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REVIEW 4 major objections 4 minor 12 references

This paper constructs an explicit four-dimensional body of constant width by intersecting all unit balls centered on a carefully chosen subset of a Reuleaux 4-simplex, a direct analogue of the second Meissner body.

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2026-08-04 09:10 UTC pith:CL6Y4PR2

load-bearing objection The construction is genuinely appealing and the shadow body is a nice find, but the central theorem rests on a summarized case analysis that has at least one false written step; it needs a complete proof before I'd trust it. the 4 major comments →

arxiv 2510.16672 v3 pith:CL6Y4PR2 submitted 2025-10-19 math.MG

A four-dimensional body of constant width

Marcela G. Mercado-Flores , Miguel Raggi , Edgardo Rold\'an-Pensado This is my paper
classification math.MG MSC 52A2052A38
keywords constant widthReuleaux simplexMeissner bodyspindletetrahedral symmetryorthogonal projectionconvex geometry4-dimensional body
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper attempts to construct, explicitly and in closed geometric form, a four-dimensional body of constant width. Starting from the unit Reuleaux 4-simplex, the authors select a generating set M0 lying in the simplex's 2-skeleton and define M as the intersection of all unit balls centered at points of M0. They argue that M has diameter 1, that every boundary point outside M0 lies on exactly one supporting unit sphere, and that consequently M has constant width, with smooth boundary away from M0 and every diameter ending in M0. This makes M a natural four-dimensional analogue of the second Meissner body, with tetrahedral symmetry. Projecting M onto a 3-plane yields a 3D constant-width body with six elliptical edges and a numerical volume close to the Meissner bodies, which the authors propose as a candidate in the search for the minimal-volume tetrahedrally symmetric body of constant width.

Core claim

Theorem 1.1 is the central claim: if R is a unit Reuleaux 4-simplex with vertices A, B, C, D, E, there exists a set M0 containing the vertices and contained in the 2-skeleton of R such that M, the intersection of all unit balls centered at M0, is a body of constant width. M0 is fixed by the tetrahedral symmetries of the base ABCD, lies on the boundary of M, and is the unique non-smooth stratum: every point of ∂M \ M0 is smooth and every diameter of M has at least one endpoint in M0. The construction mirrors the second Meissner body by preserving 2-faces incident to the apex E, and the proof reduces to showing the diameter of M0 is exactly 1 and that no point outside M0 is at unit distance fr

What carries the argument

M0 is the union of six 2-dimensional patches F_{X,Y}, one for each pair of base vertices, where each F_{X,Y} is the part of the spherical 2-face F_{X,Y,E} lying on the E-side of the plane span({X,Y}). The body is M = ∩_{P∈M0} B(P,1). The proof machinery consists of Lemma 4.1, which identifies the farthest point from a given point on a sphere; the spindle lemma, Lemma 4.3, which characterizes the intersection of a spindle generated by a subset S with a unit sphere centered at x∈S as a half-space cap; a diameter-1 computation for M0; and a uniqueness proof that reduces to finitely many linear-programming checks. The spindle description converts the continuous intersection problem into a finite

Load-bearing premise

The load-bearing assumption is the unshown computational step in Section 4.2.2: for all but one of the listed relative-position configurations, minimizing a linear functional over a polyhedral region shows that its intersection with M is either empty or a single vertex, and the paper explicitly says these computations are summarized rather than displayed; if any of those minimization claims were false, the uniqueness property (and hence smoothness, diameter-endpoint, and cons

What would settle it

Find two distinct points Q, R in M0 and a point P in ∂M \ M0 with ||P−Q|| = ||P−R|| = 1. Concretely, for the residual configuration 3(a), solve the two distance equations ||P−Q(s)|| = 1 and ||P−R(t)|| = 1 together with the stationarity equations αB cos s − αA sin s = 0 and αB cos t − αC sin t = 0 for s,t in (0, π/2) and nonnegative coefficients αA, αB, αC, αD not all zero; the paper claims the only solution forces αD = 1 and all other coefficients zero, so any alternative nonnegative solution with P outside M0 would refute the theorem. Independently, running the listed linear programs for the

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If Theorem 1.1 is correct, there is an explicit, symmetrically described 4D constant-width body whose boundary is smooth except for the 2-complex M0, extending the Meissner construction to one higher dimension.
  • The orthogonal projection onto the base hyperplane is a 3D body of constant width with tetrahedral symmetry and six elliptical edges of eccentricity 1/√2; it is not a Minkowski average of Meissner or pea bodies, since those averages are smooth where this body has non-smooth edges.
  • The shadow's numerical volume, about 0.420, is only slightly above the Meissner volume (approximately 0.419) and far below the width-1 ball (approximately 0.524), making it a new concrete candidate for minimal volume among tetrahedrally symmetric constant-width bodies in R3.
  • Because every diameter has an endpoint in M0, the 2-complex M0 carries all extremal information about the body, a structural encoding of a constant-width body that is new in higher dimensions.

