Pith. sign in

REVIEW 3 major objections 8 minor 17 references

Paul levy's Isoperimetric Problems on cyclic polygons

T0 review · 3 major / 8 minor · reviewed 2026-07-09 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Lévy's polygon ratio conjecture verified for curved, Macnab, and bounded-side n-gons

desk verdict Partial verifications of Lévy's polygon conjecture for special families, but monotonicity claims rest on Maple without analytical proofs read the letter →

arxiv 2607.07095 v1 pith:PZLP5ZDO submitted 2026-07-08 math.CA

Abd Raouf Chouikha (LAGA) This is my paper · ORCID
classification math.CA MSC 51M1051M2552A40
keywords isoperimetricinequalitycyclicpolygonLévyconjecturepseudo-areaBonnesenHeronformulaBrahmaguptaMacnab
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 a 1966 conjecture of Paul Lévy about a ratio φ_n = A_n / P_n comparing the true area A_n of a cyclic n-gon to a 'pseudo-area' P_n built from its side lengths via a generalized Heron–Brahmagupta product. Lévy conjectured that φ_n is always sandwiched between the corresponding ratio for a regular n-gon (φ_n^0) and 1. The author verifies this conjecture for several families: curved (n+1)-gons and (n+2)-gons inscribed in circular arcs, generalized Macnab polygons with k sides of one length and m of another, and n-gons whose sides satisfy a boundedness condition a_i < k·L_n/n with 1 < k < 3/2. In each case the ratio φ_n is shown to be strictly decreasing in n, interpolating between 1 (for n=3,4) and e^π (as n→∞), matching the regular-polygon bounds.

What carries the argument

The pseudo-area P_n = (L_n^2/4) · ∏(1 - 2a_i/L_n)^{1/2}, generalizing Heron's formula (triangles) and Brahmagupta's formula (quadrilaterals); the ratio φ_n = A_n/P_n compared against the regular n-gon baseline φ_n^0 = 1/(n tan(π/n)) · (1 - 2/n)^{n/2}; Bonnesen-type polygonal isoperimetric inequalities L_n^2 - 4n tan(π/n) A_n ≥ (L_n - 2nR sin(π/n))^2 providing area bounds via circumradius R; and computer-algebra-assisted (Maple) verification of monotonicity for finite n.

What would settle it

A single cyclic n-gon (for some n ≥ 5) with side lengths violating the boundedness condition a_i < k·L_n/n, for which φ_n ≤ φ_n^0 or φ_n ≥ 1, would falsify Lévy's conjecture. More immediately, a counterexample to monotonicity—some n-gon family where φ_n increases between consecutive n values—would break the interpolation argument from φ_3 = 1 to the e^π limit.

Watch

Extended reading notes

Core claim

For each special class of cyclic polygon examined, the ratio φ_n = A_n / P_n is strictly decreasing as a function of n, with φ_3 = φ_4 = 1 and lim_{n→∞} φ_n = e^π, which places it in the interval (φ_n^0, 1) required by Lévy's conjecture. The key mechanism is that the pseudo-area P_n, defined as (L_n^2/4) times the product of (1 - 2a_i/L_n)^{1/2} over all sides, captures enough geometric information about side-length distribution that the true area A_n always exceeds the regular-polygon-scaled version but falls short of P_n itself. For bounded-side polygons, the author derives explicit double-sided inequalities on φ_n using Bonnesen-type isoperimetric deficits, showing that the side-length-bL

Load-bearing premise

The monotonicity of φ_n with respect to n for finite n is verified computationally (via Maple) rather than by analytical proof, so the strict-decreasing claim for each family rests on symbolic computation that the reader must trust without seeing a closed-form derivative argument.

Editorial extensions

If this is right

  • If Lévy's conjecture holds for all cyclic n-gons, then P_n provides a universal upper bound on the area of any cyclic polygon in terms of its side lengths alone, extending Heron's and Brahmagupta's formulas to arbitrary n.
  • The decreasing-from-1-to-e^π pattern of φ_n across all tested families suggests e^π may play a universal role as the asymptotic area-deficiency constant for cyclic polygons, analogous to how 4π appears in the smooth isoperimetric inequality.
  • The bounded-side condition a_i < k·L_n/n with k near 1 quantifies how close a polygon must be to regular for the conjecture to be provable by current Bonnesen-type methods, delineating the boundary between tractable and open cases.
  • The generalized Macnab polygon results (k sides of length a, m sides of length b) extend the conjecture verification to a two-parameter family, suggesting that the conjecture may be approachable through a density argument over side-length configurations.

