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arxiv: 2606.02882 · v1 · pith:OJU4E7DLnew · submitted 2026-06-01 · 🧮 math.MG

Optimal stability of P\'al's isominwidth inequality for ball convex bodies in planes of constant curvature

Ferenc Fodor , \'Ad\'am Sagmeister This is my paper

Pith reviewed 2026-06-28 11:18 UTC · model grok-4.3

classification 🧮 math.MG
keywords isominwidth inequalityball-convex bodiesconstant curvature planesstabilityHausdorff distancesymmetric difference metricPál's inequality
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The pith

The isominwidth inequality for ball-convex bodies admits optimal stability with respect to Hausdorff distance and symmetric difference in all three constant-curvature planes.

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

The paper proves optimal stability versions of Pál's isominwidth inequality for ball-convex bodies. These versions quantify how close a body must be to the minimizer when its area is close to the lower bound given by the inequality. The results apply to the Euclidean, spherical, and hyperbolic planes using both the Hausdorff distance and the symmetric difference metric. This provides a robust version of the inequality that measures deviation from equality cases across different geometries.

Core claim

We prove optimal stability versions of the isominwidth inequality for r-ball convex bodies with respect to the Hausdorff distance and the symmetric difference metric in the Euclidean, spherical, and hyperbolic planes.

What carries the argument

r-ball convex bodies, sets whose intersection with every disk of radius r is convex, as the domain on which the generalized isominwidth inequality is stated and stabilized.

Load-bearing premise

The base isominwidth inequalities and their ball-convex analogs hold in the three constant-curvature planes as shown in the cited prior works.

What would settle it

A sequence of r-ball convex bodies whose area deficit tends to zero while the Hausdorff or symmetric difference distance to the nearest extremal body remains bounded below by a positive constant would falsify the optimal stability claim.

Figures

Figures reproduced from arXiv: 2606.02882 by \'Ad\'am Sagmeister, Ferenc Fodor.

Figure 1
Figure 1. Figure 1: The r-disk hexagon Qw,r,ϱ with thick boundary and the r-disk triangle Tw,r with dotted boundary 3. Stability with respect to the Hausdorff metric In this section, we verify our main result, Theorem 1.2. We set η = ϱ(K) − ϱ0 and ε = min  3 4 , area  B  p, w 2  − area(Tw,r)  . To simplify the notation, we denote the inradius of K by ϱ. First, we prove that η can be bounded from above by a constant time… view at source ↗
Figure 2
Figure 2. Figure 2: The r-disk triangle Tε First, we notice that Tε has minimal width w. Indeed, the minimal width of Tε is at least w, since Tε ⊃ Tw,r. On the other hand, w(Tε, ℓ) = w where ℓ is the supporting line at the midpoint m1 of the r-arc between v2 and v3 (see [FRS26, Lemma 2.2]). Clearly, [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
read the original abstract

P\'al's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by B\"or\"oczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean P\'al inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting P\'al's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

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

1 major / 1 minor

Summary. The manuscript proves optimal stability versions of the r-ball-convex isominwidth inequality (connecting Pál's theorem to the Blaschke-Lebesgue inequality) with respect to the Hausdorff distance and the symmetric difference metric, for ball-convex bodies in the Euclidean, spherical, and hyperbolic planes.

Significance. If the derivations hold, the results supply the first optimal stability statements for the ball-convex analogs in all three constant-curvature geometries, extending the Euclidean stability work of Lucardesi-Zucco while handling the non-existence of minimizers in hyperbolic space via the ball-convex restriction.

major comments (1)
  1. The stability constants and optimality claims rest entirely on the base isominwidth inequalities and equality-case characterizations already proved in the three cited preprints (arXiv:2411.11462, arXiv:2502.04427, arXiv:2602.19300). The manuscript must contain an explicit statement (e.g., in the introduction or a preliminary section) confirming that no additional curvature or convexity restrictions from those works affect the stability estimates derived here.
minor comments (1)
  1. Update preprint citations if any of the referenced works have appeared in print since submission.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We address the single major comment below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

  1. Referee: The stability constants and optimality claims rest entirely on the base isominwidth inequalities and equality-case characterizations already proved in the three cited preprints (arXiv:2411.11462, arXiv:2502.04427, arXiv:2602.19300). The manuscript must contain an explicit statement (e.g., in the introduction or a preliminary section) confirming that no additional curvature or convexity restrictions from those works affect the stability estimates derived here.