Where Pith is reading between the lines

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

  • A natural follow-up is to turn the summarized linear-programming verifications into an explicit, machine-checkable certificate; that would make the construction independently verifiable and potentially automated.
  • The same spindle-plus-linear-programming scheme could in principle be adapted to Reuleaux simplices in higher dimensions, though the number of cases would grow; the method itself is dimension-agnostic rather than relying on 4D-specific geometry.
  • The elliptical-edge shadow suggests that orthogonal projections of higher-dimensional constant-width bodies may be a productive source of new low-dimensional extremal bodies, directly feeding the minimal-volume question for 3D bodies with prescribed symmetry.
  • The golden-ratio placement of the apex E relative to the orthogonal base suggests a hidden arithmetic substructure in the construction; making that explicit might reveal a family of related bodies rather than a single isolated example.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper defines a 4-dimensional convex body M as the intersection of all unit balls centered at points of a set M0 contained in the 2-skeleton of the unit Reuleaux 4-simplex R. M0 is the union of six spherical patches F_{X,Y} associated with pairs of vertices in {A,B,C,D}. The main theorem asserts that M has constant width, M0 lies on ∂M, ∂M is smooth outside M0, and every diameter has an endpoint in M0. The proof proceeds via a diameter analysis of M0 and a uniqueness property for boundary points. The paper also considers the orthogonal projection of M onto the hyperplane spanned by {A,B,C,D}, obtaining a 3-dimensional constant-width body with tetrahedral symmetry and elliptical edges, and reports a numerical volume estimate.

Significance. The construction is explicit, parameter-free, and produces a natural candidate for a 4-dimensional analogue of the second Meissner body, as well as a new 3-dimensional constant-width body via projection. If the proofs are completed, this would be a notable contribution to the sparse collection of explicit constant-width bodies in higher dimensions. The spindle lemma (Lemma 4.3) is a useful geometric tool. However, the proof of the central uniqueness property is currently incomplete and rests on unverified computational summaries.

major comments (4)
  1. [§4.2.2, Configuration 3(a)] The claim that the stationarity system has only the solution α_A=α_B=α_C=0, α_D=±1 is false. For s=t=π/6, set u=(√11−2)/7, α_A=α_C=u√3, α_B=u, α_D=0. Then 7u²+4u−1=0, and both distance equations and both derivative equations are satisfied. The point P is not in M (e.g., ∥P−E∥²≈2.11>1), but the text does not invoke this or any other global constraint to exclude it. Since uniqueness underpins §§4.3–4.5, this is a load-bearing gap.
  2. [§4.2.2] The proof of the uniqueness property for the remaining configurations (1(a)–4(b)) is delegated to an unshown linear-programming computation: 'we focus on the key arguments and summarize the results of the computations.' No code, certificates, or explicit inequalities are provided, so a reader cannot verify the central step. This is not a cosmetic omission; the constant-width claim depends on this uniqueness assertion. Please provide full calculations or a verifiable certificate.
  3. [§4.2.2, Case 3] The assertion that if T is a circle then 'it is necessary that at least two of the vertices of F_{A,B} also lie on the circle T' is not generally true for a spherical triangle contained in a cap and touching its boundary at two points. A proof specific to the geometry of F_{A,B} is required.
  4. [§4.4] The step 'since M is strictly convex and M0 ⊂ ∂M, this unique diametrically opposite point must be P′. Therefore the diameter of M is 1' appears to assume part of what must be proved. The existence of a boundary point at distance 1 from P along the direction Q−P has not been established; it does not follow from the single ball B(Q,1). An additional argument using all ball constraints is needed before invoking Pál's theorem.
minor comments (4)
  1. [§4.2] 'compacity' should be 'compactness'.
  2. [§5] The volume 0.420 is described as 'Numerical modeling'; please state explicitly that this is a numerical estimate, not a proved value.
  3. [§5] The assertion that π(M) is not a Minkowski average of Meissner bodies or peabodies is based on a smoothness heuristic; as written it is an observation, not a theorem. If intended as a rigorous claim, a proof is needed.
  4. [§1.3] The abstract mentions a recent construction of the same shadow from a different 4D body, but this is not discussed in the text; please clarify the relation.

