arxiv: 2605.00299 · v1 · submitted 2026-05-01 · 🧮 math.MG · math.NT

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On convex bodies with constant non-central sections

D. Ryabogin, J. Haddad

Pith reviewed 2026-05-09 15:30 UTC · model grok-4.3

classification 🧮 math.MG math.NT

keywords convex bodiessectionsconstant areabodies of revolutionuniquenessDiophantine conditionsHausdorff dimensionfour dimensions

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The pith

A symmetric convex body of revolution in four dimensions that contains the unit ball and has constant-area sections by all hyperplanes tangent to the ball must itself be a ball, provided a number derived from the area A satisfies specificDi

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

The paper proves a uniqueness theorem for symmetric convex bodies of revolution in R^4 that contain the unit Euclidean ball and whose sections by hyperplanes tangent to this ball all have the same area A greater than zero. Under the additional requirement that one over pi times the arctangent of the cube root of three A over four pi has arithmetic properties visible in its continued-fraction expansion, the only such body is the Euclidean ball. A reader would care because the result gives a partial characterization of the ball by the constancy of these non-central three-dimensional sections, and because the admissible values of A form a set of positive Hausdorff dimension rather than a sparse collection. The proof therefore applies to a rich family of possible section areas.

Core claim

We prove that if C is a symmetric convex body of revolution in R^4 containing the unit Euclidean ball B_4, such that the sections of C by hyperplanes tangent to B_4 have constant area A>0, then C is a Euclidean ball, provided 1/π arctan((3A/4π)^{1/3}) satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values A satisfying these properties has positive Hausdorff dimension.

What carries the argument

The constancy of area for all three-dimensional sections cut by hyperplanes tangent to the inscribed unit ball, together with the revolution symmetry in four dimensions and the continued-fraction arithmetic condition on the angle whose tangent is the cube root of three A over four pi.

If this is right

For every A whose associated arctangent expression meets the arithmetic criteria, the Euclidean ball is the only symmetric convex body of revolution with the constant-tangent-section property.
The admissible areas A form an uncountable set of positive Hausdorff dimension.
The characterization holds only when the body is a solid of revolution and the ambient space is four-dimensional.
The result supplies a concrete Diophantine filter that selects a large class of areas for which the uniqueness statement is valid.

Where Pith is reading between the lines

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

The arithmetic condition is likely an artifact of the present proof method; without it, non-ball examples may exist.
Analogous statements could be investigated in dimensions other than four or for bodies lacking revolution symmetry.
The constancy condition on tangent sections might be combined with other section or projection data to obtain uniqueness results without Diophantine restrictions.
Numerical checks could examine whether small non-radial perturbations of the ball produce varying tangent-section areas precisely when the arctangent condition holds.

Load-bearing premise

The number one over pi times arctan of the cube root of three A over four pi must have continued-fraction properties that permit the uniqueness argument to succeed.

What would settle it

A non-spherical symmetric convex body of revolution in R^4 containing the unit ball whose tangent hyperplane sections all have identical area A, where one over pi arctan of (3A/4pi) to the one-third does satisfy the required continued-fraction arithmetic properties.

Figures

Figures reproduced from arXiv: 2605.00299 by D. Ryabogin, J. Haddad.

**Figure 1.** Figure 1: The values t(s) and z(s) are the critical points of the polynomial Ps, whose graph is depicted in blue. tangent to B2 at the upper half-plane and has slope s, then Ps(b(s)) − Ps(a(s)) = − 4 3 (R2 − 1)3/2 s(1 + s 2) , where R is given by equation (4) and (7) Ps(x) = − 2 3 √ 1 + s 2 s x 3 − 1 + 2s 2 s 2 x 2 − 2 √ 1 + s 2 s x. Notice that Ps is a third-degree polynomial in x. Consider s ∈ R and a tangent line… view at source ↗

**Figure 2.** Figure 2: The function TK for some points in the plane. Here ε(p) = 1, s(p) < 0, ε(p ′ ) = −1, s(p ′ ) > 0 and ε(p ′′) = −1, s(p ′′) > 0. respect to the origin (see view at source ↗

