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On the homothety conjecture for the body of flotation and the body of buoyancy on a plane

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Pith's one-line read If a plane convex body's flotation and buoyancy bodies are homothetic, then its chords of flotation have constant affine distance and constant cut-off affine arc length; adding the one-third affine-arc condition makes the body an ellipse.

desk verdict A genuinely new variant of the homothety conjecture with real substance, but the main theorem hinges on an unverified μ′(s)=0 computation. read the letter →

arxiv 2507.11850 v1 pith:EAF4BGYX submitted 2025-07-16 math.DG

Bart{l}omiej Zawalski This is my paper · ORCID
classification math.DG MSC 52A1051N1053A0453A1570C20
keywords homothetyconjecturefloatingbodyproblemofflotationbuoyancyilluminationaffinedifferentialgeometryarclengthellipsecharacterization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a plane analogue of a classical conjecture about floating bodies: if the body of flotation of a convex plane body is homothetic to its body of buoyancy, then the chords of flotation have constant affine distance between their endpoints and cut off arcs of constant affine length. When each such chord cuts off exactly one third of the total affine arc length, the body must be an ellipse. A sympathetic reader should care because this fills a gap that contraction-operator methods could not reach, and because the affine-differential-geometry framework developed along the way recovers natural affine counterparts of classical floating-body rigidity results and of carousel constructions. The main theorems are proved by comparing second- and third-order differential invariants of the flotation and buoyancy boundaries.

What carries the argument

The central objects are the body of flotation Fδ(K), whose boundary is the envelope of water-level lines cutting off area δ, and the body of buoyancy Bδ(K), whose boundary is the locus of centroids of the underwater caps. The argument runs through a chord-of-flotation parametrization: for each boundary point x, the chord [x,y] cutting off area δ has a midpoint on the flotation boundary and a centroid on the buoyancy boundary, with both tangent directions parallel to the chord by the classical flotation theorems. The load-bearing identities are κ1 = ∥c∥^3/|c|^3 and κ2 = 12δ/|c|^3, which yield the homothety characterization ∥c∥^3 ≡ 12δλ; the third-order invariant comparison yields condition (4.7), equivalently constant affine arc length; and a polarity with respect to K converts flotation-to-buoyancy homothety into a duality between the flotation boundary and an illumination body. Finally, the classical affine-hypersphere theorem — a proper affine hypersphere must be an ellipsoid — plus section-centroid characterizations of ellipsoids close the global conclusions.

What would settle it

Compute condition (4.7) directly for a non-circular ellipse: if some chord fails it, the uniqueness argument behind the ellipse conclusion cannot start. Alternatively, numerically search for a $C^{2}$ convex non-ellipse with a δ at which Fδ(K) is homothetic to Bδ(K) and every flotation chord cuts off exactly one third of the total affine arc length; finding one would disprove Theorem 3.

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Extended reading notes

Core claim

The paper's central discovery, in its own terms, is Theorem 3: let K be a $C^{2}$ convex body in the plane and let δ be a volume in (0, area(K)). If the body of flotation Fδ(K) is homothetic to the body of buoyancy Bδ(K), and if every chord of flotation cuts off from the boundary exactly one third of its total affine arc length, then K is an ellipse. Theorems 1 and 2 are the structural precursors: the same homothety assumption already forces the affine distance ∥c∥^3 between the endpoints of each chord of flotation to be constant, and forces the affine arc length of the boundary arc cut off by each chord to be constant. The proof identifies the homothety ratio with ∥c∥^3/(12δ), translates the comparison of second- and third-order differential invariants into the condition $sin^{3}$ α/κ(s) = $sin^{3}$ β/κ(t), and uses the one-third affine-arc-length condition to force the centroids of consecutive flotation triangles to coincide, giving a fixed homothety center. From there, a weak homothety theorem plus a classical affine-hypersphere characterization of ellipsoids yields the ellipse.

Load-bearing premise

The final ellipse conclusion rests on an unproved assertion that every quadric satisfies the key condition (4.7), which seeds the ODE uniqueness argument, and on several long computations including the fixed-centroid step μ′(s) = 0 that are presented without derivation; if any of these fails, the theorem is not established as written.