Circularity Check

0 steps flagged

No significant circularity: the construction is explicit and self-contained; constant width follows from the external Pál criterion, not from the definition of M0.

full rationale

M is defined in Section 3 as the intersection of all unit balls centered at an explicitly described subset M0 of the Reuleaux 2-skeleton; M0 is fixed by the simplex geometry (the components F_{X,Y} and arcs S_{X,Y}), so no parameter is fitted to the conclusion. The constant-width proof uses the standard Pál characterization (diameter 1 plus every boundary point at unit distance from another boundary point), cited to [MMO19], and the properties of M0 are proved by direct Euclidean estimates. The citations to the authors' earlier [MRP17] and [MPRRP20] are historical or methodological and are not load-bearing: 'From here we follow the approach taken in [MRP17]' is followed by an external theorem, not by a self-cited uniqueness or construction result. The one notable weakness is a proof gap in §4.2.2, where the uniqueness property rests on an unshown LP case analysis ('we focus on the key arguments and summarize the results of the computations') and where the written treatment of Configuration 3(a) asserts an algebraic conclusion ('the only solutions satisfy α_A = α_B = α_C = 0 and α_D = ±1') without justifying exclusion of other stationary solutions. This is an omitted-proof/correctness concern, not a circular dependency: the uniqueness property is not assumed as input and does not reduce to the definition of M.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The proof leans on two standard convex-geometry theorems and two elementary spherical-distance lemmas with omitted proofs. The most fragile input is the summarized linear-programming verification in §4.2.2; no free parameters are fitted and no new entities are postulated.

axioms (5)
  • standard math Pál's theorem: a compact convex set with diameter 1 and the property that every boundary point is at distance 1 from some point of the set is a body of constant width.
    Invoked in §4.5 to conclude M has constant width from diameter and endpoint properties.
  • standard math Orthogonal projection of a constant-width body onto a subspace is a constant-width body of the same width.
    Used at the start of §5 to assert π(M) is a 3D body of constant width.
  • standard math Lemma 4.1: distance from a point to a sphere increases monotonically along the sphere in the direction of the center, with the proof omitted.
    Load-bearing for the diameter analysis of M0 (§4.1) and for the spindle characterization (§4.2.1); proof is stated to be a straightforward exercise but not shown.
  • standard math Lemma 4.3: for a spindle generated by a relatively open subset of a k-sphere, points at unit distance from a generator x are exactly J_x^+ ∩ S(x,1); proof omitted.
    Used to confine intersection of unit spheres with M to half-spaces in §4.2; proof omitted for brevity.
  • ad hoc to paper Linear-programming assertions: for all but one of the relative-position configurations, minimizing a linear functional over the constructed polyhedral region shows the intersection with M is empty or a single vertex.
    Stated in §4.2.2 without showing the computations; this underpins the uniqueness property and therefore smoothness, diameter endpoints, and constant width.

pith-pipeline@v1.3.0-alltime-deepseek · 12881 in / 11267 out tokens · 88281 ms · 2026-08-04T09:10:48.482387+00:00 · methodology

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read the original abstract

The study of bodies of constant width is a classical subject in convex geometry, with the 3-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $\mathbb R^4$, addressing the challenge of constructing such bodies in higher dimensions. Our method produces a natural analogue of the second Meissner body, by modifying a 4-dimensional Reuleaux simplex. The resulting body possesses tetrahedral symmetry and has a boundary composed of both smooth surfaces and a non-smooth subset of the Reuleaux 4-simplex. Furthermore, we analyze the orthogonal projection of this body onto the 3-dimensional hyperplane of its base. This "shadow" is a 3-dimensional body of constant width with tetrahedral symmetry. It has six elliptical edges and its volume is only slightly larger than that of the Meissner bodies. This body was recently constructed as a projection of a different 4-dimensional body, however the construction presented here is new and gives additional properties.

