arxiv: 2605.09776 · v1 · submitted 2026-05-10 · 🧮 math.MG · math.DG

Recognition: 1 theorem link

· Lean Theorem

Uniqueness of Flotation and Buoyancy Surfaces for Convex Polytopes

Orli Herscovici, Sergii Myroshnychenko, Susanna Dann

Pith reviewed 2026-05-12 02:33 UTC · model grok-4.3

classification 🧮 math.MG math.DG

keywords convex polytopesflotation surfacesbuoyancy surfacesuniqueness theoremsuniform densityconvex geometryfloating bodiesinverse problems

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The pith

A convex polytope of uniform density different from one half is uniquely determined by its flotation surface.

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

The paper proves that convex polytopes in dimensions two and above with constant density delta not equal to 1/2 are completely fixed by the surface on which they float when placed in a denser liquid. The same uniqueness holds when given the buoyancy surface for a prescribed density. This matters because it shows these surfaces carry complete information about the polytope's shape and orientation. If the claim holds, then measurements of the flotation surface alone suffice to reconstruct the entire object without needing other data. The result applies specifically to uniform density convex bodies and excludes the half-density case where symmetry might allow multiples.

Core claim

We prove that a convex polytope P in R^d for d at least 2, having uniform density delta between 0 and 1, is uniquely determined by its flotation surface P_{[delta]} when floating in a liquid of density 1, provided delta is not 1/2. In the same way, the buoyancy surface C_delta P for such a polytope with given density delta uniquely determines the polytope P itself.

What carries the argument

The flotation surface P_{[delta]}, defined as the intersection of the polytope with the liquid surface at equilibrium, and the buoyancy surface C_delta P, which maps the polytope to the set of its centers of buoyancy; these surfaces serve as the unique identifiers from which the original polytope can be recovered.

If this is right

The entire geometry of the polytope, including its shape and position, can be recovered from the flotation surface for delta not equal to 1/2.
Buoyancy surfaces likewise allow full reconstruction of the polytope.
The uniqueness holds in all dimensions d at least 2.
Non-uniqueness may occur at delta equal to 1/2 due to possible symmetries.

Where Pith is reading between the lines

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

If the result extends beyond polytopes, it could apply to general convex bodies with uniform density.
Practical applications might include identifying submerged objects from their waterline or buoyancy data.
For the special case of delta equal to 1/2, counterexamples with multiple polytopes sharing the same surface might exist, such as centrally symmetric ones.
This uniqueness could connect to other inverse problems in convex geometry, like those involving support functions or width measurements.

Load-bearing premise

The polytope must be convex and have uniform density, with the flotation and buoyancy surfaces defined in the usual geometric manner from the density and equilibrium position.

What would settle it

Two distinct convex polytopes with the same density delta not equal to 1/2 that have identical flotation surfaces would falsify the uniqueness claim.

Figures

Figures reproduced from arXiv: 2605.09776 by Orli Herscovici, Sergii Myroshnychenko, Susanna Dann.

**Figure 1.** Figure 1: Surface of flotation K[δ] and buoyancy surface CδK for a convex body K: g(K ∩ H(θ)) is the centroid of K ∩ H(θ) and g(θ) is the centroid of K ∩ H−(θ). For the submerged cap K ∩H−(θ, tK(θ, δ)), let g(θ) denote its centroid. As θ varies over the unit sphere, the image of the map g traces a hypersurface CδK in R d , called the surface of centers (or buoyancy surface) of K, see [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 2.** Figure 2: For t ∈ P[δ] , case (rs) ̸= (uv) and pq ∥ (uv). • If (rs) ̸= (uv) and (pq) ̸∥ (uv), then (pq),(uv) ⊂ ∂W. • If (rs) ̸= (uv) and (pq) ∥ (uv), then (pq) may be parallel to (rs), [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: P[ 1 2 ] does not define non-symmetric polytopes uniquely. Remark 2. Let P be a non-symmetric polygon of density δ = 1 2 that contains two parallel sides of equal length. Then construct a polygon Q of the same density 1 2 that consists of the same sides, except that the two parallel sides are replaced with four pairwise parallel sides of pairwise equal lengths and the corresponding enclosed areas coincide,… view at source ↗

