arxiv: 2604.06553 · v1 · submitted 2026-04-08 · 🧮 math.MG

Recognition: no theorem link

A characterization of the sphere in terms of the stereographic projection

Efr\'en Morales-Amaya

Pith reviewed 2026-05-10 18:33 UTC · model grok-4.3

classification 🧮 math.MG

keywords convex bodyspherestereographic projectionaxial symmetryhomothetysectioncharacterizationEuclidean space

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The pith

A convex body in three-space is a sphere precisely when the cones from one boundary point to its plane sections are axially symmetric and the symmetry rotations send each section to a homothetic copy of its stereographic image.

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 of the sphere among convex bodies in Euclidean 3-space. It assumes a unique supporting plane at one boundary point S and considers stereographic projection from another boundary point N onto that plane. The key result is that K is a sphere if and only if every cone spanned by N and a section of K is unchanged by a 180-degree rotation around its axis, and moreover that same rotation takes the section to a figure homothetic to the projection of the section, with the homothety centered at N. This links the classical fact that stereographic projection sends circles to circles on the sphere to these symmetry conditions. A supporting characterization of the circle via a stereographic property is used in the proof.

Core claim

Let K be a convex body in E^3 with boundary points N ≠ S, where the supporting plane Π_S at S is unique. Define the stereographic projection Ψ from bd K ∖ {N} to Π_S by intersecting the line through N and x with Π_S. Then K is a sphere if and only if, for every section K_Γ of K, the cone with vertex N and base K_Γ is invariant under rotation by angle π (axial symmetry), and this rotation maps K_Γ to a set that is homothetic to Ψ(K_Γ) with center of homothety at N.

What carries the argument

The axial symmetry of cones from N to sections of K combined with the condition that the π-rotation maps sections to homothetic images of their stereographic projections from N.

If this is right

The sphere fulfills both properties because its stereographic projection preserves circles and the relevant rotations align with the geometry.
Every section of such a K must itself be a circle, following from the stereographic property of circles.
This gives a purely geometric test for the sphere using only symmetry and homothety in projections, without explicit curvature computations.
The result extends the classical circle-to-circle mapping property to a full characterization of the sphere.

Where Pith is reading between the lines

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

One could test whether similar projection-based symmetries characterize spheres in higher dimensions or on other manifolds.
This approach might connect to other known characterizations of the sphere, such as those involving constant brightness or floating bodies.
If the conditions hold, it implies that the body has no flat parts or irregularities that would break the homothety mapping.

Load-bearing premise

The supporting plane at S must be unique; otherwise the stereographic projection is not a single-valued map onto a fixed plane and the definitions break down.

What would settle it

A convex body other than a sphere, such as a non-spherical ellipsoid, that still has all its N-cones axially symmetric under π-rotations and satisfies the homothety mapping of sections to projections would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.06553 by Efr\'en Morales-Amaya.

**Figure 2.** Figure 2: The Stereographic Property characterizes the circle. Case N ̸= N′ , S ̸= S ′ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The Stereographic Property characterizes the circle. Case N = N′ , S = S ′ [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Let $K$ be a convex body in the 3-dimensional Euclidian space $\mathbb{E}^3$ and let $N,S$ in the boubdary bd$K$ of $K$, $N\not=S$. Suppose that the support plane $\Pi_S$ of $K$ at $S$ is unique. For every point $x$ in bd$ K$, different than $N$, we define the stereographic projection $\Psi:\textrm{bd}K\backslash \{N\} \rightarrow \Pi_S$ of $x$ onto $\Pi_S$ as the point $y:=L(N,x)\cap \Pi_S$. It is a well known property of the sphere $\mathbb{S}^2$ in $\mathbb{E}^3$ that the stereographic projection maps circles onto circles (see \cite{Hilbert} pag. 248). In this work we investigate what geometric elements determines that this property is fulfilled. Here we demonstrate that the following two properties of a convex body $K\subset \mathbb{E}^3$ in terms of the stereographic projection characterize the sphere in $\mathbb{E}^3$: (1) The cones defined by the sections of $K$ and the point $N$ are axially symmetric (that is, they are invariant under a rotation by an angle of $\pi$). (2) given a section $K_\Gamma$ of $K$, the rotation that leaves the cone defined by $K_\Gamma$ and $N$ invariant is such that it maps $K_\Gamma$ into a homothetic figure to $\Psi(K_\Gamma)$ by a homothety with center of homothety at $N$. An important element in the proof of the main theorem of this work is a cha\-racterization of the circle based on a geometric property, which will be called the stereographic property. It is worth highlighting that the stereographic projection defined on the sphere maps circles onto circles is intimately linked to the conditions (1) and (2) and the stereographic property of the circle.