Figures

Figures reproduced from arXiv: 2510.16672 by Edgardo Rold\'an-Pensado, Marcela G. Mercado-Flores, Miguel Raggi.

Figure 1
Figure 1. Figure 1: The second Meissner body, the non-smooth circular edges are marked in red. 1.3. The shadow of M. Projecting the body M onto the affine hyperplane spanned by the vertices {A, B, C, D} yields a three-dimensional body of constant width, which we refer to as the shadow of M. By construction, this shadow body inherits the symmetries of the regular tetrahedron; that is, it is invariant under any isometry that pe… view at source ↗
Figure 2
Figure 2. Figure 2: The shadow of M. 2. Notation To facilitate readability, we summarize here the notation that will be used throughout the rest of the paper. • B(x, r): Closed ball of radius r centered at the point x. • S(x, r): Sphere of radius r centered at x; that is, the boundary of B(x, r). • span(X): The linear space spanned by a point set X. • pos(X): The positive cone spanned by point set X. • X◦ : The interior of a … view at source ↗
Figure 3
Figure 3. Figure 3: Three views of the face FA,B,E being cut by the plane span({A, B}) in order to produce FA,B, as seen in the 3-space gener￾ated by A, B and E. 3. Definition of the Body We begin with a regular 4-simplex with vertices A, B, C, D, E and side length 1. For convenience, we assume that the vectors {A, B, C, D} form an orthogonal basis, each with norm 1/ √ 2, and that E = φ 2 (A + B + C + D), where φ = 1+√ 5 2 is… view at source ↗
Figure 4
Figure 4. Figure 4: The distance from P to Q is less than the distance from P to Q′ . • Finally, we use the preceding properties to show that M has constant width, which completes the proof of Theorem 1.1. 4.1. M0 is a subset of M. First, note that it is enough to show that the diameter of M0 is 1. Indeed, assume that the diameter is 1. If there is a point P ∈ M0 \M, then by the definition of M, there is a point Q ∈ M0 such t… view at source ↗
Figure 5
Figure 5. Figure 5: The sets R, M and M0 cut by (left) and projected onto (right) the plane span({MA,B, MC,D}). The component FC,D lies on the 2-sphere S with center MA,B and radius √ 3/2, which is contained in the 3-space H = span({C, D, E}). The set {MA,B, C, D} forms an orthogonal basis for H. To apply Lemma 4.1, we project P onto H. The projection PH is given by: PH = P · MA,B ∥MA,B∥ 2 MA,B + P · C ∥C∥ 2 C + P · D ∥D∥ 2 D… view at source ↗
Figure 6
Figure 6. Figure 6: On the left is a spindle Sp1 (S, 1) in R 2 , where S is a subset of circle of radius 1/3. On the right is another example of a spindle Sp1 (S, 1) in R 3 where S is contained in a circle of radius 3/4. complement to H passing through the center of Sk. For any point x ∈ S, let Jx be the affine space spanned by J and x, and let J + x be the closed half-space of Jx bounded by J that does not contain x. Then th… view at source ↗
Figure 7
Figure 7. Figure 7: Projections of the set M onto a 2-dimensional plane. The red regions are the projections of a vertex or a set that generate spindles. The cyan region is the resulting polyhedral region that must contain P. • The closed half-space bounded by the hyperplane through B, C, D, E that does not contain A. By exploiting the symmetries of M0, the remaining analysis can be reduced to the representative configuration… view at source ↗
Figure 8
Figure 8. Figure 8: The elliptical arcs of the shadow of M. area centered at the origin that passes through the points π(X) and π(Y ). A direct calculation shows that each of these six ellipses has an eccentricity of e = 1/ √ 2. For completeness, we provide explicit parametrizations for the six elliptical arcs EX,Y . To express these symmetrically, we choose an orthonormal basis for the subspace HE in which the projected vert… view at source ↗
Figure 9
Figure 9. Figure 9: A representation of the relevant objects in the projection. we know there exists another point R ∈ π(M0) on the opposite arc EC,D such that ∥Q − R∥ = 1. Our initial assumption requires that this point R must lie inside the ball B 3 (P, 1) and in the supporting half-space HQ. However, there is no point in the set HQ ∩ B3 (P, 1) at distance of 1 from Q. This contradicts the existence of R. Since both cases l… view at source ↗

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Reference graph

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