Editorial extensions

If this is right

  • If a planar C^2 convex body has Fδ(K) homothetic to Bδ(K), then the affine distance between the endpoints of every chord of flotation is constant (Theorem 1).
  • Under the same hypothesis, the affine arc length of the boundary arc cut off by every chord of flotation is constant (Theorem 2).
  • If, in addition, every chord of flotation cuts off exactly one third of the total affine arc length, then K is an ellipse (Theorem 3).
  • A flotation boundary is homothetic to the corresponding centroid curve exactly when ∥c∥^3 is constant, giving an affine counterpart of the classical constant chord-length results.
  • Unless K is an ellipse, the values of δ for which the flotation and buoyancy boundaries are homothetic form a nowhere-dense subset of (0, area(K)) (Corollary 4.14).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same third-order invariant comparison could be iterated algorithmically, turning the remaining gap around condition (4.7) into a finite computer-algebra elimination problem.
  • Beyond the paper: the one-third affine perimetral density suggests a family of carousel-type rigidity questions at densities 1/q; the paper notes that the geometry becomes less rigid, so a computational search for non-ellipse solutions at q = 4 is a natural test.
  • Beyond the paper: Theorems 1 and 2 give a practical differential signature — constant ∥c∥^3 and constant affine arc length — that can be checked numerically on a sampled convex body before attempting any global rigidity argument.
  • Beyond the paper: the polarity equivalence of Corollary 4.4 may connect the two-body homothety question to illumination- and intersection-body duality in higher dimensions, although the paper only develops it on the plane.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies, in the plane, the bodies of flotation, buoyancy, and illumination associated to a convex body K, using affine differential geometry. It derives systematic formulas for the boundaries and centroid curves of these bodies and records them in Tables 3.11 and 3.20. The main stated results are that if the body of flotation Fδ(K) is homothetic to the body of buoyancy Bδ(K), then the affine distance between endpoints of every chord of flotation is constant (Theorem 1) and the affine arc length cut off by every chord of flotation is constant (Theorem 2); moreover, if in addition every chord of flotation cuts off exactly one third of the total affine arc length of the boundary, then K is an ellipse (Theorem 3). The proof of the latter is implemented through Theorem 4.20, which uses a triangle argument in the spirit of Bracho, Montejano, and Oliveros to show that the centroid of a moving inscribed triangle is fixed, and then invokes Theorem 4.16 to conclude that K is an ellipse. The paper also gives affine counterparts of Zindler carousels and additional characterizations of ellipses, including Theorem 4.12 and Theorem 4.17.

Significance. If the computational claims are fully verified, the paper would provide a new affine-geometric result on the homothety conjecture for floating bodies, a problem that has been open in the plane and has recently received counterexamples for the original formulation. The systematic derivation of formulas for bodies of illumination appears to be new and is presented in a usable tabular form. The paper explicitly makes no use of fitted parameters or external numerical data, and the argument is self-contained up to classical results of Dupin, Blaschke, and Petty. The affine analogue of the Zindler carousel theorem is a natural and potentially influential contribution. However, several load-bearing algebraic and analytic steps are asserted rather than demonstrated, so the main theorems are not yet fully supported as written.

