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arxiv: 1906.10439 · v2 · pith:OC4NYDF6new · submitted 2019-06-25 · 🧮 math.MG

Functions with isotropic sections

Ioannis Purnaras , Christos Saroglou This is my paper

Pith reviewed 2026-05-25 16:07 UTC · model grok-4.3

classification 🧮 math.MG
keywords isotropic sectionscosine transformFunk transformgreat spheresconvex hypersurfaceseven functionslocal rigidity
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The pith

Local isotropy on great-sphere sections forces the cosine transform to be affine and the Funk transform constant on open sets of the sphere.

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

The paper proves a local version of an earlier global theorem: for n at least 3, if an even bounded measurable function g on the sphere has the property that its restriction to every great sphere perpendicular to a given open set U is isotropic, then the cosine transform of g restricted to U must be of the form constant plus inner product with a fixed vector, and the Funk transform restricted to U must be a constant. This conclusion holds without forcing g itself to be constant almost everywhere on the union of those perpendicular spheres. The argument relies on a new generalization of a classical differential-geometry fact for convex hypersurfaces that is obtained along the way.

Core claim

If n ≥ 3, g : S^{n-1} → R is even and bounded measurable, U is open in S^{n-1}, and the restriction of g to any great sphere perpendicular to U is isotropic, then C(g)|_U = c + ⟨a, ·⟩ and R(g)|_U = c' for fixed c, c' in R and a in R^n.

What carries the argument

The isotropy condition on all great-sphere sections perpendicular to the open set U, which is used to constrain the cosine and Funk transforms of g on U.

If this is right

  • The forms of the cosine and Funk transforms on U are completely determined by the isotropy assumption on the perpendicular sections.
  • The result holds without any global constancy requirement on g in the perpendicular directions.
  • A new differential-geometric identity for convex hypersurfaces is available for use in related integral-transform problems.

Where Pith is reading between the lines

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

  • The local rigidity may extend to other integral transforms whose kernels involve the same great-sphere geometry.
  • The differential-geometry lemma obtained in the proof could apply to curvature questions on convex bodies beyond the sphere setting.
  • One could test whether dropping evenness of g still yields the same conclusion on U.

Load-bearing premise

Every great-sphere section perpendicular to the open set U is isotropic.

What would settle it

An even bounded measurable g on S^{n-1} together with an open U such that all perpendicular great-sphere sections are isotropic yet C(g) restricted to U fails to be affine or R(g) restricted to U fails to be constant.

read the original abstract

We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if $n\geq 3$, $g:\mathbb{S}^{n-1}\to\mathbb{R}$ is an even bounded measurable function, $U$ is an open subset of $\mathbb{S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)|_U=c'$, for some fixed constants $c,c'\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$. Here, ${\cal C}(g)$ denotes the cosine transform and ${\cal R}(g)$ denotes the Funk transform of $g$. However, we show that $g$ does not need to be equal to a constant almost everywhere in $U^\perp:=\bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp)$. For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.

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 a local version of a theorem of Myroshnychenko–Ryabogin–Yaskin: for n≥3, even bounded measurable g:S^{n-1}→R and open U⊂S^{n-1}, if every great-sphere section perpendicular to U is isotropic, then C(g)|_U=c+⟨a,·⟩ and R(g)|_U=c′ for constants c,c′∈R and vector a∈R^n. The conclusion holds without the global assumption that g is constant a.e. on U^⊥; the argument relies on a new generalization, in the setting of convex hypersurfaces, of a classical differential-geometry result.

Significance. The local characterization strengthens the prior global result by removing the constancy requirement on U^⊥ while still recovering the precise linear/constant forms of the cosine and Funk transforms. The new convex-hypersurface lemma is presented as being of independent interest and appears to be the key technical ingredient that permits the local-to-global distinction. The proof is described as direct and non-circular.

minor comments (3)
  1. [Abstract] Abstract: the statement refers to 'the restriction (section) of f' being isotropic, yet the function whose transforms are analyzed is g; the precise relation between f and g (or whether they coincide) must be stated explicitly in the abstract and introduction.
  2. The manuscript would benefit from a short dedicated paragraph (perhaps in the introduction) that contrasts the local hypothesis and conclusion with the global theorem being extended, including the precise point at which the new lemma replaces the constancy assumption.
  3. Notation: the symbols C(g) and R(g) are introduced without an immediate reminder of their integral definitions; a one-line recall of the kernels would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report contains no specific major comments requiring point-by-point replies.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new lemma

full rationale

The paper proves a local implication from the isotropy hypothesis on sections to the stated linear/constant forms of C(g) and R(g) on U. This rests on a new generalization of a classical differential-geometry result for convex hypersurfaces, which is presented as independent content. The argument explicitly distinguishes the local case from the global theorem of Myroshnychenko-Ryabogin-Saroglou without reducing the target conclusion to a redefinition or fit of the input data. Self-citation occurs but is not load-bearing for the local result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard properties of the cosine and Funk transforms and on a new generalization of a classical differential-geometry fact; no free parameters or invented entities are mentioned.

axioms (2)
  • standard math Standard properties of even bounded measurable functions and the cosine/Funk transforms on the sphere
    The statement invokes these transforms and their known behavior in convex geometry.
  • domain assumption A new generalization of a classical differential-geometry result for convex hypersurfaces
    The paper obtains this lemma for use in the proof and states it is of independent interest.

pith-pipeline@v0.9.0 · 5754 in / 1375 out tokens · 60803 ms · 2026-05-25T16:07:32.555418+00:00 · methodology

discussion (0)

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Reference graph

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