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On the average scale-invariant Cassinian metric

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Pith's one-line read This paper proves that the balls of the scale-invariant Cassinian metric in a once-punctured Euclidean space are convex precisely for radius r ≤ log 3 and not convex for r > log 3, and it sharpens comparisons between this metric and other…

desk verdict The paper's central convexity claim is false — there is a concrete counterexample at r = log 3 — but the comparison inequalities are decent and could be salvaged. read the letter →

arxiv 2506.00810 v2 pith:DOR7S232 submitted 2025-06-01 math.MG math.CV

Manas Mohapatra , Antti Rasila , Matti Vuorinen This is my paper
classification math.MGmath.CV MSC 30F4551M0551M10
keywords scale-invariantCassinianmetrichyperbolic-typemetricsballsconvexitythresholdpuncturedEuclideanspacequasiconformalmappingsdistanceratioGromovhyperbolicity
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 studies the average scale-invariant Cassinian metric, a hyperbolic-type distance defined on spaces with finitely many punctures by averaging logarithms of Cassinian ratios to the punctures. Its main geometric result is a sharp convexity threshold: in R^n \ {z}, every ball of the one-puncture metric τ_p is convex when its radius is at most log 3 and fails to be convex when the radius exceeds log 3. The paper also proves sharp two-sided comparison inequalities relating τ_p and its average to the u, j, ~j, j*, and triangular-ratio metrics, gives density limits for the metric, and shows that L-bilipschitz maps distort τ_p by at most $L^{2}$. These results matter because they give a concrete Euclidean picture of a metric used in geometric function theory and quasiconformal geometry, where such sharp ball-shape information is often hard to obtain.

What carries the argument

The central object is the scale-invariant Cassinian metric τ_p(x,y) = log(1 + 2 d(x,y)/$\sqrt$(d(x,p)d(y,p))) and its k-puncture average τ̂_D. For the convexity theorem the paper reduces to the plane with puncture at 0 and center x = 1; a boundary point y of B_{τ_p}(1,r) satisfies |1-y| = (e^r - 1)/2 $\sqrt$(|y|), and writing t = |y| and θ = angle(y,0,1), the law of cosines gives t as one of two branches parametrized by θ. The sign of the derivative of the tangent slope of these branches is controlled by α(θ) = (2 cos θ + (e^r - 1)^2/4)^2 - 4; the unproved Lemma 6.2 is the inequality that makes m'_1(θ) ≤ 0 and m'_2(θ) ≥ 0 for 0 < r ≤ log 3, yielding convexity.

What would settle it

Evaluate the expression in Lemma 6.2, 4 $sin^{2}$ θ $\sqrt$(α(θ)) + sin θ α'(θ) - 2 cos θ α(θ) + (α(θ))^{3/2}, on a fine grid with 0 < r ≤ log 3 and 0 < θ < arccos(1 - (e^r - 1)^2/8); any positive value is a counterexample to the lemma and invalidates the proof of convexity. Plotting the boundary of B_{τ_p}(1,r) in $R^{2}$ \ {0} for r slightly below and above log 3 should show a concavity dip appear exactly as r crosses log 3.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the geometry of the scale-invariant Cassinian metric in once-punctured Euclidean space is governed by a sharp logarithmic threshold. For D = R^n \ {z} and the metric τ_p(x,y) = log(1 + 2|x-y|/$\sqrt$(|x-p||y-p|)), the metric ball B_{τ_p}(x,r) is convex exactly for 0 < r ≤ log 3, while for every r > log 3 it is not convex (Theorems 6.1 and 6.3). Alongside this, the paper establishes sharp constants in comparisons with other hyperbolic-type metrics: for instance τ_p ≤ u_p ≤ 2τ_p, (1/2) ~j_p ≤ τ_p ≤ 2 ~j_p, and j_p ≤ τ_p ≤ 2j_p in once-punctured spaces, with analogous bounds for the averaged metric in multiply punctured spaces. It also proves inclusion relations between Euclidean and Cassinian balls, identifies the local density 2/d(x,p), and shows L-bilipschitz maps are $L^{2}$-bilipschitz in the τ_p metric.

