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arxiv: 1906.11965 · v1 · pith:LCGQMBC2new · submitted 2019-06-27 · 🧮 math.MG

Some inequalities for tetrahedra

Jin-ichi Itoh , Jo\"el Rouyer , Costin V\^ilcu This is my paper

Pith reviewed 2026-05-25 13:26 UTC · model grok-4.3

classification 🧮 math.MG
keywords tetrahedraintrinsic radiusextrinsic radiusdiametergeometric inequalitiesmetric geometrypolyhedra
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The pith

Tetrahedra satisfy inequalities between intrinsic and extrinsic radii and diameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves several inequalities that relate the intrinsic radii and diameters of tetrahedra to their extrinsic versions. Intrinsic quantities are defined using the intrinsic metric on the surface of the tetrahedron, whereas extrinsic ones use distances in the surrounding Euclidean space. If these inequalities are true, they would give precise bounds on how the surface geometry compares to the embedded geometry in three dimensions. This matters for understanding the possible configurations of tetrahedra and for applications in geometry where both surface and space distances are relevant.

Core claim

The authors establish inequalities involving the intrinsic and extrinsic radii and diameters for tetrahedra, showing how these quantities are related under the standard geometric definitions.

What carries the argument

Inequalities linking intrinsic and extrinsic radii and diameters of tetrahedra.

If this is right

  • Relations hold between the radii and diameters in both intrinsic and extrinsic senses for every tetrahedron.
  • The inequalities apply uniformly to all non-degenerate tetrahedra.
  • These results extend comparisons of radii and diameters to the tetrahedron setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of comparison between surface and space measures could apply to other convex polyhedra.
  • Numerical checks on sample tetrahedra would provide direct verification of the bounds.
  • The relations might inform distance computations in geometric modeling software.

Load-bearing premise

The tetrahedra under consideration are non-degenerate with well-defined positive intrinsic and extrinsic radii and diameters.

What would settle it

Identifying a non-degenerate tetrahedron where any of the proved inequalities between the radii or diameters fails would disprove the result.

Figures

Figures reproduced from arXiv: 1906.11965 by Costin V\^ilcu, Jin-ichi Itoh, Jo\"el Rouyer.

Figure 1
Figure 1. Figure 1: Unfoldings of T and T 0 . 8 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The development of T in the proof of Theorem 2. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

We prove inequalities involving intrinsic and extrinsic radii and diameters of tetrahedra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to prove a collection of inequalities relating intrinsic and extrinsic radii and diameters of tetrahedra, under standard non-degeneracy conditions for such geometric objects in Euclidean 3-space.

Significance. If the derivations are correct and the inequalities are new, the results would add to the body of metric inequalities for tetrahedra, potentially aiding comparisons between different size measures in discrete and metric geometry. The work is framed as a pure proof paper with no data fitting or ad-hoc parameters.

minor comments (1)
  1. The abstract is extremely brief and supplies no explicit statements of the claimed inequalities, proof techniques, or verification methods, which hinders immediate assessment of novelty and correctness from the front matter alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their time and for summarizing our manuscript on inequalities relating intrinsic and extrinsic radii and diameters of tetrahedra. The recommendation is listed as uncertain, yet no specific major comments or concerns are raised in the report. We stand by the correctness of the derivations under the stated non-degeneracy conditions and believe the inequalities are new contributions to metric geometry of tetrahedra.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript is a pure proof paper establishing inequalities for intrinsic/extrinsic radii and diameters of tetrahedra under standard non-degeneracy conditions. No equations, definitions, or claims in the abstract or described structure reduce any result to a fitted parameter, self-referential definition, or load-bearing self-citation. The central claims are derived from geometric axioms and metric properties rather than from the paper's own inputs or prior author work invoked as uniqueness theorems. This is the expected outcome for a self-contained mathematical proof in metric geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5518 in / 854 out tokens · 27776 ms · 2026-05-25T13:26:17.760469+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    P. K. Agarwal, B. Aronov, J. O’Rourke and C. A. Schevon, Star un- folding of a polytope with applications , SIAM J. Comput. 26 (1997), 1689-1713

    work page 1997

  2. [2]