Reading between the lines

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

  • The universal appearance of e^π as the n→∞ limit across structurally different polygon families hints that φ_n might satisfy a universal asymptotic expansion φ_n = e^π + c/n + O(1/n^2) with a universal leading constant c, which if proven would reduce the conjecture to a finite-n verification problem.
  • The reliance on Maple for monotonicity checks suggests that a clean analytic proof of monotonicity for finite n might require a convexity or log-concavity argument on φ_n viewed as a function of both n and the side-length parameters, which the paper does not attempt.
  • The bounded-side approach via Bonnesen inequalities could potentially be sharpened by using the improved inequality L_n^2 - 4n tan(π/n) A_n ≥ 2R tan(π/n)(L_n - 2nR sin(π/n)) to push the admissible range of k beyond 3/2, progressively expanding the class of polygons for which the conjecture is rigorously confirmed.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. The paper revisits a conjecture of Paul Lévy concerning the ratio φ_n = A_n / P_n for cyclic n-gons, where A_n is the area and P_n is the pseudo-area defined by a generalized Brahmagupta-type product. The conjecture states that φ_n^0 < φ_n < 1, where φ_n^0 is the ratio for the regular n-gon. The author verifies this conjecture for several special classes of polygons: curved n-gons approximating circular arcs, (n+1)-gons with n−1 equal sides, generalized Macnab polygons with two alternating side lengths, and polygons whose side lengths satisfy a boundedness condition (C_1). The approach combines explicit computation of areas and pseudo-areas with Bonnesen-type isoperimetric inequalities.

Significance. Lévy's conjecture is a long-standing open problem in the geometry of polygons, and partial verifications for non-trivial polygon classes contribute to the literature. The paper provides explicit formulas for φ_n in several structured families and attempts to connect these to Bonnesen-type inequalities. However, the reliance on computer algebra for monotonicity claims limits the analytical depth of the contribution.

major comments (3)
  1. Throughout Sections 2.1.1, 2.2.1, 2.2.2, and 2.3, key monotonicity and inequality claims for finite n are stated as verified by Maple (e.g., §2.1.1: 'By Maple it is verified that φ_{n+1} is strictly decreasing'; §2.3: 'By Maple we may prove φ_n is decreasing when n≥3'). No code, intermediate symbolic expressions, or analytical arguments are provided. These claims are load-bearing for the verification of Conjecture (L) in each family. Without reproducible derivations or rigorous proofs, these results cannot be independently verified from the text alone. The author should either provide the analytical monotonicity arguments or include sufficient computational detail (code, expressions, error bounds) to make the verification reproducible.
  2. The paper verifies Conjecture (L) only for highly constrained polygon classes: (n+1)-gons with n−1 equal sides, (k+m)-gons with two alternating side lengths, and polygons satisfying condition (C_1) with 1<k<3/2. As noted in §2.4, condition (C_1) explicitly restricts to polygons 'fairly close to the regular one.' None of the examples test polygons with many distinct side lengths or irregular angular distributions. The abstract and introduction should clarify that the conjecture is verified only for these specific families, not in general, to avoid any impression that the conjecture is settled.
  3. In §2.3, the limit lim_{n→∞} φ_n ≤ e^π is derived using the Bonnesen-type inequality (6) to bound the area. However, the subsequent claim that 'φ_n is decreasing when n≥3' is again delegated to Maple. The logical flow is unclear: if the area bound from (6) is used to establish the limit, is the same bound used for the monotonicity, or is monotonicity purely computational? The relationship between the analytical bound and the computational claim should be clarified.
minor comments (8)
  1. The title capitalization is inconsistent: 'Paul levy's Isoperimetric Problems on cyclic polygons' should be 'Paul Lévy's Isoperimetric Problems on Cyclic Polygons.'
  2. §1.2, Eq. (5): the condition is written as 'a_n < a_1 + a_2 + ... + a_{n−1}, ⇔ 2a_i/L_n < 1.' The indexing is confusing — the condition should hold for all i, not just a_n. The equivalence notation (⇔) is also non-standard here.
  3. §1.2: the reference to Lemma 1 cites [10], but Lemma 1 as stated appears to be a standard approximation result; the citation should be verified for accuracy.
  4. §2.1.1: the expression for φ_{n+1} is extremely lengthy and difficult to parse. Consider simplifying or providing intermediate steps to improve readability.
  5. §2.2.2: the limit lim_{n→∞} φ_{n+1} is stated as e^π, but the intermediate expression involving f(α) is convoluted. The simplification to e^π should be shown more explicitly.
  6. §2.4, Proposition 2.1: the condition (C_1) states 1<k<3/2, but later in Remarks 2.2 it is stated that 1<k<log_π<3/2. The notation 'log_π' is unclear — is this log(π)? This should be clarified.
  7. Several references have formatting issues (e.g., [16] Zeng and Dong lists 'Acta Math. Sinica, Vol 41' without clear volume/issue separation; [2] Chouikha has inconsistent capitalization in the title).
  8. The manuscript would benefit from a concluding section that explicitly summarizes which families have been verified and what remains open, rather than the brief §3 remark.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; self-citations are used for context but the paper's computations are direct and independent.

