athanase Papadopoulos (IRMA , MPI) This is my paper · ORCID
classification math.GT MSC 32G1553C7051K0551K1053B4053C60
keywords asymmetricmetricEuclideantrianglessingularstructuresFinslerstructuregeodesicsbestLipschitzmapTeichmüllertheoryconvexpolygons
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

This paper introduces an asymmetric distance $\eta$ on the space of unit-area marked Euclidean triangles: writing the three edge lengths through semiperimeter-defect coordinates $A_1,A_2,A_3$, the distance from $X$ to $Y$ is $\log \max_i\{A'_i/A_i\}$. It proves that this metric is Finsler, with infinitesimal norm $\max_i v_i/A_i$, that geodesics are exactly the paths on which one coordinate's logarithmic growth dominates the other two on every subinterval, and that the space is complete. The same formula defines an asymmetric metric on the space $E(S,T)_1$ of unit-area singular Euclidean structures on a compact surface with a fixed triangulation, where it is again Finsler and bigeodesic, and complete after arithmetic or max symmetrisation. A two-triangle example shows the asymmetric metric on triangulated surfaces need not have the convergence-symmetry property and can be forward incomplete. The metric is, in the abstract's phrase, a restriction of a non-symmetric version of the classical Thompson distance, and the paper presents it as a Euclidean analogue of the classical asymmetric metric on Teichmüller space.

What carries the argument

The central object is the change of coordinates $A_i=(l_j+l_k-l_i)/2$ on the edge lengths of a Euclidean triangle, the semiperimeter-defect coordinates, in which Heron's formula becomes $\mathrm{area}^2=(A_1+A_2+A_3)A_1A_2A_3$ and the unit-area condition is a smooth hypersurface. The metric compares two triangles by the logarithm of the maximum ratio of these coordinates, which is also the logarithm of the best Lipschitz constant of a label-preserving map between boxes in $\mathbb{R}^3$ with side lengths $A_i$. The Finsler norm is $F(A,v)=\max_i v_i/A_i$, whose unit ball in the equivalent quadrant model is a right triangle. The geodesic criterion, one coordinate's logarithmic growth dominating the others on every subinterval, makes bigeodesics explicit: paths of the form $A_i(t)=A_i^{1-t}(A'_i)^t$, rescaled to unit area, work for any pair of endpoints. For triangulated surfaces the same coordinates are taken face by face, and the image of the parametrization is asserted to be cut out by exactly one linear equality per interior edge.

What would settle it

For a fixed triangulation of a compact surface with at least one interior edge, enumerate all positive assignments $A_{ijk}$ satisfying the one-linear-equality-per-interior-edge condition and try to build the corresponding Euclidean surface by gluing the triangles; exhibit one assignment that satisfies every equality but cannot be realized by any gluing. Finding such an assignment would disprove the 'only restraint' assertion on which Sections 4-7 rest.

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Extended reading notes

Core claim

The paper's central claim is that $\eta(X,Y)=\log \max_i\{A'_i/A_i\}$ on the unit-area hypersurface $T_1=\{(A_1,A_2,A_3):A_i>0,\ (A_1+A_2+A_3)A_1A_2A_3=1\}$ is a complete asymmetric Finsler metric. A path is a geodesic precisely when, after rescaling each point to unit area, some face coordinate $A_j$ has logarithmic growth at least as large as the other two coordinates on every subinterval; any two points are joined by a bigeodesic. The infinitesimal norm is $F(A,v)=\max_i v_i/A_i$, and the same formula defines a Finsler metric on the unit-area slice $E(S,T)_1$ of Euclidean structures on a fixed triangulation. The paper proves $T_1$ is complete and has the convergence-symmetry property; for $E(S,T)_1$, completeness holds for the symmetrised metrics, while the asymmetric metric itself can fail: a disc triangulated by two triangles admits a forward-Cauchy sequence of unit-area quadrilaterals that degenerates to a segment. Along the way the paper develops a completion theory for asymmetric metric spaces based on forward and backward Cauchy sequences and the convergence-symmetry property.

Load-bearing premise

The load-bearing premise for the triangulated-surface results is the assertion after Proposition 4.3 that the image of the face-coordinate parametrization is cut out exactly by one linear equality per interior edge; if hidden constraints existed, $E(S,T)_1$ and the metric $\eta(T)$ would be different objects and the geodesic and Finsler theorems would not follow.