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 characterizes the sphere among convex bodies by axial symmetry of projection cones plus a homothety condition under the preserving rotation.

read the letter

Colleague, the main thing to know is that this work shows two properties of the stereographic projection from a boundary point N on a convex body K force K to be a sphere, assuming a unique support plane at S. The properties are axial symmetry of the cones from N to plane sections under π-rotations, and the condition that the rotation maps each section to a homothetic copy of its stereographic image with center at N. This is presented as a new characterization that recovers the classical circle-to-circle behavior of the projection on spheres. The paper also supplies an auxiliary characterization of the circle via a stereographic property that serves as the bridge in the argument. This specific pairing of cone symmetry and homothety is not in the cited prior literature, and the approach stays inside standard Euclidean convex geometry without fitted parameters or circular definitions. It does a clean job linking the symmetries directly to the shape. The uniqueness of the support plane is stated up front as necessary for the projection to be well-defined, which is fair. The central argument appears to hold without internal contradictions or unjustified steps, though the full proof details would need checking to confirm the auxiliary circle result follows strictly from the two conditions and that no hidden assumption creeps into the homothety mapping. This is for readers in metric or convex geometry who work on characterizations of standard bodies via projections and symmetries. Someone who knows the classical facts from Hilbert and likes minimal geometric conditions would get value from it. The paper shows honest engagement with the literature and clear thinking on its own terms, so it deserves a serious referee. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to characterize the sphere among convex bodies K in E^3 (with unique support plane at boundary point S) via two properties of the stereographic projection Ψ from another boundary point N: (1) axial symmetry (invariance under π-rotations) of the cones formed by N and plane sections K_Γ of K, and (2) the condition that these rotations map K_Γ to a figure homothetic to Ψ(K_Γ) with homothety center at N. The argument proceeds by establishing an auxiliary 'stereographic property' characterization of circles that follows from (1) and (2), which in turn forces K to be a sphere.

Significance. If the derivation holds, the result supplies a new, axiomatically grounded characterization of the sphere that directly ties the classical circle-preserving property of stereographic projection to symmetry and homothety conditions on convex bodies. It is parameter-free, relies only on standard Euclidean convexity and the explicit uniqueness assumption for the support plane, and avoids self-referential or fitted quantities. This could be of interest in convex geometry for understanding invariants that distinguish quadrics.

minor comments (3)

Abstract contains multiple typographical errors: 'boubdary' for 'boundary', 'Euclidian' for 'Euclidean', and a hyphenation artifact in 'cha-racterization'. These should be corrected for readability.
The notation K_Γ for sections and the precise definition of the cones are introduced in the abstract without a preliminary sentence clarifying the plane Γ; a short definitional sentence at the start of the main text would improve accessibility.
The auxiliary characterization of the circle is described as 'important' but its statement is not isolated as a numbered lemma or proposition; doing so would make the logical structure of the main theorem easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and for recognizing the potential interest of this characterization in convex geometry. The recommendation for minor revision is noted, and we will incorporate any editorial or presentational improvements in the revised manuscript. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a characterization theorem: among convex bodies K in E^3 with unique support plane at S, the two stated properties (axial symmetry of cones from N under π-rotations, and the rotation mapping K_Γ to a homothetic copy of Ψ(K_Γ) centered at N) force K to be a sphere. This rests on an auxiliary geometric characterization of circles via the stereographic property, derived from the given conditions using standard Euclidean axioms and convexity. No equations reduce the conclusion to the inputs by construction, no parameters are fitted and renamed as predictions, and the sole external citation (Hilbert) is to a classical fact about spheres, not a self-citation chain. The derivation is self-contained and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on the standard axioms of Euclidean 3-space and convexity of K, plus the uniqueness of the support plane at S; no free parameters or invented entities are introduced.

axioms (2)

domain assumption K is a convex body in Euclidean 3-space E^3
Stated at the beginning of the setup.
domain assumption The support plane at S is unique
Required for the stereographic projection to be well-defined onto Π_S.

pith-pipeline@v0.9.0 · 5692 in / 1423 out tokens · 38523 ms · 2026-05-10T18:33:11.024875+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 3 canonical work pages

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