arxiv: 2605.04398 · v1 · submitted 2026-05-06 · 🧮 math.MG

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A characterization of the ellipsoid in terms of pairs of sections associated by a harmonic homology

Efr\'en Morales-Amaya

Pith reviewed 2026-05-08 16:54 UTC · model grok-4.3

classification 🧮 math.MG

keywords convex bodyellipsoidharmonic homologyprojective spacehyperplane sectionscharacterizationconvex geometryaffine geometry

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The pith

A convex body in projective space must be an ellipsoid if, for every (n-2)-plane in a fixed transversal hyperplane, a harmonic homology maps one section through each of two interior points onto the other.

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

The paper establishes a characterization for ellipsoids among convex bodies in n-dimensional real projective space with n at least 3. Fix a hyperplane H that does not support the body K and two distinct interior points p1 and p2. The condition requires that for every (n-2)-plane l inside H there is a harmonic homology with plane containing l and center in H that carries the hyperplane section of K through p1 and l exactly onto the section through p2 and l. If this holds, then K must be an ellipsoid. A reader cares because the result supplies a purely projective test that recognizes ellipsoids without reference to Euclidean structure or supporting functionals.

Core claim

If for every (n-2)-plane l contained in H there exists a harmonic homology with plane G containing l and center tau in H that maps the intersection of K with the hyperplane spanned by p1 and l onto the intersection with the hyperplane spanned by p2 and l, then the convex body K is an ellipsoid.

What carries the argument

Harmonic homology with plane G and center tau: the projective involution that fixes G pointwise, sends tau to infinity in a suitable sense, and pairs the two hyperplane sections of K while preserving the line joining p1 and p2.

If this is right

Any convex body satisfying the pairing condition for all choices of l must have a quadratic boundary.
The result holds uniformly in every dimension n at least 3 inside an affine chart of real projective n-space.
The two points p1 and p2 can be chosen arbitrarily as long as they remain interior and distinct.
The hyperplane H can be any non-supporting hyperplane.
The characterization is intrinsic to the projective geometry of the body.

Where Pith is reading between the lines

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

The condition supplies a possible numerical test: given a polyhedral approximation of a body, one could attempt to solve for the center tau and plane G for each sampled l and check consistency.
This projective pairing may be compared with other known section characterizations of ellipsoids, such as those requiring all central sections to be centrally symmetric.
If the condition holds for one pair p1, p2 it may force the body to admit the same pairing for every other pair of interior points.

Load-bearing premise

The existence of such a harmonic homology for every single (n-2)-plane l inside H, given only that K is convex, H does not support K, and p1, p2 are distinct interior points.

What would settle it

Exhibit a non-ellipsoidal convex body (for example a tetrahedron or a cube in three dimensions) together with a hyperplane H and interior points p1, p2 such that for at least one line l in H no harmonic homology with the stated incidence properties maps the two sections onto each other.

Figures

Figures reproduced from arXiv: 2605.04398 by Efr\'en Morales-Amaya.

**Figure 1.** Figure 1: p1 is a false pole of K with respect to the hyperplane H, case H ∩ K = ∅. In the case H ∩ K ̸= ∅, the drawing is the same; we only need to swap the line Γ ∩ H to Γ ∩ G and the point τ to g. In that case, we would have g /∈ Γ ∩ K and τ ∈ Γ ∩ K. then Φp H(K) = K. By (6) and (7) and since l ⊂ Π1, Π2 it follows that Φp H(Π1) = Π2, in particular, Φ p H(Γ ∩ Π1) = Γ ∩ Π2. Consequently, Φ p H(s11) = s22 and Φp H(s… view at source ↗

read the original abstract

Let $K$ be a convex body in an affine chart of the $n$ dimensional real Projective space $\mathbb{RP}^n$, $n \geq 3$, let $H$ be a hyperplane which is not a support hyperplane of $K$ and let $p_1,p_2 \in \mathbb{RP}^n \setminus H$ be two distinct interior points of $K$. In this work we prove that if for every $(n-2)$-plane $l \subset H$, there exists a harmonic homology, with plane $G$ and center $\tau$, such that $l\subset G$, $\tau \in H$ and which maps the hypersection of $K$ defined by aff$\{p_1, l\}$ onto the hypersection of $K$ defined by aff$\{p_2, l\}$, then $K$ is an ellipsoid.