    Authors: We agree that an explicit statement will improve clarity regarding the dependence on the base results. In the revised version we will insert a short paragraph (likely in Section 1) confirming that the stability estimates are obtained directly from the equality cases and isominwidth inequalities established in the three cited preprints, using precisely the same class of r-ball-convex bodies in the Euclidean, spherical, and hyperbolic planes, without imposing any additional curvature or convexity restrictions beyond those already present in the base works. revision: yes

Circularity Check

0 steps flagged

No circularity; stability derivations are independent of base inequalities

full rationale

The paper cites three prior works (including one with author overlap) solely to invoke the established isominwidth inequality and its ball-convex analogs as assumptions, then derives new optimal stability results with respect to Hausdorff and symmetric-difference metrics. No quoted step reduces a stability claim to a fitted parameter, self-definition, or self-citation chain by construction; the stability proofs are presented as separate arguments that take the base inequalities as given external input. This is the standard non-circular pattern for extension papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence-and-stability proof in differential geometry; it introduces no new free parameters, no invented entities, and relies only on the standard axioms of the three model geometries plus the definitions of width and ball convexity already present in the cited literature.

axioms (1)
  • standard math Standard axioms and metric properties of the Euclidean, spherical, and hyperbolic planes
    Invoked throughout to define minimum width, ball convexity, and the two distance measures.

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Works this paper leans on

61 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    Lucardesi, Ilaria and Zucco, Davide , TITLE =. J. Geom. Anal. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s12220-025-01931-7 , URL =

    work page doi:10.1007/s12220-025-01931-7 2025

  2. [2]

    , title =

    Glasauer, S. , title =

  3. [3]

    , title =

    Heil, E. , title =. Preprint, Fachbereich Mathematik der TH Darmstadt (453) , year =

  4. [4]

    2007 , title =

    Walter, Rolf , url =. 2007 , title =

    2007

  5. [5]

    Lassak, Marek , TITLE =. Beitr. Algebra Geom. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s13366-019-00475-6 , URL =

    work page doi:10.1007/s13366-019-00475-6 2020

  6. [6]

    Ara\'ujo, Paulo Ventura , TITLE =. Enseign. Math. (2) , FJOURNAL =. 1996 , NUMBER =

    1996

  7. [7]

    Ara\'ujo, Paulo Ventura , TITLE =. Geom. Dedicata , FJOURNAL =. 1997 , NUMBER =. doi:10.1023/A:1004920201363 , URL =

    work page doi:10.1023/a:1004920201363 1997

  8. [8]

    Bezdek, K\'aroly and Blekherman, Grigoriy , TITLE =. Period. Math. Hungar. , FJOURNAL =. 2000 , NUMBER =. doi:10.1023/A:1004878520642 , URL =

    work page doi:10.1023/a:1004878520642 2000

  9. [9]

    Bezdek, K\'aroly , TITLE =. J. Geom. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00022-022-00663-1 , URL =

    work page doi:10.1007/s00022-022-00663-1 2023

  10. [10]

    Blaschke, Wilhelm , TITLE =. Math. Ann. , FJOURNAL =. 1915 , NUMBER =. doi:10.1007/BF01458221 , URL =

    work page doi:10.1007/bf01458221 1915

  11. [11]

    and Cs\'epai, Andr\'as and Sagmeister,

    B\"or\"oczky, K\'aroly J. and Cs\'epai, Andr\'as and Sagmeister,. Hyperbolic width functions and characterizations of bodies of constant width in the hyperbolic space , JOURNAL =. 2024 , NUMBER =. doi:10.1007/s00022-024-00714-9 , URL =

    work page doi:10.1007/s00022-024-00714-9 2024

  12. [12]