Load-bearing premise

The convexity claim for r ≤ log 3 rests on Lemma 6.2, a technical inequality about the boundary curve's slope derivative that is stated without proof; if that inequality fails for some allowed r and θ, the convexity theorem does not follow.

Editorial extensions

If this is right

  • The convexity threshold log 3 is sharp, so any radius above log 3 produces a non-convex ball in R^n \ {z}; this gives a complete convexity criterion for once-punctured Euclidean space.
  • The comparison inequalities transfer questions about Cassinian balls to questions about better-known hyperbolic-type metrics, with best-possible constants in the sharp cases D = R^n \ {0} and D = R^n \ {-e_1, e_1}.
  • The inclusion B(x,r) ⊆ B_{τ_p}(x,t) ⊆ B(x,R) with r = (e^t - 1)/(e^t + 1)d(x) and R = (e^t - 1)/(3 - e^t)d(x) quantifies how Cassinian balls sit inside Euclidean balls for t < log 3.
  • The density limit τ_p(x,y)/d(x,y) → 2/d(x,p) means the metric is asymptotically a rescaling of Euclidean distance near each point.
  • L-bilipschitz maps preserve the τ_p metric up to factor L^2, and the theorem gives a linear dilatation bound L^2 for quasiconformality.

Reading between the lines

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

  • The same polar-curve method could plausibly yield a convexity threshold for the average metric τ̂_D in multiply punctured Euclidean spaces; the paper proves convexity only for the one-puncture metric, but the average's density and inclusion estimates are already established.
  • The sharp comparison constants suggest the scale-invariant Cassinian metric is quantitatively the same as the hyperbolizing metric u_D up to factor 2; one might test whether τ̂_D and u_D are bi-Lipschitz equivalent with constant 2 in more general domains.
  • Since the convexity threshold comes from a discriminant condition α(θ) > 0, one could test numerically whether the threshold log 3 persists for other centers or under small perturbations of the puncture geometry; the proof fixes the center at distance 1 from the puncture.
  • A proof or refutation of Lemma 6.2 by analytic means would settle the convexity result independently of the present sign analysis; the lemma is the only unproved step in the argument.
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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 studies the average scale-invariant Cassinian metric \hat\tau_D, defined as an average of one-point metrics \tau_{p_i}. Sections 3 and 4 prove two-sided comparison inequalities with Ibragimov's metric u_D, the distance-ratio metrics j_D and \tilde j_D, the tanh-transformed metric j_D^*, and the triangular-ratio metric, together with density estimates for \tau_p and \hat\tau_D. Section 5 gives Euclidean ball inclusion relations and a Lipschitz/quasiconformality result. Section 6 addresses the convexity of \tau_p-metric balls in once-punctured Euclidean space and claims the sharp threshold that balls are convex for r \le \log 3 and non-convex for r > \log 3.

Significance. If the convexity threshold were correct, it would provide a sharp geometric description of the balls of the scale-invariant Cassinian metric in once-punctured Euclidean space and would be a natural counterpart to known results for the triangular-ratio metric. The comparison estimates in Sections 3 and 4 are elementary, mostly correct, and are accompanied by concrete sharpness examples; these parts are a useful contribution independent of Section 6. However, the advertised convexity theorem is false, and the unproved Lemma 6.2 on which its proof rests is also false. The paper therefore cannot be accepted in its present form; the positive portions might form the basis of a separate article without the convexity claims.