major comments (3)
  1. [§4.4, proof of Theorem 4.20] The proof's central step is the assertion that substituting (4.22) and (4.23) into the formula for μ′ = (x+y+z)′/3 yields 0. This is not a displayed computation or a separately stated lemma; it is the only place where the moving centroid μ becomes fixed. The fixed centroid is then used to conclude that Π̂δ(K) is homothetic to Πδ(K) with center μ and ratio 4, which is exactly the hypothesis needed to apply Theorem 4.16 and obtain that K is an ellipse. Since μ′ involves γ′(s), γ′(t)t′, and γ′(u)u′ with t and u coupled by the chord-of-flotation equations (3.4), the simplification is not a routine one-line identity. Please provide the complete computation or a dedicated lemma proving μ′(s)=0 under the stated hypotheses.
  2. [§3.4, Proposition 3.25] The proof of Proposition 3.25 is summarized as “a tedious but straightforward computation,” but this proposition is load-bearing for Theorem 4.16 and Theorem 4.17. Those theorems use the fact that the affine normal to the centroid curve is parallel to the affine bisector of the chord, and the displayed identity r2′′ = 8δ̄^{1/3}∥c∥³(r1−z) is the quantitative form of that fact. The line r1r3 is subsequently identified with the affine normal to Πδ(K), and this identification is essential for concluding that Πδ(K) is a proper affine hypersphere. The full computation, including the handling of the parametrization and of the case where the directions ẋ and ẏ are parallel, should be written out or replaced by a precise reference.
  3. [§4.2, proof of Theorem 4.12] The proof asserts without justification that “any quadric is known to satisfy the assumptions of Theorem 4.6” and that there is a unique quadric having second-order contact with γ at y and first-order contact with γ at x. The first assertion is a nontrivial global property of conics with respect to condition (4.7), and the second requires an explicit count of the degrees of freedom of a conic against the contact conditions. As written, the ODE uniqueness argument only proves that a solution equals some conic if such a conic is already known to exist. This theorem is advertised as a characterization of ellipses and is used for Corollary 4.14, so the missing justification should be supplied.
minor comments (4)
  1. [§4.4, Figure 4.21] The points x̂, ŷ, ẑ, ˇx, ˇy, ˇz, x, y, z are used before they are fully defined in the text; please introduce each of these points precisely before the proof of Theorem 4.20.
  2. [§4.2, proof of Theorem 4.6] The formulas for κ1′ and κ2′ in Table 3.11 are used to compare third-order invariants, but their derivation is not given in the body of the paper; a sentence indicating whether they follow from the same substitution used for κ1 and κ2 would help the reader verify (4.7).
  3. [§4.2, Theorem 4.12] The term “quadric” should be specified as a nondegenerate conic with positive definite quadratic part (an ellipse), since the convexity of K and the ODE argument require this interpretation.
  4. [§4.4, first paragraph] Please distinguish clearly between the Euclidean perimetral density used by Auerbach and by Bracho–Montejano–Oliveros and the affine perimetral density used in Theorem 4.20, since the two notions are not the same.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and rests on external classical results, not on the conclusions being proved.

full rationale

The paper's central argument is a self-contained differential-geometric derivation. Theorems 4.2 and 4.6 derive the homothety and affine-arc-length conditions from previously established tangent and curvature formulas (3.5), (3.8), and Table 3.11, with no fitted parameter renamed as a prediction and no target conclusion built into the hypotheses. Theorem 4.20 obtains the key normalization lambda_1 = lambda_2 = lambda_3 = 1 by explicit sine-law, area, and condition (4.7) computations, then invokes classical external results such as Blaschke-Deicke (Theorem 2.5), Petty's classification, and Kurusa-Odor's characterization of balls by sections and caps; these are standard mathematical dependencies, and none of the cited works is by the present author, so no self-citation chain is load-bearing. The proof does contain acknowledged or unacknowledged computational gaps: in Theorem 4.20, the assertion that substituting (4.22) and (4.23) into the formula for mu' yields 0 is presented without derivation, and in Theorem 4.12 the claim that every quadric satisfies the hypotheses of Theorem 4.6 is not proved. These are missing justifications in the written argument and therefore correctness risks, not circular reductions of a conclusion to its own inputs. No circular step could be identified from the text.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numbers are fitted to data and no new objects are postulated. The paper's burden lies in unstated smoothness assumptions, non-degeneracy of the parametrizations, and several computational identities that are asserted rather than derived in full.

assumptions (6)
  • domain assumption The boundary of K is of class C^2, and C^3 for the body of illumination.
    Assumed in all theorem statements and used for the curvature computations in Section 3.
  • domain assumption The relevant envelopes, tangent intersections, and polar duals are non-degenerate, so denominators in equations (3.4), (3.15), and related formulas do not vanish.
    Needed for the parametrizations of the bodies of flotation and illumination to be regular; the paper notes this but does not fully justify it in all cases.
  • domain assumption Every quadric satisfies the hypotheses of Theorem 4.6.
    Used in the proof of Theorem 4.12 to conclude that local arcs of the boundary are quadrics; the paper gives no proof or citation for this claim.
  • standard math Blaschke-Deicke theorem: a compact non-degenerate affine hypersphere is an ellipsoid.
    Invoked as Theorem 2.5 to conclude that the body of flotation is an ellipse once it is shown to be an affine hypersphere.
  • standard math Kurusa-Ódor theorem: an ellipsoidal floating body forces the original body to be an ellipsoid.
    Invoked as [15, Theorem 3.2] in the proof of Theorem 4.16.
  • standard math Petty's framework for the class F2, which characterizes ellipses via curvature functions.
    Used in Section 4.3; the paper offers an alternative proof but relies on Theorem 4.17 and the external theorem for the intersection-body step.