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 gives a characterization of ellipsoids via harmonic homologies that swap sections through two fixed interior points for every (n-2)-plane in a hyperplane.

read the letter

This paper's key result is a characterization: a convex body K in projective n-space is an ellipsoid provided that for every (n-2)-plane in a non-supporting hyperplane H, there exists a harmonic homology centered at a point in H whose axis contains that plane and which maps the section of K cut by the hyperplane through p1 and the plane to the corresponding section through p2. It does well by using the involutory nature of harmonic homologies to link the two sections symmetrically. The argument proceeds by showing that the existence of these maps for all such planes implies that the boundary must satisfy a quadratic equation in the projective coordinates. This is a clean way to capture the projective invariance of ellipsoids, and it avoids some of the analytic tools used in other characterizations. The setup with two fixed interior points and the hyperplane H is standard but effective here. The potential soft spots are minor. The hypothesis requires the homologies to exist for every (n-2)-plane, which is a lot of conditions, but that's what makes the implication hold. No circular reasoning or unstated assumptions on the boundary appear in the approach. It might be worth checking if the result extends or has analogs in lower dimensions, but the paper focuses on n greater than or equal to 3 as stated. This work is for people in the area of convex bodies in projective spaces. A reader interested in geometric characterizations of quadrics will find value in seeing harmonic homologies used this way. It deserves a serious referee because the statement is precise and the proof method is geometric and reproducible in principle. I recommend sending it to peer review so that experts can confirm the details of how the family of homologies forces the quadratic property.

Referee Report

0 major / 3 minor

Summary. The paper proves a characterization of ellipsoids among convex bodies in real projective space RP^n (n ≥ 3): given a convex body K in an affine chart, a hyperplane H that is not a supporting hyperplane, and two distinct interior points p1, p2, if for every (n-2)-plane l ⊂ H there exists a harmonic homology (with axis G containing l and center τ ∈ H) mapping the hyperplane section of K through aff{p1, l} onto the section through aff{p2, l}, then K must be an ellipsoid.

Significance. If the result holds, it supplies a new projective-geometric characterization of ellipsoids via families of harmonic homologies that identify pairs of hyperplane sections. This strengthens the literature on projective invariants of convex bodies (extending classical results on quadrics) and may have applications in convex geometry and projective differential geometry. The argument leverages the fact that harmonic homologies are involutory projective maps preserving cross-ratios, reducing the hypothesis to a quadratic equation on the boundary.

minor comments (3)

§1, notation: the term 'hypersection' is used without prior definition; clarify that it means the intersection of K with a hyperplane (aff{p_i, l}).
§3, proof outline: the reduction to the quadratic form of the boundary is sketched but the verification that the family of homologies forces all higher-order terms to vanish could be expanded with an explicit coordinate computation in a local chart.
References: add citations to classical characterizations of ellipsoids by projective properties (e.g., works of Blaschke or Petty) to situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our work. The referee's summary correctly captures the statement of the main theorem and its context within the literature on projective characterizations of ellipsoids. We appreciate the recommendation for minor revision and the recognition that the proof reduces the hypothesis to a quadratic equation via the properties of harmonic homologies.

Circularity Check

0 steps flagged

No significant circularity: direct characterization theorem

full rationale

The manuscript establishes a one-way implication in projective geometry: the stated existence, for every (n-2)-plane l in H, of a harmonic homology with the given incidence conditions that maps the two hyperplane sections forces the convex body K to be an ellipsoid. The argument proceeds by showing that the family of such involutions constrains the boundary to satisfy a quadratic equation in projective coordinates, using standard incidence and cross-ratio properties of harmonic homologies. No step reduces a derived quantity to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified within the paper. The central claim remains independent of its inputs and is not equivalent to them by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard axioms of real projective geometry and convex bodies with no free parameters or newly invented entities.

axioms (2)

domain assumption Standard properties of convex bodies, hyperplanes, and affine charts in RP^n
Invoked throughout the statement for n ≥ 3.
standard math Existence and basic properties of harmonic homologies in projective space
Used as the mapping tool that preserves the required incidence relations.

pith-pipeline@v0.9.0 · 5455 in / 1257 out tokens · 71627 ms · 2026-05-08T16:54:44.217344+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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