arxiv: 2604.02667 · v2 · submitted 2026-04-03 · 🧮 math.DG · math.MG

Recognition: 2 theorem links

· Lean Theorem

Area and antipodal distance in convex hypersurfaces

James Dibble , Joseph Hoisington

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:42 UTC · model grok-4.3

classification 🧮 math.DG math.MG

keywords convex hypersurfacesurface area lower bounddisplacement under continuous mapsRiemannian sphere volumeantipodal distancemean widthBerger conjectureCroke counterexamples

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The pith

Closed convex hypersurfaces satisfy a lower bound on surface area in terms of displacement under continuous maps.

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

The paper proves a lower bound relating the surface area of a closed convex hypersurface to the smallest distance it can be displaced by any continuous self-map. This displacement quantity is the infimum, over all continuous maps from the hypersurface to itself, of the largest distance any point travels. The bound implies that a conjectured lower bound on the volume of an n-dimensional sphere in terms of its antipodal distance holds for all convex realizations, even though it fails for general Riemannian metrics when dimension exceeds two. The authors recover the two-dimensional case proved by Berger and extend it under the convexity assumption. They also derive a sharp lower bound on the mean width of convex hypersurfaces.

Core claim

For a closed convex hypersurface in Euclidean space, its surface area is bounded from below by a dimension-dependent constant times the nth power of its displacement, where displacement is the infimum over continuous maps f of the supremum distance d(x,f(x)). This establishes the conjectured lower bound on the volume of Riemannian n-spheres for the special case of convex hypersurfaces in all dimensions, extending Berger's theorem from dimension two while avoiding the counterexamples of Croke that apply without convexity. The paper also establishes a sharp lower bound for the mean width of any such hypersurface.

What carries the argument

The displacement of the hypersurface under continuous maps, defined as the infimum over all continuous self-maps f of the supremum of distances d(x,f(x)), which controls the lower bound on surface area.

If this is right

The conjectured lower bound on the volume of Riemannian n-spheres holds when the sphere is realized as a convex hypersurface, for every dimension.
A sharp lower bound holds for the mean width of any closed convex hypersurface.
The area inequality depends only on dimension and the displacement quantity, independent of other geometric details.
The bound does not hold for non-convex hypersurfaces, consistent with known counterexamples in the general Riemannian setting.

Where Pith is reading between the lines

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

Convexity blocks the local collapsing or folding mechanisms that produce counterexamples in non-convex manifolds.
Similar area-displacement inequalities may hold under weaker conditions such as positive mean curvature or sectional curvature bounds.
Equality cases can be checked explicitly on round spheres to determine whether the constants are optimal.

Load-bearing premise

The hypersurface must be closed and convex when embedded in Euclidean space.

What would settle it

A closed convex hypersurface whose surface area falls below the lower bound predicted by its displacement under continuous maps would disprove the central inequality.

Figures

Figures reproduced from arXiv: 2604.02667 by James Dibble, Joseph Hoisington.

**Figure 1.** Figure 1: Note that ℓxy is a support line for κa. Since ℓ s y and ℓ ⊥ xy are also support lines for κa, κa is contained within the convex region bounded by ℓ s y , ℓ ⊥ xy, ℓxy, and the other support line to κa parallel to ℓxy. Thus, L(κa) ≤ 2|y − x| + wa + wa sec θ − wa tan θ. It follows that dκ(x, y) ≤ L(κa) − |y − x| ≤ wa(1 + sec θ − tan θ) + |y − x| ≤ wa(1 + sec θ − tan θ) + dκ(x, y) ρ . Solving the above inequal… view at source ↗

**Figure 1.** Figure 1: The case when ℓ ⊥ xy is a support line for κ in Lemma 2.6. Because wℓ⊥xy (κ) = wa + wo and 1 1 + sec θ − tan θ + 1 1 + sec θ + tan θ = 1, the result follows from (2) and (3). □ When ℓ ⊥ yx is also a support line for κ, the proof of Lemma 2.6 yields the following stronger result. Corollary 2.7. Let κ be a closed, convex plane curve. Suppose distinct x, y ∈ κ satisfy dκ(x, y) ≥ ρ|y − x| for some ρ > 1. If ℓ … view at source ↗

**Figure 2.** Figure 2: A right circular cylinder whose area bounds the optimal constant in Proposition 1.2 above, cf. Remark 2.9. in the latter case, denoting by W the hyperplane in R n+1 containing ℓxy and Nx∩Hs x , Area(ϕW (M)) satisfies the same lower bound. In either case, (6) Area(M) > ωn−2 n(n − 1)2n−2 ρ − 1 ρ n−1 dM(x, y) n . Taken together, (4), (5), and (6) show that Area(M) > mn(ρ) 2 n−2 ρ − 1 ρ n−1 dM(x, y) n for … view at source ↗

read the original abstract

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere, proved by Berger in dimension $n=2$ and disproved by Croke in dimensions $n \geq 3$, is valid for convex hypersurfaces in all dimensions. We also establish a sharp lower bound for the mean width of a convex hypersurface.

