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arxiv: 2501.06337 · v2 · pith:KDWZLREOnew · submitted 2025-01-10 · 🧮 math.MG

On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry

M. Angeles Alfonseca , B. Zawalski This is my paper

Pith reviewed 2026-05-23 05:46 UTC · model grok-4.3

classification 🧮 math.MG
keywords Bezdek conjectureconvex bodiesreflection symmetryplanar sectionshigh-dimensional geometryellipsoidsbodies of revolutionaffine transformations
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The pith

If central sections of a convex body have reflection axes whose complementary invariant subspaces are all parallel to one fixed hyperplane, then the body is an ellipsoid or a body of revolution in every dimension n at least 3.

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

Bezdek conjectured that convex bodies in three dimensions are ellipsoids or bodies of revolution exactly when every planar section has a reflection axis. This paper proves the statement in all dimensions n at least 3, but adds the requirement that the complementary invariant subspaces of those axes must be parallel to a single fixed hyperplane. The argument works in both the orthogonal and affine categories. A sympathetic reader would care because the alignment condition lets the higher-dimensional case reduce to the known three-dimensional case.

Core claim

Assuming every section through a fixed point has an axis of reflection and that the complementary invariant subspaces are all parallel to a fixed hyperplane, any convex body in R^n for n greater than or equal to 3 must be an ellipsoid or a body of revolution. The same conclusion holds in the affine setting.

What carries the argument

The alignment condition that all complementary invariant subspaces are parallel to a fixed hyperplane, which reduces the problem in dimension n to the three-dimensional case.

If this is right

  • The characterization of ellipsoids and bodies of revolution by reflective central sections extends from three dimensions to arbitrary dimensions n at least 3.
  • The result holds in both the orthogonal and affine settings.
  • The shape of the body is completely determined by the reflection axes of its central sections once the alignment condition is satisfied.
  • Any potential counterexample to the original conjecture without the alignment must have non-parallel complementary invariant subspaces.

Where Pith is reading between the lines

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

  • Removing or weakening the alignment condition would resolve the full original conjecture in all dimensions.
  • The common parallel direction in the symmetry data may allow a projection or slicing argument that reduces dimension while preserving the reflection property.
  • Analogous alignment assumptions might characterize other families of convex bodies defined by section properties.
  • A search for four-dimensional examples with deliberately misaligned subspaces could test whether the alignment is necessary.

Load-bearing premise

The complementary invariant subspaces of the reflection axes must all be parallel to one fixed hyperplane.

What would settle it

A convex body in dimension 4 or higher whose central sections all have reflection axes, whose complementary invariant subspaces are not all parallel to one hyperplane, and which is neither an ellipsoid nor a body of revolution.

Figures

Figures reproduced from arXiv: 2501.06337 by B. Zawalski, M. Angeles Alfonseca.

Figure 4.4
Figure 4.4. Figure 4.4: Reducing the problem of a (−1)-aligned quasi-center to a many-body problem in codimension 1 Theorem 2 (i)], where n = 3 and k = 1. However, this concept of alignment is somewhat dual to the one proposed in this paper. In [3], the subrepresentation assumed to be trivial is (V +o)∩ H, while in our paper it is the quotient representation. The dual concept emerged in the context of a centrally symmetric body… view at source ↗
Figure 6.12
Figure 6.12. Figure 6.12: Notations used in the proof of Claim 6.11 which implies that 1 − cos(α1 − α2) 1 − cos β is bounded, and, in consequence, that sin2 (α1 − α2) sin2 β = 1 − cos(α1 − α2) 1 − cos β · 1 + cos(α1 − α2) 1 + cos β is bounded. Now, let q ∈ T ⊥ be any point in the orthogonal complement of T and let P12 = ⟨n1, n2⟩ + q be the 2- dimensional affine plane parallel to both n1 and n2 and passing through q (see fig. 6.1… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: The implication graph representing relations between different variants of Question 3.19 In particular, this would mean that the two families of representations from Definition 2.9 are undoubtedly the most interesting ones. But Conjecture 7.2 has much more far-reaching consequences. Without loss of generality, we may assume that G ≤ O(R n). Denote by V ⊥ i the hyperaxis of ki-revolution for Bi and suppos… view at source ↗
read the original abstract

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in $\mathbb R^3$, ellipsoids and bodies of revolution are characterized by the fact that all their planar sections have an axis of reflection. We prove Bezdek's conjecture in arbitrary dimension $n\geq 3$, assuming only that sections passing through a fixed point have an axis of reflection, provided that the complementary invariant subspaces are all parallel to a fixed hyperplane. The result is proven in both orthogonal and affine settings.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves Bezdek's 1999 conjecture in dimensions n≥3: convex bodies in R^n whose planar sections through a fixed point possess an axis of reflection are ellipsoids or bodies of revolution, provided the complementary invariant subspaces are all parallel to a fixed hyperplane. The result is established in both the orthogonal and affine settings under this alignment hypothesis.

Significance. If the derivation holds, the work supplies the first higher-dimensional extension of Bezdek's characterization by isolating a geometrically natural alignment condition on the reflection symmetries that makes the proof feasible. The simultaneous treatment of the orthogonal and affine cases, together with the explicit statement of the extra hypothesis, strengthens the contribution relative to the original three-dimensional conjecture.

minor comments (3)
  1. The statement of the alignment condition in the introduction and in the main theorem should be cross-referenced with a precise definition of 'complementary invariant subspaces' (currently introduced only in §2).
  2. Notation for the fixed hyperplane and the common direction of the subspaces is introduced inconsistently between the orthogonal and affine sections; a single global symbol would improve readability.
  3. The proof of the affine case (§4) invokes a reduction to the orthogonal case without spelling out the change-of-basis argument; adding one diagram or a short paragraph would clarify the passage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its significance as the first higher-dimensional extension of Bezdek's conjecture under a geometrically natural alignment hypothesis, and the recommendation for minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is a conditional mathematical proof

full rationale

The paper states an explicit conditional result: Bezdek's conjecture is proved in dimension n≥3 only under the additional hypothesis that complementary invariant subspaces are parallel to a fixed hyperplane. This premise is presented upfront as an extra restriction beyond the 1999 conjecture, and the work is framed as holding in both orthogonal and affine settings under that restriction. No equations, definitions, or steps reduce by construction to fitted inputs, self-referential definitions, or load-bearing self-citations. The central claim does not rename known results or smuggle ansatzes via prior work; it is a direct proof under stated assumptions and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are identifiable. The work relies on background facts from convex geometry.

axioms (2)
  • standard math Convex bodies are compact convex sets in Euclidean or affine space
    Standard definition invoked by any statement about sections of convex bodies.
  • domain assumption Existence and properties of reflection symmetries in planar sections
    Core hypothesis of Bezdek's conjecture and the paper's extension.

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