full rationale

The paper verifies Lévy's Conjecture (L) for several restricted families of cyclic polygons by direct computation of the ratio φ_n = A_n / P_n. The derivations in Sections 2.1–2.3 proceed by explicitly computing perimeter, area, and pseudo-area for each polygon family, then forming the quotient and checking monotonicity. The author cites prior work [2], [3] for intermediate results (e.g., Lemma 4-6, Proposition 4-1, Proposition 4-2), but these citations provide supporting lemmas rather than defining the target quantity in terms of itself. The Bonnesen-type inequalities (Eqs. 6–7) are cited from [16], [17] and provide independent upper bounds on area. Proposition 2.1 improves the author's own prior bound [2, Prop. 4-2] by tightening the constant, which is a genuine refinement, not a circular restatement. The key weakness of the paper is the heavy reliance on Maple for monotonicity verification (e.g., 'By Maple it is verified that φ_{n+1} is strictly decreasing'), which is a reproducibility and rigor concern, not a circularity concern. No step in the derivation chain reduces to its own inputs by construction. The self-citations are contextual and do not form a load-bearing chain that forces the conclusion. The paper is self-contained against external benchmarks for the specific families considered, and the central computations are independent of the cited prior work in the sense that they do not assume the conjecture to prove it. Score 1 reflects one minor self-citation that is not load-bearing for the main computational results.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper introduces one free parameter (k) for bounding side lengths. It relies on standard mathematical background (Wirtinger) and domain-specific inequalities (Bonnesen-type). The most notable ad-hoc reliance is on Maple computations treated as proof.

free parameters (1)
  • k
    A real number 1 < k < 3/2 introduced in condition (C1) to bound the side lengths of polygons, measuring deviation from regularity.
assumptions (3)
  • standard math Wirtinger inequality
    Used in the introduction to derive the classical isoperimetric inequality (Eq. 1).
  • domain assumption Bonnesen-type isoperimetric inequalities for polygons
    Inequalities (6) and (7) from Zhang [17] and Zeng-Dong [16] are used as starting points for deriving bounds in Section 2.3 and Proposition 2.3.
  • ad hoc to paper Maple-verified monotonicity
    The paper repeatedly states 'By Maple we may verify...' (Sections 2.1.1, 2.2.1, 2.3.2) to assert that certain functions are strictly decreasing, treating computational output as a mathematical axiom without analytical proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Paul levy's Isoperimetric Problems on cyclic polygons." pith.science (2026). https://pith.science/paper/PZLP5ZDO

@misc{pith2026260707095,
  author       = {Pith},
  title        = {Pith review of: Paul levy's Isoperimetric Problems on cyclic polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZLP5ZDO}},
  note         = {Machine review of arXiv:2607.07095}
}
read the original abstract

In this paper we are interested in isoperimetric inequalities for plane n-gons in relation with a old conjecture proposed by P. Levy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    BergerPolygons, polyhedra, polytopes, Geometry Revealed