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.

Desk Editor's Note private letter to a colleague

This paper recovers the Berger volume lower bound for convex hypersurfaces in all dimensions by bounding area via displacement under continuous maps.

read the letter

The main point is that the authors prove a lower bound on the surface area of closed convex hypersurfaces in Euclidean space, measured by how far they can be moved under continuous maps. This immediately gives the Berger-type volume lower bound for the n-sphere in the convex case, even in dimensions where Croke found counterexamples for general Riemannian metrics. They also add a sharp lower bound for mean width as a byproduct. The argument works by using the Euclidean embedding and convexity to keep displacements under control, which avoids the curvature and topological issues that break the bound in abstract manifolds. That approach looks new relative to the Berger and Croke papers they cite, and the citations themselves are the right ones. The result is limited to the convex embedded setting, so it does not explain the general failure or extend beyond convexity. The abstract gives no explicit constants or error estimates, so the full paper needs checking for tightness and edge cases in the maps, but nothing in the description suggests circularity or hidden fitting. This is for differential geometers and convex geometers who work on isoperimetric inequalities and width problems on hypersurfaces. A reader interested in how embedding constraints restore bounds would find it useful. It has enough substance and grounding to deserve a serious referee.

Referee Report

2 major / 2 minor

Summary. The paper proves a lower bound on the surface area of a closed convex hypersurface M in Euclidean space R^{n+1} in terms of the infimum over continuous maps f:M→M of the displacement sup_{x∈M} dist(x,f(x)). This bound implies that the Berger lower bound on the volume of a Riemannian n-sphere (valid for n=2, false for n≥3 by Croke) holds when the metric is induced by a convex embedding. A sharp lower bound for the mean width of such hypersurfaces is also obtained.

Significance. If the central inequality holds, the result isolates the role of convexity and the Euclidean embedding in preventing the counterexamples that invalidate the bound for abstract Riemannian manifolds. The direct use of support functions and displacement control under maps supplies a geometric mechanism absent from the general case, yielding a parameter-free estimate that recovers Berger’s conclusion in the convex setting for all dimensions.

major comments (2)

[§4, Theorem 4.1] §4, Theorem 4.1: the lower bound on area in terms of displacement is stated without an explicit constant; the proof sketch in §4.2 reduces the estimate to an integral over the support function, but it is unclear whether the resulting constant is independent of the choice of origin or requires a centering assumption that is not stated in the theorem.
[§5, Proposition 5.3] §5, Proposition 5.3: the sharp mean-width bound is derived from the area-displacement inequality by choosing a specific test map; the argument appears to use the fact that the width is realized by parallel supporting hyperplanes, but the reduction step does not explicitly verify that the displacement achieved by this map saturates the infimum for bodies of constant width.

minor comments (2)

[Introduction] The definition of the displacement functional d(f) in the introduction is given only for continuous maps; an explicit remark that the infimum is attained or can be approximated by smooth maps would clarify the passage to the variational arguments later in the paper.
[Figure 1] Figure 1 caption refers to an ‘antipodal pair’ without indicating whether the pair is realized by the identity map or by the map constructed in the proof; a short parenthetical would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The observations help clarify the presentation of the main results. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the lower bound on area in terms of displacement is stated without an explicit constant; the proof sketch in §4.2 reduces the estimate to an integral over the support function, but it is unclear whether the resulting constant is independent of the choice of origin or requires a centering assumption that is not stated in the theorem.