Editorial extensions

If this is right

  • Between any two unit-area marked triangles there is a bigeodesic, and the geodesic condition is explicit: one coordinate's logarithmic growth dominates the others on every subinterval.
  • $T_1$ is complete as a metric space and has the convergence-symmetry property, so every Cauchy sequence converges and no nontrivial completion is needed there.
  • For every $t\in(0,1)$, the space $E(S,T)_1$ is complete with respect to the max-symmetrised metric $\eta(T)^m_t$, hence also with respect to the arithmetic symmetrisation $\eta(T)^a_t$.
  • The formulas push forward to the space of unit-area convex Euclidean polygons with $n$ distinguished boundary points, where they define Finsler metrics $\eta_m$ and $\eta_a$; the paper leaves geodesics and completeness of these metrics open.
  • The two-triangle disc example separates the asymmetric theory from its symmetrisation: the symmetrised space is complete while the asymmetric metric is forward incomplete.

Reading between the lines

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

  • If the 'only restraint' premise holds, then $E(S,T)_1$ is a linear section of a positive orthant, so the difference between complete and incomplete examples should be governed by which boundary directions of that cone are reachable by forward-Cauchy sequences; classifying triangulations by this boundary combinatorics is a natural next step.
  • Example 7.7 suggests the completion of $E(S,T)_1$ for $\eta(T)$ adds flat structures in which some face coordinates collapse to zero, with forward and backward convergence distinguishing the two sides of the collapse; computing the metric completion for the two-triangle disc would test this.
  • The box-Lipschitz reformulation indicates that $\eta$ can be compared with other Thompson-type metrics on cones of labelled convex bodies; if that comparison holds, the geodesic and Finsler formulas should transfer to spaces of labelled polytopes with the same coordinate ratios.
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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 / 4 minor

Summary. The paper introduces an asymmetric distance η on the space T1 of unit-area marked Euclidean triangles, parametrized by semiperimeter-defect coordinates A_i = (a_j + a_k − a_i)/2, defined by η(X,Y) = log max_i(Y_i/X_i). It proves that η is a complete Finsler metric on T1, gives a box-Lipschitz equivalent formulation, characterizes geodesics by a coordinate-dominance condition, constructs bigeodesics, and gives the infinitesimal norm F(A,v) = max_i(v_i/A_i). It then extends the construction to E(S,T)_1, the space of unit-area singular Euclidean structures on a fixed triangulation, claiming the same geodesic, Finsler, and completeness properties, and develops a theory of completion for asymmetric metrics. A final section studies spaces of convex Euclidean polygons with several triangulations.

Significance. The single-triangle portion of the paper is clean and well executed; the metric is a non-symmetric Thompson-type distance in logarithm coordinates, and the explicit formulas for geodesics and the Finsler norm are useful and clearly motivated by Thurston's theory. If the triangulated-surface results are repaired as indicated below, the paper would provide a natural Euclidean analogue of aspects of Thurston's metric, together with a concrete forward-incomplete example (Example 7.7) that is both instructive and valuable. The completion theory for asymmetric metrics is a reasonable contribution, though the proof of one of its key steps needs repair.