major comments (3)
  1. [§6, Theorem 6.3] The central convexity claim is false. In D = R^2 \ {0}, take p = 0, x = 1, and r = \log 3. For z_1 = 0.381 + 0.1i and z_2 = 0.381 - 0.1i, one has |z_i| = \sqrt{0.381^2 + 0.1^2} = 0.393904 and |1 - z_i|^2 = 0.619^2 + 0.1^2 = 0.393161, so |1 - z_i| / \sqrt{|z_i|} = 0.627026 / 0.627618 = 0.99906 < 1. Hence both points belong to B_{\tau_0}(1, \log 3). Their midpoint m = 0.381 satisfies 2|1 - m| / \sqrt{m} = 2 * 0.619 / 0.617213 = 2.0057 > 2, so \tau_0(1,m) > \log 3. The chord between two points of the ball leaves the ball, contradicting Theorem 6.3 and the claimed threshold r \le \log 3.
  2. [§6, Lemma 6.2] Lemma 6.2 is stated without proof and is numerically false. At r = \log 3 and \theta = 0, with \alpha(0) = (2 + (e^r - 1)^2/4)^2 - 4 = 5, the left-hand side of the claimed inequality equals -2\alpha(0) + \alpha(0)^{3/2} = -10 + 5\sqrt{5} \approx 1.18 > 0, contradicting the asserted non-positivity on (0, \arccos(1 - (e^r - 1)^2/8)). By continuity the expression is positive for sufficiently small \theta > 0 as well. Since the proof of Theorem 6.3 depends directly on this lemma, the convexity result is unsupported independently of the explicit counterexample.
  3. [§6, Theorem 6.1] The proof of non-convexity for r > \log 3 contains a sign error. For g(\theta) = 2\sin\theta - \tan\theta \sqrt{\alpha(\theta)}, one has g'(0) = 2 - \sqrt{\alpha(0)}, not 2 + \sqrt{\alpha(0)} as stated; for r > \log 3 this derivative is negative, not positive. Moreover, the text first says one needs g(\theta) < 0 near \theta = 0, then derives g(\theta) > 0, and concludes that the slope is negative; the chain of signs is inconsistent. Thus the non-convexity direction is not established as written, even though the conclusion may be true.
minor comments (4)
  1. [§3, Theorems 3.2 and 3.4] The symbol \zeta_D appears in the sharpness computations but is never defined; it should be \hat\tau_D, the average scale-invariant Cassinian metric.
  2. [§5, Theorem 5.4 proof] The lower bound \tau_p(x,y) \ge \log((1 - 3t)/(1 + t)) is claimed for t \in (0,1/2), but the right-hand side is undefined when t \ge 1/3; the argument should restrict to t \in (0,1/3) or use a different lower bound.
  3. [§6, display after Theorem 6.1] The printed formula for m_1'(\theta) has misplaced parentheses and inconsistently mixes \theta and \alpha(\theta) factors, so the expression is not readable as written and should be re-derived carefully.
  4. [Figure 1] The caption lists radii r = \log 3, \log 5, \log 7, \log 8.8 but does not indicate the numerical counterexample points; adding those points would help readers see the claimed threshold.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the comparison theorems and convexity analysis derive from the metric definitions via elementary inequalities; the unproved Lemma 6.2 is a rigor gap, not a circular reduction.