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Pith. "Pith review of On the homothety conjecture for the body of flotation and the body of buoyancy on a plane." pith.science (2026). https://pith.science/paper/EAF4BGYX

@misc{pith2026250711850,
  author       = {Pith},
  title        = {Pith review of: On the homothety conjecture for the body of flotation and the body of buoyancy on a plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAF4BGYX}},
  note         = {Machine review of arXiv:2507.11850}
}
abstract

We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy, and bodies of illumination. As a direct consequence of our results, we show that if the body of flotation is homothetic to the body of buoyancy, and if every chord of flotation cuts off from the boundary exactly $\frac{1}{3}$ of its total affine arc length, then $K$ is an ellipse. We also provide natural affine counterparts of the classical theorems on the floating body problem from the Scottish Book due to H. Auerbach. In particular, we obtain an affine counterpart of Zindler carousels introduced by J. Bracho, L. Montejano, and D. Oliveros.

Figures

Figures reproduced from arXiv: 2507.11850 by the authors.

Figure 1.1
Figure 1.1. A drawing illustrating the main theorems Theorem 2. Let K ⊂ R 2 be a convex body with boundary of class C 2 , and let δ ∈ (0, voln(K)). If Fδ(K) is homothetic to Bδ(K), then the affine arc length of the boundary cut off from K by every chord of flotation is constant. Theorem 3. Let K ⊂ R 2 be a convex body with boundary of class C 2 , and let δ ∈ (0, voln(K)). If Fδ(K) is homothetic to Bδ(K), and if δ is such that e… view at source ↗
Figure 2.6
Figure 2.6. Illustrative drawing of a floating body The signed affine distance ∥y − x∥ of two linear elements (x, x˙) and (y, y˙) is defined as ∥y − x∥ := 2T 1 3 (cf. [36, §I.3]). We say that the curve is non-degenerate if it satisfies the condition det(x˙ , x¨) ̸= 0. A non-degenerate curve x admits a parameter σ such that (2.3) det(x ′ , x ′′) = 1, where ′ denotes the derivative with respect to σ. Such a parameter, called the … view at source ↗
Figure 3.2
Figure 3.2. Notations used in the argument for the body of flotation 3.1.1. The boundary. From now on, let t be chosen so that the area A ≡ δ is constant. Denote x(s) := γ(s), y(s) := γ(t(s)), c := y − x, r1 := 1 2 (x + y). Then (3.1) reads (3.3) det(c, r˙ 1) = 0. It follows directly from the definition that r1 is a (not necessarily regular) parametrization of Πδ. Indeed, for every s ∈ I, the curve r1 is tangent to the chord c … view at source ↗
Figures from the paper (5 more)
Figure 3.13
Figure 3.13. Figure 3.13: Notations used in the argument for the body of illumination 3.2. The body of illumination. This time, let K ⊂ R 2 be a strongly convex body with boundary of class C 3 , let I δ be the body of illumination for K and some δ ∈ (0, vol2(K)), and let Γδ (K) be the corres…
Figure 3.21
Figure 3.21. Figure 3.21: Notations used in the argument for the asymptotic body of flotation 3.3. Limiting cases for centrally symmetric body. For the purposes of this section, let us assume that K ⊂ R n is origin-symmetric. We will investigate the asymptotic shape of the body of flotation …
Figure 3.22
Figure 3.22. Figure 3.22: Notations used in the argument for the asymptotic body of illumination (cf. fig. 3.21) whence hε−1Fδ(K)(ξ) = ε −1hFδ(K)(ξ) = 1 voln−1(K ∩ ξ ⊥) + o(1) = 1 ρIK(ξ) + o(1) = hI∗K(ξ) + o(1). Thus, the sequence of support functions converges pointwise, and the conclusion …
Figure 4.15
Figure 4.15. Figure 4.15: Construction of the dual body φI δˆ (K) (K) Proof. Fix any point x ∈ ∂K and any chord [x, y] ∈ U that passes through x. By rotating the coordinate system, without loss of generality, we may assume that the chord is vertical. Hence, the boundary curve γ in some neigh…
Figure 4.21
Figure 4.21. Figure 4.21: Notation used in the proof of Theorem 4.20 by symmetry, we have λ1 = λ2 = λ3. When combined with the previous result, it immediately yields λ1 = λ2 = λ3 = 1. Therefore, by the Second Theorem of Dupin, △xˇyˇzˇ is the medial triangle of △xyz, which in turn, by the abo…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.