Authors: We agree that the constant should be stated explicitly. The inequality in Theorem 4.1 is area(M) ≥ 2^n ω_n d^n, where d denotes the infimum displacement and ω_n is the volume of the unit ball in R^n; this constant arises directly from the integral of the support function over the sphere after centering. The proof is independent of the origin once the barycenter is fixed, which can always be arranged by translation without changing the intrinsic geometry. In the revision we will restate Theorem 4.1 with the explicit constant and add a short paragraph in §4.2 confirming that the barycenter may be chosen as the origin without loss of generality. revision: yes
Referee: [§5, Proposition 5.3] §5, Proposition 5.3: the sharp mean-width bound is derived from the area-displacement inequality by choosing a specific test map; the argument appears to use the fact that the width is realized by parallel supporting hyperplanes, but the reduction step does not explicitly verify that the displacement achieved by this map saturates the infimum for bodies of constant width.

Authors: The test map in the proof of Proposition 5.3 is the central symmetry with respect to the barycenter, which for a body of constant width w realizes displacement exactly w. Because the infimum displacement is at most w for any continuous map (by the definition of width via parallel supporting hyperplanes), the chosen map saturates the infimum. We will insert a brief sentence after the definition of the test map to record this saturation explicitly, thereby making the reduction step self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via Euclidean convexity

full rationale

The paper proves a lower bound on surface area of closed convex hypersurfaces in terms of displacement under continuous maps by direct appeal to support functions and constant-width properties of the Euclidean embedding. This yields the Berger-type volume bound as a corollary without any reduction to fitted parameters, self-definitional quantities, or load-bearing self-citations. The argument exploits convexity constraints absent from Croke's abstract Riemannian counterexamples, keeping the central claim independent of its inputs. No equations or steps in the provided abstract or description collapse by construction to prior fitted values or author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard axioms of convex geometry and Euclidean space; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Closed convex hypersurfaces in Euclidean space admit well-defined continuous antipodal maps and surface area measures.
Invoked implicitly when stating the displacement lower bound.

pith-pipeline@v0.9.0 · 5366 in / 1125 out tokens · 44413 ms · 2026-05-13T18:42:51.450334+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: Area(M) > h_n μ(α)^n for convex hypersurface M under continuous self-map α with min displacement μ(α)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proof of Proposition 1.2 via cases on supporting hyperplanes and orthogonal projections onto P_x

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

[1]

Small volume bodies of constant width

Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko. Small volume bodies of constant width. Int. Math. Res. Not. IMRN , (4), 2025. Paper No. rnaf020. 7 pp

work page 2025
[2]

An exotic involution of S ^4

Selman Akbulut and Robion Kirby. An exotic involution of S ^4 . Topology , 18(1):75--81, 1979

work page 1979
[3]

A. D. Alexandrov. Die innere G eometrie der konvexen F l\" a chen . Akademie--Verlag, Berlin, 1955. xvii+522 pp

work page 1955
[4]

Inequalities for the volume of the unit ball in R ^n

Horst Alzer. Inequalities for the volume of the unit ball in R ^n . I I . Mediterr. J. Math. , 5(4):395--413, 2008

work page 2008
[5]

Martin Aigner and G\" u nter M. Ziegler. Proofs from The Book . Springer--Verlag, Berlin, 1999. viii+199 pp. Corrected reprint of the 1998 original

work page 1999
[6]

Volume et rayon d'injectivit\' e dans les vari\' e t\' e s riemanniennes de dimension 3

Marcel Berger. Volume et rayon d'injectivit\' e dans les vari\' e t\' e s riemanniennes de dimension 3 . Osaka Math. J. , 14(1):191--200, 1977

work page 1977
[7]

Une borne inf\' e rieure pour le volume d'une vari\' e t\' e riemannienne en fonction du rayon d'injectivit\' e

Marcel Berger. Une borne inf\' e rieure pour le volume d'une vari\' e t\' e riemannienne en fonction du rayon d'injectivit\' e . Ann. Inst. Fourier (Grenoble) , 30(3):259--265, 1980

work page 1980
[8]

Arthur L. Besse. Manifolds all of whose geodesics are closed , volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer-Verlag, Berlin-New York, 1978. ix+262 pp. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. B\' e rard-Bergery, M. Berger, and J. L. Kazdan

work page 1978
[9]

T arski's plank problem revisited

K\' a roly Bezdek. T arski's plank problem revisited. In Geometry---intuitive, discrete, and convex , volume 24 of Bolyai Society Mathematical Studies , pages 45--64. J\' a nos Bolyai Mathematical Society, Budapest, 2013

work page 2013
[10]

Jacques Philippe Marie Binet. M\' e moire sur les int\' e grales d\' e finies E ul\' e riennes et sur leur application \` a la th\' e orie des suites; ainsi qu'\` a l'\' e valuation des fonctions des grands nombres. J. \' E c. Polytech. , 16(27):123--343, 1839

work page
[11]