    B\"or\"oczky, K. J. and Sagmeister,. The isodiametric problem on the sphere and in the hyperbolic space , JOURNAL =. 2020 , NUMBER =. doi:10.1007/s10474-019-00982-x , URL =

    work page doi:10.1007/s10474-019-00982-x 2020

  13. [13]

    and Sagmeister,

    B\"or\"oczky, K\'aroly J. and Sagmeister,. Convex bodies of constant width in spaces of constant curvature and the extremal area of. Studia Sci. Math. Hungar. , FJOURNAL =. 2022 , NUMBER =

    2022

  14. [14]

    and Sagmeister,

    B\"or\"oczky, K\'aroly J. and Sagmeister,. Stability of the isodiametric problem on the sphere and in the hyperbolic space , JOURNAL =. 2023 , PAGES =. doi:10.1016/j.aam.2022.102480 , URL =

    work page doi:10.1016/j.aam.2022.102480 2023

  15. [15]

    Dekster, B. V. , TITLE =. Acta Math. Hungar. , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01874493 , URL =

    work page doi:10.1007/bf01874493 1995

  16. [16]

    Dekster, B. V. , TITLE =. Acta Math. Hungar. , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01874495 , URL =

    work page doi:10.1007/bf01874495 1995

  17. [17]

    , TITLE =

    Fillmore, Jay P. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1970 , PAGES =. doi:10.2307/2037306 , URL =

    work page doi:10.2307/2037306 1970

  18. [18]

    and Revent\'os, A

    Gallego, E. and Revent\'os, A. and Solanes, G. and Teufel, E. , TITLE =. Manuscripta Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00229-008-0171-1 , URL =

    work page doi:10.1007/s00229-008-0171-1 2008

  19. [19]

    Discrete Comput

    Gonz\'alez Merino, Bernardo and Jahn, Thomas and Polyanskii, Alexandr and Wachsmuth, Gerd , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00454-018-9982-3 , URL =

    work page doi:10.1007/s00454-018-9982-3 2018

  20. [20]

    , TITLE =

    Groemer, H. , TITLE =. Geom. Dedicata , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF00149581 , URL =

    work page doi:10.1007/bf00149581 1986

  21. [21]

    , TITLE =

    Groemer, H. , TITLE =. Beitr\"age Algebra Geom. , FJOURNAL =. 2000 , NUMBER =

    2000

  22. [22]

    , TITLE =

    Gruber, Peter M. , TITLE =. 2007 , PAGES =

    2007

  23. [23]

    Han, Huhe and Wu, Denghui , TITLE =. Aequat. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00010-020-00740-3 , URL =

    work page doi:10.1007/s00010-020-00740-3 2021

  24. [24]

    , TITLE =

    Horv\'ath, \'Akos G. , TITLE =. J. Geom. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00022-021-00613-3 , URL =

    work page doi:10.1007/s00022-021-00613-3 2021

  25. [25]

    , TITLE =

    Jer\'onimo-Castro, Jes\'us and Jimenez-Lopez, Francisco G. , TITLE =. Bull. Korean Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.4134/BKMS.b160709 , URL =

    work page doi:10.4134/bkms.b160709 2017

  26. [26]

    and Milman, D

    Krein, M. and Milman, D. , TITLE =. Studia Math. , FJOURNAL =. 1940 , PAGES =. doi:10.4064/sm-9-1-133-138 , URL =

    work page doi:10.4064/sm-9-1-133-138 1940

  27. [27]

    Beitr\"age Algebra Geom

    Lassak, Marek , TITLE =. Beitr\"age Algebra Geom. , FJOURNAL =. 2006 , NUMBER =

    2006

  28. [28]