**Figure 3.** Figure 3: The lines Ls0 and L(X(M),0) are symmetric with respect to the X axis. If s < s0 the segment of Ls bounded by the red lines is between the X axis and Ls0 . which is valid (i.e. compatible with other properties of K). That is the content of Proposition 5. But first we need a lemma. Lemma 4. Let M ⊆ R 2 be a convex body, symmetric with respect to the X and Y axes, containing B2 in its interior, and let X(M) b… view at source ↗

**Figure 4.** Figure 4: Replace this value of x in (10) to obtain 2s 2 3(s 2 + 1) − 2s 2 + 1 s 2 + 1 + 2 < − 4(R2 − 1)3/2 3s(s 2 + 1) + 2 √ s 2 + 1 s r R2 − 1 s 2 + 1 + s √ s 2 + 1! + 2 √ s 2 + 1 3s r R2 − 1 s 2 + 1 + s √ s 2 + 1!3 + 2s 2 + 1 s 2 r R2 − 1 s 2 + 1 + s √ s 2 + 1!2 which after some computations simplifies to (b), as required. Finally let us consider condition (c). First notice that −s(X(K), 0), −s(R, 0) < 0. As befo… view at source ↗

**Figure 5.** Figure 5: The function GA is computed as the difference between the green angle arctan(√ x 2 − 1) and the red angle, given by arctan(p R2 A − 1). It is important to notice that δA,k is exactly the angle in (0, π/2) formed by the X axis and the line 0, uA,k. The values of αA,k, βA,k are the angles corresponding to the two points from {uA,1, . . . , uA,k} which are closest to the X axis. They will be used to bound the… view at source ↗

**Figure 6.** Figure 6: The points uA,1, . . . , uA,k bound the position of X(K) which belongs to the red segment. The orange arc contains all the points p ∈ SR in the upper half-plane, with s(p) between −s(X(K), 0) and −s(R, 0). The blue arc is the points with angle [−FA,−(αk), FA,+(αk, βk)], as in condition C(A, k). Then by Proposition 5, u2 = TR(u1) belongs to SR ∩ ∂K. Now we proceed by induction and assume that u1, . . . , u… view at source ↗

**Figure 7.** Figure 7: Some values of RA obtained using numerical simulations. The green lines represent the values of RA for which the conditions C(A, k) are verified for all k ≥ 2. The red lines represent the values of RA for which Theorem 14 failed to guarantee the conditions. All the computations were done in Wolfram Mathematica 14.2. Finally, take any k ∈ N with k ∈ [˜q2 − 1, q˜3 − 2]. Since k + 1 ∈ [˜q2, q˜3) we have δk… view at source ↗

read the original abstract

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac 1{\pi} \arctan((\frac{3A}{4\pi})^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.

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

Haddad and Ryabogin prove a conditional uniqueness result for symmetric revolution convex bodies in R^4 with constant tangent hyperplane section areas, under an explicit continued-fraction condition on A, plus a positive Hausdorff-dimension statement for the admissible A's.

read the letter

The main thing to know is that this paper gives a conditional rigidity theorem: a symmetric convex body of revolution in four dimensions that contains the unit ball and has constant area A for all hyperplanes tangent to that ball must be the Euclidean ball, provided (1/π) arctan((3A/4π)^{1/3}) satisfies certain arithmetic properties visible in its continued-fraction expansion. They also show the set of such A has positive Hausdorff dimension. That combination is the actual new piece. Earlier work on section problems has uniqueness results, but tying the geometric conclusion to this specific Diophantine restriction and then measuring the size of the good set in the Hausdorff sense looks fresh. The reduction via revolution symmetry to a one-variable problem is standard and effective here, and the arithmetic condition is used to control approximations in whatever integral or Fourier identity arises from the constant-area assumption. That part reads as honest progress inside the subfield. The soft spot is the condition itself. It is necessary for their argument, which means the result is not unconditional uniqueness, and the paper does not construct or rule out counterexamples when the continued-fraction property fails. The restriction to bodies of revolution and to dimension four also keeps the scope narrow, though that is explicit in the statement. Overall the claim is precise, the setup is clean, and there is no visible circularity or internal contradiction. This is for specialists working on convex bodies, section rigidity, or Diophantine conditions in geometric analysis. A reader already following results like Busemann-Petty type problems or profile equations for revolution bodies will get concrete value from the arithmetic-geometric link. It deserves a serious referee because the result is non-trivial, the evidence is formally grounded, and the Hausdorff-dimension add-on gives it a quantitative edge.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that if C is a symmetric convex body of revolution in R^4 containing the unit Euclidean ball B_4 such that all sections of C by hyperplanes tangent to B_4 have the same 3-volume A > 0, then C must be the Euclidean ball, provided that the number (1/π) arctan((3A/4π)^{1/3}) satisfies certain Diophantine properties readable from its continued-fraction expansion. The paper further shows that the set of admissible values of A has positive Hausdorff dimension.