major comments (3)
  1. [Section 4, Corollary 4.8] The construction of the bigeodesic is invalid because the path α(t) = (A_{ijk}^{1−t} B_{ijk}^t) is defined coordinate-wise in the ambient space (R_+^*)^J, while E(S,T) is cut out by the linear equalities of Proposition 4.3. Coordinate-wise geometric interpolation does not preserve those linear equalities: from u_0+v_0 = u'_0+v'_0 and u_1+v_1 = u'_1+v'_1 it does not follow that u_0^{1−t}u_1^t + v_0^{1−t}v_1^t = u'_0^{1−t}u'_1^t + v'_0^{1−t}v'_1^t (for example, take u_0=v'_0=1, v_0=u'_0=2, u_1=v'_1=1, v_1=u'_1=8, where the sums match at t=0 and t=1 but differ at t=1/2). Hence λ(t)α(t) is not a path in E(S,T)_1, so Corollary 4.8 does not prove the existence of a bigeodesic, and the proofs of Proposition 4.10 and Theorem 4.13 that invoke it are unsupported. The statement itself is true and repairable: the affine path β(t) = (1−t)A + tB stays in E(S,T) because the defining equalities are linear, and if j maximizes B_j/A_j, then the logarithmic derivative of β_i is pointwise dominated by that of β_j, so the normalized affine path is a bigeodesic by Theorem 4.6. Please replace the construction and amend the proofs that depend on it.
  2. [Section 4, after Proposition 4.3] The sentence 'Conversely, this is the only restraint' identifies the image of Ψ with the intersection of the positive orthant with the hyperplanes A_{jki}+A_{kij} = A_{j'k'i'}+A_{k'i'j'} for interior edges. This identification is load-bearing: it underlies the manifold structure in Proposition 4.9, the completeness proof in Theorem 7.6, and the Finsler results of Section 4.1, but the converse is not proved. Please add the proof: given positive A_{ijk} satisfying the equalities, define l_i := A_{jki}+A_{kij} for any face containing edge i; the equalities make l_i independent of the chosen face, positivity of the A-coordinates gives the triangle inequalities in each face, and gluing the resulting Euclidean triangles along edges yields a Euclidean structure on (S,T).
  3. [Section 5, step (5) in the completion construction] The proof that (X^*, Δ) has the convergence-symmetry property is not valid as written. The argument applies the convergence-symmetry hypothesis of X to the double-indexed pairs (p_{n(k),m(k)}) and (q_{n(k),m(k)}) without ensuring that the selected diagonal subsequence satisfies d(p_{n(k),m(k)}, q_{n(k),m(k)}) → 0; the assumption lim_n lim_m d(p_{n,m}, q_{n,m}) = 0 does not by itself imply this for the subsequence chosen from the condition on d(q_{n,m}, p_{n,m}). A correct proof can be obtained by choosing, for each k, representatives and indices m_k, m'_k large enough that d(p_{n_k,m'_k}, q_{n_k,m'_k}) < 1/k and such that the Cauchy-diagonal terms d(p_{n_k,m'_k}, p_{n_k,m_k}) and d(q_{n_k,m'_k}, q_{n_k,m_k}) are small, then applying the convergence-symmetry property of X and a triangle-inequality argument. Please rewrite this step.
minor comments (4)
  1. [Proposition 2.1] The asymmetry check uses the points (1,1,1) and (√3/2, √3/2, 1−√3/2), which do not lie in T1; please either normalize them to unit area or note that asymmetry on (R_+^*)^3 descends to T1 by the scaling formula exp(η(λX, λ'Y)) = (λ'/λ) exp(η(X,Y)).
  2. [Abstract and body] The Abstract states that the metric is 'a restriction of a non-symmetric version of the classical Thompson distance' and that the paper builds a bridge to Thompson's metrics, but the body does not discuss Thompson distances; please add a remark or soften the claim.
  3. [Section 8, open problems (4) and (5)] The formulations conflate the metrics: problem (4) refers to 'ηm and ηm' and problem (5) to 'ηa and ηa'; presumably these should compare ηm with ηa, and the inequalities should be stated between the two different metrics.
  4. [Throughout] There are several typos: 'Cleary' before Definition 4.4, 'W need' in the proof of Lemma 2.13, 'a convex Euclidean structures' in the title of Section 4, 'l_{FP}' in Section 8 (should presumably be l_{Fm}), and in Example 7.7 the sentence after defining Q'_n says 'the area of Qn is also equal to 1' where Q'_n is meant.

Circularity Check

0 steps flagged · score 2.0 of 10

The triangle-space derivation is self-contained; the triangulated-surface extension has an unproved 'only restraint' premise, an invalid bigeodesic interpolation in Corollary 4.8, and a same-author citation [21], but none of these reduces a prediction to its inputs by construction.