full rationale

Walking the derivation chain, every load-bearing result is obtained by applying elementary inequalities (AM-GM, Bernoulli, the triangle inequality) to the defining formulas of the metrics, which are stated in Section 1 and are not themselves consequences of the target inequalities. For example, Theorem 3.1 derives τ_p ≤ u_p ≤ 2τ_p from the definitions together with Bernoulli's inequality (2.1); Theorem 3.8 is essentially the AM-GM inequality rewritten through the identity e^{τ_p} − 1 = 2d(x,y)/√(d(x,p)d(y,p)), where the compared quantities are independently defined metrics, not fitted values. Theorems 5.1 and 5.2 deduce ball inclusions from the two-sided bounds of Theorems 4.1 and 4.2, which in turn follow from the metric definitions and the triangle inequality; no parameter is fitted and no prediction is recycled as an input. The convexity analysis of Section 6 is also non-circular: Theorem 6.1 computes the tangent slope of the boundary curve from the defining equation τ_0(1,y) = r, and the threshold r > log 3 emerges from the angular range over which the polar representation exists, not from an assumed conclusion. Theorem 6.3 relies on Lemma 6.2, which is stated without proof; this omitted proof is a genuine rigor gap, and the skeptical counterexample, if numerically correct, would invalidate Theorem 6.3 rather than reveal a circular step, because Lemma 6.2 is a sufficient calculus condition and not an assumption equivalent to convexity. Self-citations are present but not load-bearing: [35] is cited for prior study of the same metric (whose definition is restated in the paper), and [19] for the polar-coordinate slope technique (a standard calculus method restated in the paper); neither imports an unverified theorem as the central premise of an argument. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters, no invented entities, and no fitted constants. The paper relies only on standard facts (triangle inequality, AM-GM, Bernoulli) and the domain assumption that D is a punctured metric or Euclidean space. The Ptolemaic condition is implicit for the metric interpretation in Euclidean spaces.

assumptions (3)
  • standard math The metric space axioms and the standard Euclidean geometry of R^n.
    Used throughout: triangle inequality, AM-GM, Bernoulli inequality in (2.1), polar coordinates in Section 6.
  • domain assumption The domain D is a (possibly multiply) punctured metric or Euclidean space with D = X \ {p1,...,pk}.
    The metrics τ_p and τ̂_D are defined only away from the punctures; all theorems embed this assumption.
  • domain assumption The scale-invariant Cassinian metric is a metric, which requires the underlying space to be Ptolemaic; this holds for the Euclidean spaces used in Sections 5 and 6.
    The paper does not verify the Ptolemaic condition explicitly; for R^n it is standard, but general metric space statements only use the formula.

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Pith. "Pith review of On the average scale-invariant Cassinian metric." pith.science (2026). https://pith.science/paper/DOR7S232

@misc{pith2026250600810,
  author       = {Pith},
  title        = {Pith review of: On the average scale-invariant Cassinian metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DOR7S232}},
  note         = {Machine review of arXiv:2506.00810}
}
read the original abstract

We establish geometric relationships between the average scale-invariant Cassinian metric and other hyperbolic type metrics. In addition, we study the local convexity properties of the scale-invariant metric balls in Euclidean once punctured spaces.

Figures

Figures reproduced from arXiv: 2506.00810 by the authors.

Figure 1
Figure 1. The τp-metric balls in R 2 \ {0} centred at e1 = (1, 0) with radii r = log 3, log 5, log 7, and log 8.8. References [1] A. G. Aksoy, Z. Ibragimov, and W. Whiting, Averaging one-point hyperbolic-type metrics Proc. Amer. Math. Soc. 146 (2018), 5205–5218. [2] S¸. Altınkaya, M. Fujimura, and M. Vuorinen, Hilbert metric and quasiconformal mappings, arXiv2502.18109. [3] G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorin… view at source ↗

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Works this paper leans on

36 extracted references · 33 canonical work pages

  1. [1]

    A. G. Aksoy, Z. Ibragimov, and W. Whiting, Averaging one-point hyperbolic-type metrics Proc. Amer. Math. Soc. 146 (2018), 5205–5218

    work page 2018

  2. [2]

    Altınkaya, M

    S ¸. Altınkaya, M. Fujimura, and M. Vuorinen , Hilbert metric and quasiconformal mappings, arXiv2502.18109

    arXiv

  3. [3]

    G. D. Anderson, M. K. V amanamurthy, and M. K. Vuorinen, Conformal Invariants, In- equalities, and Quasiconformal Maps , John Wiley & Sons, Inc., 1997

    work page 1997

  4. [4]

    A. F. Beardon , Geometry of discrete groups , Springer-Verlag, New York, 1995

    work page 1995

  5. [5]

