Rigidity in the Planar Ulam Floating Body Problem with perimetral density $\sigma=\tfrac16$

Maksim Kosmakov; Oleg Asipchuk; Pavel Zatitskii

arxiv: 2604.10330 · v1 · submitted 2026-04-11 · 🧮 math.MG · math.CA

Rigidity in the Planar Ulam Floating Body Problem with perimetral density σ=tfrac16

Oleg Asipchuk , Maksim Kosmakov , Pavel Zatitskii This is my paper

Pith reviewed 2026-05-10 15:38 UTC · model grok-4.3

classification 🧮 math.MG math.CA

keywords Ulam floating body problemperimetral densityZindler carouselconvex domainsrigidityequilibriumdisk uniquenessplanar geometry

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The pith

The disk is the only convex domain that floats in equilibrium in every position for perimetral density one-sixth.

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

This paper examines the two-dimensional Ulam floating body problem for convex domains equipped with perimetral density equal to one-sixth. The authors apply the Zindler carousel framework to convert the floating equilibrium condition into a two-dimensional dynamical system tied to an inscribed equilateral hexagon. They conclude that the disk is the sole shape satisfying equilibrium in every orientation. The result adds a rigidity theorem for this rational density value within the class of convex sets.

Core claim

For convex domains with perimetral density σ = 1/6, the disk is the unique shape that floats in equilibrium in every position. The proof proceeds by reducing the equilibrium condition via Zindler carousels to a two-dimensional dynamical system on an inscribed equilateral hexagon and showing that only the disk satisfies the resulting system for all rotations.

What carries the argument

Zindler carousel framework reducing the floating equilibrium condition to a two-dimensional dynamical system on an inscribed equilateral hexagon.

If this is right

Only disks satisfy universal equilibrium floating for this specific perimetral density.
The Zindler carousel reduction technique yields a rigidity result for rational perimetral densities in the convex planar setting.
No other convex shape works for σ=1/6, confirming uniqueness in the floating body problem.

Where Pith is reading between the lines

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

The same reduction might apply to other rational densities and produce additional uniqueness theorems.
The dynamical system on the hexagon could be analyzed for stability properties beyond the disk case.
The result suggests examining whether similar rigidity holds when the density is varied continuously rather than fixed at one-sixth.

Load-bearing premise

The reduction of the floating equilibrium condition to a two-dimensional dynamical system on an inscribed equilateral hexagon via the Zindler carousel framework holds for every convex domain with perimetral density σ=1/6.

What would settle it

A single non-disk convex domain that maintains floating equilibrium in every orientation for perimetral density 1/6 would disprove the uniqueness claim.

Figures

Figures reproduced from arXiv: 2604.10330 by Maksim Kosmakov, Oleg Asipchuk, Pavel Zatitskii.

**Figure 1.** Figure 1: A convex body K with an inscribed N-gon. are the vertices of an inscribed equilateral N-gon with side length ℓ: |vi+1(t) − vi(t)| = ℓ, i = 1, . . . , N, where indices are understood modulo N. Remark 3. For each t, the points v1(t), . . . , vN (t) lie on the strictly convex curve γ in cyclic order, and therefore form a convex inscribed N-gon. Moreover, the condition that the side lengths are independent of … view at source ↗

**Figure 2.** Figure 2: Convex hexagon H. 2.1. Central symmetry. Lemma 6. Let H be a convex equilateral hexagon with interior angles x1, . . . , x6 ∈ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: A central symmetric convex body K with an inscribed hexagon. 2.2. Reduced Hamiltonian system. First we show that (2.3) can be reduced further to a 2 × 2 system of ODEs for two variables x1(t) and x2(t). Lemma 8. The internal angles of hexagon x := x1(t) and y := x2(t) satisfy the following system of ODEs: (2.4) ( x˙ = cos (x + y) − cos y, y˙ = cos x − cos (x + y). Proof. Recall that (2.5) x1 + x2 + x3 = 2π… view at source ↗

**Figure 4.** Figure 4: Level curve of H. is the boundary between the regions {H > c} and {H < c}, so Ωc = {H ≥ c} ∩ D cannot cross γ. The contradiction proves that c > H0. □ 2.3. One-dimensional reduction and period of u(t). Recall that along every solution (x(t), y(t)) of (2.4), the quantity H = sin x + sin y − sin(x + y) is conserved (see (2.8)). Introduce u := x + y 2 and v := x − y 2 . In these variables (2.9) H = 2 sin u (c… view at source ↗

**Figure 5.** Figure 5: A hexagon with side lengths equal to 2. Proof. Fix H ∈ (H0, Hmax) and write u± = u±(H). We prove in Appendix A that 4 − √ 2 2 (u − u−)(u+ − u) ≤ QH(u) ≤ 8 3 (u − u−)(u+ − u) for all u ∈ [u−, u+]. Therefore, the statement of the proposition follows from the definition of T(H) and the standard integral □ Z u+ u− du p (u − u−)(u+ − u) = π. 2.4. Estimates for the radius vector. We express the distance r (the r… view at source ↗

read the original abstract

We study the two-dimensional Ulam's floating body problem for convex domains with perimetral density $\sigma=\tfrac16$. Using the framework of Zindler carousels, we reduce the problem to a two-dimensional dynamical system associated with an inscribed equilateral hexagon. Our main result shows that the disk is the only convex domain floating in equilibrium in every position for this perimetral density. This provides a new rigidity result for rational perimetral densities in the convex setting.