full rationale

The core results for T1 are derived directly from the definition of eta(X,Y)=log max_i A'_i/A_i: Proposition 2.1 checks the metric axioms, Propositions 2.2-2.3 give the box/Lipschitz reformulation, Theorem 2.6 characterizes geodesics by the dominance condition, and Proposition 2.14 with Theorem 2.15 prove the Finsler realization F(A,v)=max_i v_i/A_i by comparing the integral of F with the defining log-ratio distance. No fitted parameter, hidden input, or prior theorem is used, so the single-triangle part is not circular. Completeness of T1 (Theorem 7.4) is proved by isometry with a closed subset of (R^3, d_infty), again independent. The triangulated-surface part is weaker as a derivation but still not circular: Proposition 4.3's 'Conversely, this is the only restraint' is asserted without proof, although it is in fact provable by defining l_i = A_{jki}+A_{kij}; and Theorems 4.6 and 4.10 omit proofs by analogy. The most serious issue is Corollary 4.8, where alpha(t)=(A^{1-t} B^t) is declared to be a path in E(S,T): coordinate-wise geometric interpolation does not preserve the linear equalities A_{jki}+A_{kij}=A_{j'k'i'}+A_{k'i'j'} defining E(S,T), so the construction does not lie in the space as written; the bigeodesic claim is nevertheless salvageable by the affine path (1-t)A+tB. This is an invalid inference, not a circular reduction. Finally, Theorem 4.13 cites the same authors' preprint [21] for the symmetrized Finsler family; that citation supports a secondary result and is not the basis of the main Finsler theorem. Since no output is equal to an input by construction, the appropriate circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's results rest on standard tools (Heron's formula, Sard's theorem, elementary inequalities) plus two asserted identifications: the exact linear description of E(S,T) by interior-edge equalities and the bijectivity between convex-polygon structures and triangulated Euclidean structures. No free parameters are fitted and no new entities are postulated. The main proofs are self-contained; the arithmetic-family Finsler theorems are deferred to the authors' preprint [21], which is the only notable self-citation.

assumptions (5)
  • standard math Heron's formula Ar(A1,A2,A3) = sqrt((A1+A2+A3)A1A2A3), and the map (A1,A2,A3) to marked triangle with edge lengths a1 = A2+A3, a2 = A3+A1, a3 = A1+A2 is a bijection from (R*+)^3 to marked triangle shapes.
    Defines T1 throughout Section 2 and powers the area-monotonicity argument in Prop 2.1 and Prop 7.2.
  • standard math Sard's theorem, applied to the homogeneous area function q(A) = (A1+A2+A3)A1A2A3 (and its E(S,T) analogue), implies 1 is a regular value, so T1 and E(S,T)1 are embedded submanifolds.
    Used in Sections 2.4 and 4.1; the Finsler structure is defined on these tangent spaces, so smoothness is load-bearing.
  • domain assumption The image of the parametrization Psi: E(S,T) to (R*+)^J is exactly the positive orthant cut by one linear equality per interior edge, stated as 'Conversely, this is the only restraint' after Prop 4.3.
    Load-bearing for all of Section 4; true (edge lengths are recovered as sums of adjacent A-coordinates and triangle inequalities follow from positivity), but asserted without proof.
  • domain assumption Bijectivity of the natural maps E(S,T)^c -> F and E(S,T)^c_1 -> F_1 between triangulated Euclidean structures and convex-polygon structures on a marked disk.
    Asserted without proof in Section 8; the metrics eta_m and eta_a on the polygon space are defined through these identifications.
  • standard math eta on T1 is the restriction of the non-symmetric Thompson distance on the positive cone (equivalently the Funk metric on the open 2-simplex).
    Stated in the abstract; immediate from the log-ratio definition, but the chain of equivalence is not spelled out in the body.

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Pith. "Pith review of On spaces of Euclidean triangles and triangulated Euclidean surfaces." pith.science (2026). https://pith.science/paper/YBRQWOSG

@misc{pith2026250417328,
  author       = {Pith},
  title        = {Pith review of: On spaces of Euclidean triangles and triangulated Euclidean surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBRQWOSG}},
  note         = {Machine review of arXiv:2504.17328}
}
read the original abstract

In this paper, we introduce an asymmetric distance function on the space of marked Euclidean triangles of normalised area, and we prove several properties of this metric, which turns out to be (a restriction of) a non-symmetric version of the classical Thompson distance. We give a description of the geodesics of this metric, we show that it is Finsler, and we give a formula for its infinitesimal Finsler structure. We then introduce and study a Finsler metric of the space of singular Euclidean structures on a surface adapted to an underlying fixed triangulation, and we also study its geodesics and its Finsler infinitesimal structure. We then develop a theory of completeness and completion of asymmetric metrics which is adapted to our setting, and we use this theory in the study of the completeness of the metric we introduced on the space of triangles. In doing so, we establish a bridgebetween one aspect of Thurston's theory of metrics on spaces of surfaces and Thompson's metrics. The final version of this paper will appear in Monatshefte f{\"u}r Mathematik

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Reference graph

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