    A. F. Beardon , The Apollonian metric of a domain in Rn. In: P. Duren, J. Heinonen, B. Osgood, and B. Palka (Eds.) Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 91–108. Springer- Verlag, New York, 1998

    work page 1995

  6. [6]

    Borovikova and Z

    M. Borovikova and Z. Ibragimov, Convex bodies of constant width and the Apollonian metric, Bull. Malays. Math. Sci. Soc. 31 (2) (2008), 117–128

    work page 2008

  7. [7]

    J. Chen, P. Hariri, R. Kl ´ en, and M. Vuorinen , Lipschitz conditions, Triangular ratio metric and Quasiconformal maps, Ann. Acad. Sci. Fenn. Math. 40 (2015), 683–709

    work page 2015

  8. [8]

    Chen and X

    X. Chen and X. Zhang, A New One-Point Metric on Ptolemaic Spaces. Results Math. 79 (2024), no. 4, Paper No. 176. ON THE A VERAGE SCALE-INV ARIANT CASSINIAN METRIC 19

    work page 2024

Show all 36 references
  1. [9]

    Fujimura, R

    M. Fujimura, R. Kargar, and M. Vuorinen , Formulas for the visual angle metric, J. Geom. Anal. (to appear), arXiv:2304.04485, 14pp

    arXiv

  2. [10]

    Fujimura, O

    M. Fujimura, O. Rainio, and M. Vuorinen , The visual angle metric of the half plane, arXiv:2404.08942. 33pp

    arXiv

  3. [11]

    Fujimura and M

    M. Fujimura and M. Vuorinen , Sharp distortion results for quasiregular mappings of the half plane, Complex Var. Elliptic Equ. DOI - 10.1080/17476933.2025.2500350 arXiv:2411.16966

    2025

  4. [12]

    F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74

    1979

  5. [13]

    F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199

    1976

  6. [14]

    Hariri, R

    P. Hariri, R. Kl ´ en, and M. Vuorinen , Conformally invariant metrics and quasiconformal map- pings, Springer, 2020

    2020

  7. [15]

    Hariri, M

    P. Hariri, M. Vuorinen, and X.H. Zhang , Inequalities and bilipschitz conditions for the trian- gular ratio metric, Rocky Mountain J. Math. 47 (4) (2017), 1121–1148

    2017

  8. [16]

    H ¨ast¨o, A new weighted metric: the relative metric

    P. H ¨ast¨o, A new weighted metric: the relative metric. I, J. Math. Anal. Appl. 274 (1) (2002), 38–58

    2002

  9. [17]

    H ¨ast¨o, The Apollonian metric: uniformity and quasiconvexity, Ann

    P. H ¨ast¨o, The Apollonian metric: uniformity and quasiconvexity, Ann. Acad. Sci. Fenn. Math. 28 (2) (2003), 385–414

    2003

  10. [18]

    H ¨ast¨o and H

    P. H ¨ast¨o and H. Lind ´ en, Isometries of the half-apollonian metric, Complex Var. Theory Appl. 49 (2004), 405–415

    2004

  11. [19]

    Hokuni, R

    S. Hokuni, R. Kl ´ en, Y. Li, and M. Vuorinen , Balls in the triangular ratio metric. Complex analysis and dynamical systems VI. Part 2, 105–123, Contemp. Math. 667, Israel Math. Conf. Proc. , Amer. Math. Soc., Providence, RI, 2016

    2016

  12. [20]

    Ibragimov, The Cassinian metric of a domain in ¯Rn, Uzbek

    Z. Ibragimov, The Cassinian metric of a domain in ¯Rn, Uzbek. Mat. Zh. (2009), 1, 53–67

    2009

  13. [21]

    Ibragimov, Hyperbolizing metric spaces, Proc

    Z. Ibragimov, Hyperbolizing metric spaces, Proc. Amer. Math. Soc. 139 (12) (2011), 4401–4407

    2011

  14. [22]