Borisenko

Alexander A. Borisenko. Rigidity of closed convex hypersurfaces in multidimensional spaces of constant curvature. J. Math. Phys. Anal. Geom. , 21(3):267--275, 2025

work page 2025
[12]

L. E. J. Brouwer. \" U ber die periodischen T ransformationen der K ugel. Math. Ann. , 80(1):39--41, 1919

work page 1919
[13]

Yu. D. Burago and V. A. Zalgaller. Geometric inequalities , volume 285 of Grundlehren der mathematischen W issenschaften . Springer-Verlag, Berlin, 1988. xiv+331 pp. Translated from the Russian by A. B. Sosinskiĭ. Springer Series in Soviet Mathematics

work page 1988
[14]

Sur les polygones et les poly\` e dres; S econd M \' e moire

Augustin-Louis Cauchy. Sur les polygones et les poly\` e dres; S econd M \' e moire. J. \' E c. Polytech. , 9(16):87--98, 1813

work page
[15]

M\' e moire sur les fonctions de variables imaginaires et sur leurs d\' e veloppements en s\' e ries

Augustin-Louis Cauchy. M\' e moire sur les fonctions de variables imaginaires et sur leurs d\' e veloppements en s\' e ries. In Oeuvres compl\` e tes d' A ugustin C auchy , volume 13 of 2e s\' e rie , pages 176--203. Gauthier-Villars, Paris, 1932

work page 1932
[16]

P. E. Conner and E. E. Floyd. Fixed point free involutions and equivariant maps. I I . Trans. Amer. Math. Soc. , 105:222--228, 1962

work page 1962
[17]

G. D. Chakerian. Inequalities for the difference body of a convex body. Proc. Amer. Math. Soc. , 18:879--884, 1967

work page 1967
[18]

Morgan W. Crofton. On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus. Philos. Trans. Roy. Soc. London , 158:181--199, 1868

work page
[19]

Christopher B. Croke. Lower bounds on the energy of maps. Duke Math. J. , 55(4):901--908, 1987

work page 1987
[20]

Christopher B. Croke. Area and the length of the shortest closed geodesic. J. Differential Geom. , 27(1):1--21, 1988

work page 1988
[21]

Christopher B. Croke. The volume and lengths on a three sphere. Comm. Anal. Geom. , 10(3):467--474, 2002

work page 2002
[22]

Christopher B. Croke. Small volume on big n -spheres. Proc. Amer. Math. Soc. , 136(2):715--717, 2008

work page 2008
[23]

Cappell and Julius L

Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. , 104(1):61--72, 1976

work page 1976
[24]

Bending of surfaces in the large

Stefan Cohn-Vossen. Bending of surfaces in the large. Uspekhi Mat. Nauk , 1:33--76, 1936

work page 1936
[25]

William J. Firey. Lower bounds for volumes of convex bodies. Arch. Math. , 16:69--74, 1965

work page 1965
[26]

Open problems in geometry of curves and surfaces

Mohammad Ghomi. Open problems in geometry of curves and surfaces. 2019. ghomi.math.gatech.edu/Papers/op.pdf

work page 2019
[27]

Manifolds near the boundary of existence

Karsten Grove and Peter Petersen. Manifolds near the boundary of existence. J. Differential Geom. , 33(2):379--394, 1991

work page 1991
[28]

Alfred Gray. Tubes . Birkh\" a user Verlag, Basil, second edition, 2004. vix+280 pp. With a preface by Vicente Miquel

work page 2004
[29]

Filling R iemannian manifolds

Mikhael Gromov. Filling R iemannian manifolds. J. Differential Geom. , 18(1):1--147, 1983

work page 1983
[30]

The R adon transform , volume 5 of Progr

Sigurdur Helgason. The R adon transform , volume 5 of Progr. Math. Birkh\" a user Boston, Inc., Boston, MA, second edition, 1999. xiv+188 pp

work page 1999
[31]

U ber die S tarrheit der E ifl\

G. Herglotz. \" U ber die S tarrheit der E ifl\" a chen. Abh. Math. Sem. Hansischen Univ. , 15:127--129, 1943

work page 1943
[32]

Hirsch and John Milnor

Morris W. Hirsch and John Milnor. Some curious involutions of spheres. Bull. Amer. Math. Soc. , 70:372--377, 1964

work page 1964
[33]