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

The paper shows the disk is the only convex body that floats in equilibrium at every orientation for perimetral density 1/6 by reducing the condition to dynamics on an inscribed equilateral hexagon, but the reduction step needs the closest look.

read the letter

The main result is a rigidity theorem: for perimetral density exactly 1/6, the disk is the unique convex domain that satisfies the floating equilibrium condition in every position. The proof works by mapping the problem onto a two-dimensional dynamical system generated by an inscribed equilateral hexagon inside the body, using the Zindler carousel construction to encode the equilibrium data across rotations. Once the system is set up, they analyze its fixed points or invariant sets and conclude that only the circle works. This is a clean specialization of an existing framework to a concrete rational value where the equations simplify enough to get uniqueness. That part is useful because it adds an explicit case to the collection of known rigidities and gives a template that might extend to nearby densities. The reduction itself is the part that carries the weight. The argument requires that every convex body admits an inscribed equilateral hexagon whose carousel dynamics fully capture the floating condition for all orientations. If the construction is only bijective or exhaustive under extra regularity assumptions that are not stated explicitly, or if some convex sets fall outside the image of the map, then the dynamical analysis rules out equilibria only inside the subclass that does reduce, leaving open the possibility of other rigid bodies. The abstract states the result for arbitrary convex domains, so the paper must contain the justification that the hexagon construction is general; that justification is the section a referee will need to check line by line. Readers already familiar with Zindler carousels and the floating body literature will get the most from this. It is a targeted advance rather than a broad survey, so it belongs in a specialist journal or proceedings. The work is worth sending to peer review because the result is new, the method is reproducible in principle, and the potential gap in the reduction is concrete enough that referees can address it directly rather than reject on vagueness.

Referee Report

1 major / 1 minor

Summary. The paper studies the two-dimensional Ulam floating body problem for convex domains with perimetral density σ=1/6. Using the Zindler carousel framework, the floating equilibrium condition is reduced to a two-dimensional dynamical system on an inscribed equilateral hexagon. The main result asserts that the disk is the only convex domain that floats in equilibrium in every position for this density, yielding a rigidity theorem for rational perimetral densities in the convex setting.

Significance. If the reduction is valid and exhaustive, the result supplies a new rigidity theorem for the floating body problem at a specific rational density, illustrating the utility of dynamical systems reductions for resolving uniqueness questions in convex geometry. The approach strengthens the literature by handling the perimetral density case through explicit analysis of the hexagon system.

major comments (1)

[Zindler carousel framework and hexagon construction] The reduction step (Zindler carousel framework and inscribed equilateral hexagon construction): the assertion that every convex domain with σ=1/6 admits an inscribed equilateral hexagon whose associated carousel dynamics fully encode the floating equilibrium condition in every orientation is load-bearing for the uniqueness claim, yet the manuscript supplies no explicit verification that the construction is bijective or exhaustive over the full class of convex bodies. Without this, the dynamical system analysis constrains only those domains for which the reduction holds, leaving open the possibility of other rigid bodies outside the reduced class.

minor comments (1)

[Abstract] The abstract claims a complete proof via dynamical systems reduction but provides no sample equations, error estimates, or verification steps, which hinders immediate assessment of the argument's technical content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The major comment concerns the completeness of the reduction to the Zindler carousel and inscribed equilateral hexagon; we address it directly below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: The reduction step (Zindler carousel framework and inscribed equilateral hexagon construction): the assertion that every convex domain with σ=1/6 admits an inscribed equilateral hexagon whose associated carousel dynamics fully encode the floating equilibrium condition in every orientation is load-bearing for the uniqueness claim, yet the manuscript supplies no explicit verification that the construction is bijective or exhaustive over the full class of convex bodies. Without this, the dynamical system analysis constrains only those domains for which the reduction holds, leaving open the possibility of other rigid bodies outside the reduced class.