Significance. If the proofs are correct, this is a non-trivial rigidity result in convex geometry for bodies with constant non-central sections. The conditional uniqueness statement in dimension 4 for revolution bodies, combined with the positive Hausdorff dimension of the admissible set A, indicates that the result is not vacuous and may open avenues for studying similar problems without the revolution assumption or in other dimensions. The explicit arithmetic condition on the parameter derived from A is an unusual but potentially powerful feature of the argument.

major comments (1)

The central implication from the constant-area hypothesis plus the Diophantine condition on α = (1/π) arctan((3A/4π)^{1/3}) to the conclusion that C is a ball is the load-bearing step of the paper, yet the abstract and available description provide no visible details on how the arithmetic condition is used to control the associated functional equation or approximation properties. Without this step being verifiable, the soundness of the uniqueness claim cannot be assessed.

minor comments (2)

The precise statement of the 'certain arithmetic properties' readable from the continued-fraction expansion of α should be made fully explicit (e.g., by naming the required irrationality measure or bounded partial quotients) so that readers can check the condition for concrete A without ambiguity.
It would be helpful to include a brief comparison with known results on constant central sections or constant brightness bodies to clarify the novelty relative to the classical literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of the result. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central implication from the constant-area hypothesis plus the Diophantine condition on α = (1/π) arctan((3A/4π)^{1/3}) to the conclusion that C is a ball is the load-bearing step of the paper, yet the abstract and available description provide no visible details on how the arithmetic condition is used to control the associated functional equation or approximation properties. Without this step being verifiable, the soundness of the uniqueness claim cannot be assessed.

Authors: We agree that the abstract is concise and does not sketch the argument. In the full manuscript, the constant-area condition on tangent hyperplane sections is used in Section 3 to derive a functional equation for the radial function ρ of the body of revolution (after reducing to the generating curve in the half-plane). This equation takes the form of a cohomological equation over the irrational rotation by angle α on the circle: the deviation function satisfies Δ(θ) = f(θ + α) - f(θ) + g(θ), where g encodes the area constraint. In Section 4 we invoke the assumed Diophantine properties of α (extracted from its continued-fraction expansion, specifically that the partial quotients are bounded, making α badly approximable). Standard results on Diophantine approximation then imply that the only continuous (or L^∞) solutions are constant functions, forcing ρ to be constant and hence C to be the Euclidean ball. This is carried out in the proof of Theorem 4.1. We will revise the introduction to include a short outline of this reduction and the role of the arithmetic condition, making the central step more immediately verifiable from the front matter. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central result is a conditional implication: constant-area tangent sections plus an external Diophantine condition on the continued-fraction expansion of (1/π) arctan((3A/4π)^{1/3}) imply that the body is a ball. The arithmetic condition is stated as an independent hypothesis on a real number derived from the geometric datum A; it is not defined in terms of the conclusion and does not reduce any step of the argument to a tautology or a fitted input. No self-definitional equations, renamed empirical patterns, or load-bearing self-citations appear in the provided statement. The existence of a positive-Hausdorff-dimension set of admissible A is shown separately and does not rely on the uniqueness claim. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions of convex geometry plus an external arithmetic filter on the area parameter A. No free parameters are fitted inside the derivation, and no new entities are postulated.

axioms (2)

domain assumption C is a symmetric convex body of revolution in R^4 containing the unit ball B_4
Explicitly stated as the setting for the theorem.
domain assumption Hyperplane sections tangent to B_4 have constant area A > 0
The central hypothesis whose consequences are derived.

pith-pipeline@v0.9.0 · 5397 in / 1560 out tokens · 29282 ms · 2026-05-09T15:30:49.880660+00:00 · methodology

discussion (0)

Reference graph

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