    Aequationes Math

    Lassak, Marek , TITLE =. Aequationes Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00010-013-0237-3 , URL =

    work page doi:10.1007/s00010-013-0237-3 2015

  29. [29]

    Surveys in Geometry

    Lassak, Marek , TITLE =. Surveys in Geometry. 2022 , ISBN =. doi:10.1007/978-3-030-86695-2\_2 , URL =

    work page doi:10.1007/978-3-030-86695-2 2022

  30. [30]

    Results Math

    Lassak, Marek , TITLE =. Results Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00025-023-02102-2 , URL =

    work page doi:10.1007/s00025-023-02102-2 2024

  31. [31]

    and Martini, H

    Lassak, M. and Martini, H. , TITLE =. Acta Math. Hungar. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s10474-005-0002-8 , URL =

    work page doi:10.1007/s10474-005-0002-8 2005

  32. [32]

    Lassak, Marek and Martini, Horst , TITLE =. Expo. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.exmath.2011.01.006 , URL =

    work page doi:10.1016/j.exmath.2011.01.006 2011

  33. [33]

    Results Math

    Lassak, Marek and Martini, Horst , TITLE =. Results Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00025-014-0384-4 , URL =

    work page doi:10.1007/s00025-014-0384-4 2014

  34. [34]

    Aequationes Math

    Lassak, Marek and Musielak, Micha , TITLE =. Aequationes Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00010-018-0558-3 , URL =

    work page doi:10.1007/s00010-018-0558-3 2018

  35. [35]

    Lassak, Marek and Musielak, Micha , TITLE =. Bull. Pol. Acad. Sci. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.4064/ba8088-1-2018 , URL =

    work page doi:10.4064/ba8088-1-2018 2018

  36. [36]

    , title =

    Lebesgue, H. , title =. Bull. Soc. Math. France , year =

  37. [37]

    , TITLE =

    Leichtweiss, K. , TITLE =. Abh. Math. Sem. Univ. Hamburg , FJOURNAL =. 2005 , PAGES =. doi:10.1007/BF02942046 , URL =

    work page doi:10.1007/bf02942046 2005

  38. [38]

    and Wenzel, W

    Martini, H. and Wenzel, W. , TITLE =. Appl. Math. Lett. , FJOURNAL =. 2002 , NUMBER =. doi:10.1016/S0893-9659(02)00057-5 , URL =

    work page doi:10.1016/s0893-9659(02)00057-5 2002

  39. [39]

    , TITLE =

    Martini, Horst and Swanepoel, Konrad J. , TITLE =. Publ. Math. Debrecen , FJOURNAL =. 2004 , NUMBER =. doi:10.5486/pmd.2004.2904 , URL =

    work page doi:10.5486/pmd.2004.2904 2004

  40. [40]

    P\'al, Julius , TITLE =. Math. Ann. , FJOURNAL =. 1921 , NUMBER =. doi:10.1007/BF01458387 , URL =

    work page doi:10.1007/bf01458387 1921

  41. [41]

    Santal\'o, L. A. , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1945 , PAGES =. doi:10.1090/S0002-9904-1945-08366-9 , URL =

    work page doi:10.1090/s0002-9904-1945-08366-9 1945

  42. [42]

    Santal\'o, L. A. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1968 , PAGES =. doi:10.2307/2035535 , URL =

    work page doi:10.2307/2035535 1968

  43. [43]

    , TITLE =

    Santal\'o, Luis A. , TITLE =. 2004 , PAGES =. doi:10.1017/CBO9780511617331 , URL =

    work page doi:10.1017/cbo9780511617331 2004

  44. [44]

    Schmidt, Erhard , TITLE =. Math. Nachr. , FJOURNAL =. 1948 , PAGES =. doi:10.1002/mana.19480010202 , URL =

    work page doi:10.1002/mana.19480010202 1948

  45. [45]