    M. BergerPolygons, polyhedra, polytopes, Geometry Revealed. Springer, Berlin, Heidelberg, p.505-561, (2010). 10.1007/978-3-540-70997-8-8

    work page doi:10.1007/978-3-540-70997-8-8 2010

  2. [2]

    ChouikhaProbleme de P

    R. ChouikhaProbleme de P. Levy sur les polygones articules C. R. Math. Report, Acad of Sc. of Canada, vol 10, p. 175-180, 1988

    work page 1988

  3. [3]

    ChouikhaProblems on polygons and Bonnesen-type inequalities, Indag., Vol

    R. ChouikhaProblems on polygons and Bonnesen-type inequalities, Indag., Vol. 10, Issue 4, 1999, p. 495-506. 10.1016/S0019-3577(00)87902-1

    work page doi:10.1016/s0019-3577(00)87902-1 1999

  4. [4]

    Dulio, E

    P. Dulio, E. LaengGeneralization of Herons and Brahmaguptas equali- ties to any cyclic polygonAequat. Math. 95 (2021), 941952. doi.org/10.1007/s00010-020-00771-w

    work page doi:10.1007/s00010-020-00771-w 2021

  5. [5]

    H.T. Ku, M.C. KuAnalytic Isoperimetric Inequalities, Math. Ineq. Appl., Volume 3, Number 4 (2000), 459472. 19

    work page 2000

  6. [6]

    H.T. Ku, M.C. Ku, X.M. ZhangAnalytic and geometric isoperimetric inequalitiesJ. of Geometry, vol 53, p.100-121, 1995

    work page 1995

  7. [7]

    LevyLe probleme des isoperimetres et des polygones articules Bull

    P. LevyLe probleme des isoperimetres et des polygones articules Bull. Sc. Math., 2eme serie, 90, p.103-112, 1966

    work page 1966

  8. [8]

    D. S. MacnabCyclic polygons and related questions, Math. Gazette 65 (1981), 2228

    work page 1981

Show all 17 references
  1. [9]

    OssermanThe isoperimetric inequalities Bull

    R. OssermanThe isoperimetric inequalities Bull. Amer. Math. Soc., vol 84, p.1182-1238, 1978

    1978

  2. [10]

    OssermanBonnesen-style isoperimetric inequalities Amer

    R. OssermanBonnesen-style isoperimetric inequalities Amer. Math. Monthly, vol 1, p. 1-29, 1979

    1979

  3. [11]

    Pak,The area of cyclic polygons: Recent progress on Robbins conjec- tures, Adv

    I. Pak,The area of cyclic polygons: Recent progress on Robbins conjec- tures, Adv. in Appl. Math., Vol. 34, Issue 4, 2005, p. 690696

    2005

  4. [12]

    Pech,Computations of the Area and Radius of Cyclic Polygons Given by the Lengths of Sides, ADG2004 (Hong, H

    P. Pech,Computations of the Area and Radius of Cyclic Polygons Given by the Lengths of Sides, ADG2004 (Hong, H. and Wang, D., eds.), LNAI, 3763, Gainesville, Springer, 2006, 4458

    2006

  5. [13]

    Pinelis,Cyclic polygons with given edge lengths, J

    I. Pinelis,Cyclic polygons with given edge lengths, J. Geom. 00 (2005), P.1-16. DOI 10.1007/s00022-005-1752-8

    2005 doi

  6. [14]

    Robbins,Areas of polygons inscribed in a circleDiscrete Comput

    D.P. Robbins,Areas of polygons inscribed in a circleDiscrete Comput. Geom. 12(2), 223236 (1994)

    1994

  7. [15]

    SvrtanOn circumradius equations of cyclic polygons Proc

    D. SvrtanOn circumradius equations of cyclic polygons Proc. of the 4th Croatian Combin. Days Sept. 22 23, 2022

    2022

  8. [16]

    C. Zeng, X. DongOn Some Discrete Bonnesen-style Isoperimet- ric Inequalities, Acta Math. Sinica, Vol 41, p. 14471461, (2025) doi.org/10.1007/s10114-025-3281-8

    2025 doi

  9. [17]

    ZhangBonnesen-style inequalities and pseudo-perimeters for poly- gonsJ

    X.M. ZhangBonnesen-style inequalities and pseudo-perimeters for poly- gonsJ. of Geometry, vol 60, p.188-201, 1997. 20

    1997

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.