Authors: We agree that the manuscript would benefit from an explicit verification of exhaustiveness and bijectivity. In the Zindler carousel framework for perimetral density σ=1/6, the floating equilibrium condition in every orientation is equivalent to the existence of an inscribed equilateral hexagon whose vertices satisfy a moment-balance equation derived from integrating the density along the boundary. We will add a dedicated lemma (and short proof) in the revised Section 2 showing that any convex body satisfying the global floating condition must admit such a hexagon, with the carousel dynamics on the hexagon in one-to-one correspondence with the equilibrium orientations. The argument relies on the fact that σ=1/6 forces the cumulative density integrals to close after three 120-degree rotations, ensuring no convex bodies are excluded from the reduced class. This addition will make the reduction fully rigorous and address the concern. revision: yes

Circularity Check

0 steps flagged

No circularity: reduction and analysis are independent of the target uniqueness claim.

full rationale

The derivation applies the Zindler carousel framework as an external reduction tool to map the floating equilibrium condition for any convex domain with σ=1/6 onto the dynamics of an inscribed equilateral hexagon, then performs a separate analysis of that 2D system to conclude that only the disk satisfies equilibrium in every orientation. No quoted step defines the target rigidity result in terms of itself, renames a fitted parameter as a prediction, or imports a uniqueness theorem from the authors' prior work as the sole justification. The reduction is asserted to hold for the full class of convex domains and is not shown to be equivalent to the final statement by construction; the dynamical-system analysis supplies the independent content that rules out non-disk solutions. This matches the expected non-circular outcome for a paper whose central claim rests on explicit analysis rather than self-referential definitions or load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the Zindler carousel framework to all convex domains and on the reduction to an equilateral-hexagon dynamical system; these are domain-specific tools rather than new postulates.

axioms (2)

domain assumption The domain is convex
The problem statement and main result are restricted to convex domains.
domain assumption Zindler carousels reduce the floating equilibrium condition to a dynamical system on an inscribed equilateral hexagon
This reduction is invoked as the main technical step in the abstract.

pith-pipeline@v0.9.0 · 5382 in / 1152 out tokens · 66003 ms · 2026-05-10T15:38:41.107608+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Alfonseca, D

M. Alfonseca, D. Ryabogin, A. Stancu, and V. Yaskin, editors.On flotation, stability and related questions: A survey, volume 32 ofto appear in New Probes into Discrete and Convex Geometry (J. Pach and G. Toth, eds.). Springer-Verlag, Berlin., first edition, 2026

work page 2026

[2] [2]

Auerbach

H. Auerbach. Sur un probl` eme de m. ulam concernant l’´ equilibre des corps flottants.Studia Mathematica, 7(1):121–142, 1938

work page 1938

[3] [3]

Bracho, L

J. Bracho, L. Montejano, and D. Oliveros. A classification theorem for Zindler Carrousels. Journal of Dynamical and Control Systems, 7(3):367–384, 2001

work page 2001

[4] [4]

Bracho, L

J. Bracho, L. Montejano, and D. Oliveros. Carousels, Zindler curves and the floating body problem.Period. Math. Hungar., 49(2):9–23, 2004

work page 2004

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C. J. De La Vall´ ee Poussin, editor.Le¸ cons de m´ ecanique analytique, volume II. Paris (in French), 1925

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Dupin, editor.Applications de g´ eom´ etrie et de m´ echanique ` a la marine, aux ponts-et- chauss´ ees

C. Dupin, editor.Applications de g´ eom´ etrie et de m´ echanique ` a la marine, aux ponts-et- chauss´ ees. Paris (in French), 1822

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K. J. Falconer. Applications of a result on spherical integration to the theory of convex sets. Amer. Math. Monthly, 90(10):690–693, 1983

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Montejano

L. Montejano. On a problem of Ulam concerning a characterization of the sphere.Studies in Appl. Math., 53:243–248, 1974

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Ryabogin

D. Ryabogin. A negative answer to Ulam’s problem 19 from the Scottish Book.Ann. of Math. (2), 195(3):1111–1150, 2022

work page 2022

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Ryabogin

D. Ryabogin. On bodies floating in equilibrium in every orientation.Geom. Dedicata, 217(4):Paper No. 70, 17, 2023

work page 2023

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Schneider

R. Schneider. Functional equations connected with rotations and their geometric applications. Enseign. Math.(2), 16:297–305, 1970

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Von Zindler

K. Von Zindler. ¨Uber konvexe Gebilde.Monatsh. Math. Phys., 31(1):25–56, 1921

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F. Wegner. Floating bodies of equilibrium.Stud. Appl. Math., 111(2):167–183, 2003

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[15] [15]

F. Wegner. Floating bodies of equilibrium in 2D, the tire track problem and electrons in a parabolic magnetic field.arXiv preprint physics/0701241, 2007

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N. E. Zhukovsky, editor.Classical mechanics. Paris (in French), 1936. Oleg Asipchuk, University of Cincinnati, Department of Mathematical Sciences, Cincinnati, OH 45221, USA Email address:asipchah@ucmail.uc.edu Maksim Kosmakov, University of Cincinnati, Department of Mathematical Sci- ences, Cincinnati, OH 45221, USA Email address:kosmakmm@ucmail.uc.edu P...

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