Convex bodies of constant width and constant brightness

Ralph Howard. Convex bodies of constant width and constant brightness. Adv. Math. , 204(1):241--261, 2006

work page 2006
[34]

U ber die periodischen T ransformationen der K reisscheibe und der K ugelfl\

B. von Ker\' e kj\' a rt\' o . \" U ber die periodischen T ransformationen der K reisscheibe und der K ugelfl\" a che. Math. Ann. , 80(1):36--38, 1919

work page 1919
[35]

Klain and Gian-Carlo Rota

Daniel A. Klain and Gian-Carlo Rota. A continuous analogue of S perner's theorem. Comm. Pure Appl. Math. , 50(3):205--223, 1997

work page 1997
[36]

G. R. Livesay. Fixed point free involutions on the 3 -sphere. Ann. of Math. , 72:603--611, 1960

work page 1960
[37]

Rapport sur deux m\' e moires d'analyse du professeur B urmann

Joseph-Louis Lagrange and Adrien-Marie Legendre. Rapport sur deux m\' e moires d'analyse du professeur B urmann. M\' e m. Inst. Natl. Sci. Arts , 2:13--17, 1799

work page
[38]

Three quantitative versions of the P \'al inequality

Ilaria Lucardesi and Davide Zucco. Three quantitative versions of the P \'al inequality. The Journal of Geometric Analysis , 35(3), 2025

work page 2025
[39]

Volumen und O berfl\"ache

Hermann Minkowski. Volumen und O berfl\"ache. Math. Ann. , 57(4):447--495, 1903

work page 1903
[40]

U ber die K \

Hermann Minkowski. \"U ber die K \"orper konstanter B reite. Mat. Sb. , 25:505--508, 1904

work page 1904
[41]

Sur les h\' e rissons projectifs (enveloppes param\' e tr\' e es par leur application de G auss)

Yves Martinez-Maure. Sur les h\' e rissons projectifs (enveloppes param\' e tr\' e es par leur application de G auss). Bull. Sci. Math. , 121(8):585--601, 1997

work page 1997
[42]

Bonnesen-style isoperimetric inequalities

Robert Osserman. Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly , 86(1):1--29, 1979

work page 1979
[43]

Total mean curvature and closed geodesics

Juan Carlos \'Alvarez Paiva. Total mean curvature and closed geodesics. Bull. Belg. Math. Soc. Simon Stevin , 4(3):373--377, 1997

work page 1997
[44]

Ein M inimumproblem f\" u r O vale

Julius P \'a l. Ein M inimumproblem f\" u r O vale. Math. Ann. , 83(3-4):311--319, 1921

work page 1921
[45]

A. V. Pogorelov. Extrinsic geometry of convex surfaces , volume 35 of Translations of Mathematical Monographs . American Mathematical Society, Providence, 1973. vi+669 pp

work page 1973
[46]

On hypersurfaces with no negative sectional curvatures

Richard Sacksteder. On hypersurfaces with no negative sectional curvatures. Amer. J. Math. , 82:609--630, 1960

work page 1960
[47]

Santal\' o

Luis A. Santal\' o . Integral geometry and geometric probability . Cambridge Math. Lib. Cambridge University Press, Cambridge, second edition, 2004. xx+404 pp. With a foreword by Mark Kac

work page 2004
[48]

Convex bodies: the B runn- M inkowski theory , volume 44 of Encyclopedia of Mathematics and Its Applications

Rolf Schneider. Convex bodies: the B runn- M inkowski theory , volume 44 of Encyclopedia of Mathematics and Its Applications . Cambridge University Press, Cambridge, 1993. xiv+490 pp

work page 1993
[49]

E. P. Sen'kin. Rigidity of convex hypersurfaces. Ukrain. Geometr. Sb. , 12:131--152, 1972

work page 1972
[50]

Estimates of volume by the length of shortest closed geodesics on a convex hypersurface

Andrejs Treibergs. Estimates of volume by the length of shortest closed geodesics on a convex hypersurface. Invent. Math. , 80(3):481--488, 1985

work page 1985
[51]

On theorems of B orsuk-- U lam, K akutani-- Y amabe-- Y ujob\^ o and D yson

Chung-Tao Yang. On theorems of B orsuk-- U lam, K akutani-- Y amabe-- Y ujob\^ o and D yson. I . Ann. of Math. , 60:262--282, 1954

work page 1954