    A. D. Alexandrov, Convex Polyhedra , Springer-Verlag, Berlin, 2005. Monographs in Mathematics. Translation of the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze, and A. B. Sossinsky

    work page 2005

  3. [3]

    Aronov and J

    B. Aronov and J. O’Rourke, Nonoverlap of the star unfolding , Discrete Comput. Geom. 8 (1992), 219-250

    work page 1992

  4. [4]

    G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width , in Convexity and its Applications , P. Gruber and J. Wills (Eds.), Birkh¨ auser, Bassel 1983, 49-96

    work page 1983

  5. [5]

    V. Dods, C. Traub, J. Yang, Geodesics on the regular tetrahedron and the cube, Discrete Math. 340 (2017), 3183-3196

    work page 2017

  6. [6]

    Leech, Some Properties of the Isosceles Tetrahedron, The Mathemat- ical Gazette 34 (1950), 269-271

    J. Leech, Some Properties of the Isosceles Tetrahedron, The Mathemat- ical Gazette 34 (1950), 269-271

    work page 1950

  7. [7]

    N. P. Makuha, A connection between the inner and the outer diameters of a general closed convex surface (in Russian), Ukrain. Geometr. Sb. Vyp. 2 (1966), 49-51 13

    work page 1966

  8. [8]

    O’Rourke and C

    J. O’Rourke and C. A. Schevon, Computing the geodesic diameter of a 3-polytope, Proc. 5th ACM Symp. Comput. Geom. (1989), 370-379

    work page 1989

  9. [9]

    Rouyer, Antipodes sur un t´ etra` edre r´ egulier, J

    J. Rouyer, Antipodes sur un t´ etra` edre r´ egulier, J. Geom. 77 (2003), 152- 170

    work page 2003

  10. [10]

    Rouyer, Steinhaus conditions for convex polyhedra , in: K

    J. Rouyer, Steinhaus conditions for convex polyhedra , in: K. Adiprasito et al. (Eds.), Convexity and Discrete Geometry Including Graph The- ory, Springer Proceedings in Mathematics & Statistics 148, Springer International Publishing 2016, 77–84

    work page 2016

  11. [11]

    Rouyer and C

    J. Rouyer and C. Vˆ ılcu,Sets of tetrahedra, defined by maxima of distance functions, An. S ¸tiint ¸. Univ. “Ovidius” Constant ¸a Ser. Mat.20 (2012), 197-212

    work page 2012

  12. [12]

    Sharir and A

    M. Sharir and A. Schorr, On shortest paths in polyhedral spaces , SIAM J. Comput. 15 (1986), 193-215

    work page 1986

  13. [13]

    Vˆ ılcu and T

    C. Vˆ ılcu and T. Zamfirescu, Symmetry and the farthest point mapping on convex surfaces, Adv. Geom. 6 (2006), 345-353

    work page 2006

  14. [14]

    Vˆ ılcu and T

    C. Vˆ ılcu and T. Zamfirescu,Multiple farthest points on Alexandrov sur- faces, Adv. Geom. 7 (2007), 83-100

    work page 2007

  15. [15]

    V. A. Zalgaller, An isoperimetric problem for tetrahedra , J. Math. Sci. 140 (2007), 511-527 Jin-ichi Itoh School of Education, Sugiyama Jogakuen University 17-3 Hoshigaoka-motomachi, Chikusa-ku, Nagoya, 464-8662 Japan j-itoh@sugiyama-u.ac.jp Jo¨ el Rouyer 16, rue Philippe, 68 200 H´ egenheim - France Joel.Rouyer@ymail.com Costin Vˆ ılcu Simion Stoilow In...

    work page 2007