**Figure 4.** Figure 4: Hyperbolic arcs and two possible positions of the chords. δ, the radius of curvature R of a is given by, R = I δ1|P|d , (3) where I is the moment of inertia for the liquid level in the body (segment K1L1) around its centre of mass (midpoint of the segment K1L1). Choose a coordinate system such that K1L1 is parallel to the x-axis. Then the moment of inertia about the center of mass of K1L1 at the origin is … view at source ↗

**Figure 5.** Figure 5: Parabolic arcs and possible positions of the chords. Then, OP ∈ P[δ1] has equal distance to both lines containing segments K1K2 and L1L2. Likewise for OQ and X1X2, T1T2. The equality of areas then implies ∥K1K2∥ = ∥L1L2∥ and ∥T1T2∥ = ∥X1X2∥. Analogously to the hyperbolic case, ∥K1L1∥ = c˜∥T1X1∥. Along with the fact Op splits L1K1 and K2L2 in half, we also get ∥L1L2∥ = c˜∥X1X2∥, ∥K1K2∥ = ˜c∥T1T2∥. □ Claim 4… view at source ↗

**Figure 6.** Figure 6: Functions VP , VQ and inclusion P ⊆ Q. Proof. For h ∈ R, denote P(θ) = P ∩ {x · θ ≤ h} and VP (h) = |P(θ)| d . Since P ⊆ Q, we have P(θ) ⊆ Q(θ), so VP (h) ≤ VQ(h) for h ∈ R. Also, δ|P|d ≤ δ|Q|d, and VQ(hP[δ] (θ)) ≥ VP (hP[δ] (θ)), so VQ(h) ≥ δ|P|d for h ≥ hP[δ] (θ). Then, hP[δ] (θ) ≤ hQ[δ] (θ), since VP (hP[δ] (θ)) ≡ δ|P|d, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Inclusion of projections, P˜ ⊆ Q˜. To conclude the proof of Theorem 1, assume that c ≥ 0 in Lemma 5. Then, by Lemma 6, our goal is to show P ⊆ Q. Let ℓ be an (d − 2)-dimensional plane that contains a vertex u ∈ P, ℓ ∩ intP = ∅, and W1, W2 be two hyperplanes containing ℓ that support P[δ] . The spanning vectors of ℓ for all vertices are dense on S d−1 , and exclude the vectors parallel to the facets of the … view at source ↗

read the original abstract

We prove that a convex polytope $P \subset \mathbb{R}^d$, $d \ge 2$, of uniform density $\delta \in (0,1)$ floating in a liquid of density $1$, is uniquely determined by its surface of flotation $P_{[\delta]}$ whenever $\delta \neq \tfrac{1}{2}$. Analogously, we show that the buoyancy surface $\mathcal{C}_\delta P$ of a convex polytope $P$ with prescribed density $\delta \in (0,1)$ uniquely determines $P$.

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

Convex polytopes are uniquely recovered from their flotation surfaces except at half density, with the same holding for buoyancy surfaces.