    Ibragimov, A scale-invariant Cassinian metric, J

    Z. Ibragimov, A scale-invariant Cassinian metric, J. Anal. 24 (1) (2016), 111–129

    2016

  15. [23]

    Ibragimov, M

    Z. Ibragimov, M. R. Mohapatra, S. K. Sahoo, and X.-H. Zhang, Geometry of the Cassinian metric and its inner metric, Bull. Malays. Math. Sci. Soc. 40 (1) (2017), 361–372

    2017

  16. [24]

    Kl ´ en, M

    R. Kl ´ en, M. R. Mohapatra, and S. K. Sahoo , Geometric properties of the Cassinian metric, Math. Nachr. 290 (2017), 1531–1543

    2017

  17. [25]

    B. Maji, P. Naskar, and S. K. Sahoo , A naive generalization of the hyperbolic metric, arXiv:2505.10964

    arXiv

  18. [26]

    Mocanu , Generalizations of four hyperbolic-type metrics and Gromov hype rbolicity

    M. Mocanu , Generalizations of four hyperbolic-type metrics and Gromov hype rbolicity. J. Math. Anal. Appl. (2025) https://doi.org/10.1016/j.jmaa.2025.129729

    2025

  19. [27]

    Rainio, Inequalities for the Geometric-Mean Distance Metric

    O. Rainio, Inequalities for the Geometric-Mean Distance Metric. Ukr. Math. J. 76, 1718–1730, (2025). https://doi.org/10.1007/s11253-025-02418-y

    2025 doi

  20. [28]

    Rainio and M

    O. Rainio and M. Vuorinen , Hilbert metric in the unit ball, Studia Sci. Math. Hungar. 60 (2023), 175-191, arXiv:2303.03753

    2023 arXiv

  21. [29]

    Rasila and J

    A. Rasila and J. Talponen , Convexity properties of quasihyperbolic balls on Banach spaces. Ann. Acad. Sci. Fenn. Math. 37 (2012), 215–228

    2012

  22. [30]

    Saglam, K

    I. Saglam, K. Ohshika, and A. Papadopoulos, On families of Finsler metrics. arXiv.2506.03749

    arXiv

  23. [31]

    Seittenranta, M¨ obius-invariant metrics,Math

    P. Seittenranta, M¨ obius-invariant metrics,Math. Proc. Cambridge Philos. Soc. 125 (1999), 511– 533

    1999

  24. [32]

    Song and G

    X. Song and G. W ang, A New Metric Associated with the Domain Boundary. Comput. Methods Funct. Theory (2024). https://doi.org/10.1007/s40315-024-00545-4

    2024 doi

  25. [33]

    Vuorinen, Conformal invariants and quasiregular mappings, J

    M. Vuorinen, Conformal invariants and quasiregular mappings, J. Analyse Math. 45 (1985), 69– 115. 20 M. MOHAPATRA, A. RASILA, AND M. VUORINEN

    1985

  26. [34]

    Vuorinen, Conformal geometry and quasiregular mappings , Lecture notes in mathematics 1319, Springer-Verlag, Berlin, 1988

    M. Vuorinen, Conformal geometry and quasiregular mappings , Lecture notes in mathematics 1319, Springer-Verlag, Berlin, 1988

    1988

  27. [35]

    W ang, X

    G. W ang, X. Xu, and M. Vuorinen , Remarks on the Scale-Invariant Cassinian Metric. Bull. Malays. Math. Sci. Soc. 44 (2021), 1559–1577

    2021

  28. [36]

    Q. Zhou, Z. Zheng, S. Ponnusamy, and T. Guan, Dovgoshey–Hariri–Vuorinen’s metric and Gromov hyperbolicity, Arch. Math. 123, 319–327 (2024), https://doi.org/10.1007/s00013-024-02021 - w. Department of Mathematics, Kalinga Institute of Industria l Technology (KIIT), Pa- tia, Bhu...

    2024 doi

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