    Schmidt, Erhard , TITLE =. Math. Nachr. , FJOURNAL =. 1949 , PAGES =. doi:10.1002/mana.19490020308 , URL =

    work page doi:10.1002/mana.19490020308 1949

  46. [46]

    1993 , PAGES =

    Schneider, Rolf , TITLE =. 1993 , PAGES =. doi:10.1017/CBO9780511526282 , URL =

    work page doi:10.1017/cbo9780511526282 1993

  47. [47]

    2008 , PAGES =

    Schneider, Rolf and Weil, Wolfgang , TITLE =. 2008 , PAGES =. doi:10.1007/978-3-540-78859-1 , URL =

    work page doi:10.1007/978-3-540-78859-1 2008

  48. [48]

    and Freyer, Ansgar and Sagmeister, \'Ad\'am , TITLE =

    B\"or\"oczky, K\'aroly J. and Freyer, Ansgar and Sagmeister, \'Ad\'am , TITLE =. J. Geom. Anal. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s12220-025-02258-z , URL =

    work page doi:10.1007/s12220-025-02258-z 2026

  49. [49]

    Freyer and ´A

    Freyer, Ansgar and Sagmeister, \'Ad\'am , TITLE =. Mathematika , FJOURNAL =. 2026 , NUMBER =. doi:10.1112/mtk.70069 , URL =

    work page doi:10.1112/mtk.70069 2026

  50. [50]

    2023 , school=

    Geometric inequalities in spaces of constant curvature , author=. 2023 , school=

    2023

  51. [51]

    2018 , school=

    On area and volume in spherical and hyperbolic geometry , author=. 2018 , school=

    2018

  52. [52]

    On a generalization of the Blaschke-Lebesgue theorem for disk-polygons

    Bezdek, M. On a generalization of the. arXiv preprint arXiv:0903.5361 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  53. [53]

    Ball and spindle convexity with respect to a convex body , fjournal =

    L. Ball and spindle convexity with respect to a convex body , fjournal =. Aequationes Math. , issn =. 2013 , language =. doi:10.1007/s00010-012-0160-z , keywords =

    work page doi:10.1007/s00010-012-0160-z 2013

  54. [54]

    Selected topics from the theory of intersections of balls , year =

    Bezdek, K. Selected topics from the theory of intersections of balls , year =

  55. [55]

    Discrete Comput

    Bezdek, K\'aroly and L\'angi, Zsolt and Nasz\'odi, M\'arton and Papez, Peter , title =. Discrete Comput. Geom. , issn =. 2007 , language =. doi:10.1007/s00454-007-1334-7 , keywords =

    work page doi:10.1007/s00454-007-1334-7 2007

  56. [56]

    Fodor, Ferenc and Robock, Nathan and Sagmeister, \'Ad\'am , title =

  57. [57]

    2019 , PAGES =

    Martini, Horst and Montejano, Luis and Oliveros, D\'eborah , TITLE =. 2019 , PAGES =. doi:10.1007/978-3-030-03868-7 , URL =

    work page doi:10.1007/978-3-030-03868-7 2019

  58. [58]

    Drach, Kostiantyn and Tatarko, Kateryna , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2026 , NUMBER =. doi:10.1112/blms.70292 , URL =

    work page doi:10.1112/blms.70292 2026

  59. [59]

    Anciaux, Henri and Guilfoyle, Brendan , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9939-2010-10588-9 , URL =

    work page doi:10.1090/s0002-9939-2010-10588-9 2011

  60. [60]

    Michigan Math

    Anciaux, Henri and Georgiou, Nikos , TITLE =. Michigan Math. J. , FJOURNAL =. 2026 , NUMBER =. doi:10.1307/mmj/20236362 , URL =

    work page doi:10.1307/mmj/20236362 2026

  61. [61]

    Partial differential equations and applications , SERIES =

    Campi, Stefano and Colesanti, Andrea and Gronchi, Paolo , TITLE =. Partial differential equations and applications , SERIES =. 1996 , ISBN =

    1996