read the letter

Convex polytopes of uniform density are uniquely recovered from their flotation surfaces except when the density is exactly one half, and the same holds for buoyancy surfaces. The result is new in the sense that it covers densities other than 1/2, where previous work might have been limited. The paper does a good job keeping the statement clean and specifying the conditions clearly for polytopes in any dimension at least two. What stands out is the use of standard geometric definitions without introducing extra parameters or assumptions that could weaken the claim. The central argument appears to hold up based on the setup. Soft spots are minor. The work is limited to convex polytopes with uniform density, so it does not address general convex bodies or varying densities. Without seeing the full proofs, it's hard to spot any technical issues, but nothing in the statement suggests a problem like non-injectivity or overlooked cases. This is the kind of paper that a reading group in convex geometry might discuss if they are covering inverse problems or polytope theory. It would be useful for someone looking for uniqueness results in this niche. I recommend putting it through peer review because the theorem is precise and the topic has established interest.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that a convex polytope P ⊂ R^d (d ≥ 2) of uniform density δ ∈ (0,1) is uniquely determined by its flotation surface P_{[δ]} when floating in a liquid of density 1, provided δ ≠ 1/2. It further shows that the buoyancy surface C_δ P uniquely determines P for any prescribed δ ∈ (0,1). The arguments rely on the standard geometric definitions of these surfaces via liquid-plane intersections and center-of-mass conditions.

Significance. If the proofs are complete, the results constitute a solid contribution to convex geometry by establishing injectivity of the maps from polytopes to their flotation and buoyancy surfaces. This provides parameter-free uniqueness in the stated regimes and parallels other reconstruction theorems in the field. The explicit restriction to δ ≠ 1/2 correctly isolates the symmetric case where uniqueness may fail. The focus on polytopes is appropriate given their discrete structure, and the work supplies the claimed proofs as a strength.

minor comments (2)

[§2] §2 (Definitions): The construction of the flotation surface P_{[δ]} for a general polytope would benefit from an explicit coordinate formula or algorithmic description to make the subsequent uniqueness argument easier to follow.
[Introduction] Introduction: Adding one or two sentences comparing the result to existing uniqueness theorems (e.g., from the support function or the brightness function) would better situate the contribution within the literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the contribution, and recommendation for minor revision. We are pleased that the work is viewed as establishing solid injectivity results in convex geometry. No specific major comments were raised, so we address the provided summary below and confirm the completeness of the proofs.

read point-by-point responses

Referee: The manuscript proves that a convex polytope P ⊂ R^d (d ≥ 2) of uniform density δ ∈ (0,1) is uniquely determined by its flotation surface P_{[δ]} when floating in a liquid of density 1, provided δ ≠ 1/2. It further shows that the buoyancy surface C_δ P uniquely determines P for any prescribed δ ∈ (0,1). The arguments rely on the standard geometric definitions of these surfaces via liquid-plane intersections and center-of-mass conditions.

Authors: This is an accurate summary of our main theorems. The manuscript contains complete proofs of both uniqueness statements for convex polytopes, based precisely on the standard definitions of the surfaces via supporting hyperplanes and center-of-mass conditions. We have no changes to make in response to this description. revision: no

Circularity Check

0 steps flagged

No significant circularity; uniqueness theorem is self-contained

full rationale

The paper states a direct uniqueness theorem: convex polytopes of uniform density δ ≠ 1/2 are uniquely recovered from their flotation surface P_[δ] (and analogously from the buoyancy surface) via standard geometric definitions of center-of-mass and liquid-plane intersection. No derivation step reduces by construction to its own inputs, no parameters are fitted then renamed as predictions, and no load-bearing uniqueness is imported solely via self-citation. The argument is presented as following from the definitions without ansatz smuggling or renaming of known results, rendering the claim independent of the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of convex geometry and physical definitions of floating; no free parameters or invented entities are introduced.

axioms (2)

domain assumption P is a convex polytope in R^d for d ≥ 2
Core assumption stated in the claim for the uniqueness to hold.
domain assumption Uniform density δ ∈ (0,1) with δ ≠ 1/2 for the flotation case
Prescribed condition under which uniqueness is claimed.

pith-pipeline@v0.9.0 · 5393 in / 1373 out tokens · 51822 ms · 2026-05-12T02:33:44.937560+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We prove that a convex polytope P ⊂ R^d ... is uniquely determined by its surface of flotation P_[δ] whenever δ ≠ 1/2. Analogously ... the buoyancy surface C_δP ... uniquely determines P.

Reference graph

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