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Illumination Bodies in Projective Geometries

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Pith's one-line read The volume derivative of illumination bodies defines a surface area for convex bodies in Riemannian spaces of constant curvature and projective Finsler geometries.

desk verdict The paper extends illumination bodies to constant-curvature Riemannian and projective Finsler geometries and shows their volume derivatives yield a generalized surface area via reduction to a Euclidean weighted-volume result. read the letter →

arxiv 2605.25122 v1 pith:HY7RDUPD submitted 2026-05-24 math.MG math.DG

Rotem Assouline , Florian Besau , Elisabeth M. Werner This is my paper · ORCID
classification math.MGmath.DG
keywords illuminationbodiessurfaceareaconvexRiemanniangeometryFinsleraffineprojectivegeometriesvolumederivatives
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the definition of illumination bodies from Euclidean space to Riemannian spaces of constant curvature and to projective Finsler geometries. It proves that the derivative of the volume of these bodies with respect to a parameter produces a surface area measure on the boundary of convex bodies. This construction generalizes the affine surface area known in flat Euclidean space. The argument first establishes a general differentiability result for weighted volumes of weighted illumination bodies in Euclidean space and then transfers the result to the new geometric settings. The appendix supplies explicit examples of illumination bodies in the non-Euclidean cases.

What carries the argument

Illumination bodies whose volume derivative produces a surface area measure that generalizes affine surface area.

What would settle it

An explicit convex body in hyperbolic space or a projective Finsler geometry for which the volume of its illumination body is not differentiable with respect to the scaling parameter.

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Extended reading notes

Core claim

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space. The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space.

Load-bearing premise

Illumination bodies can be defined in a sufficiently regular way in these geometries so that their volumes are differentiable.

Editorial extensions

If this is right

  • The volume of an illumination body remains differentiable when the ambient space is a Riemannian manifold of constant curvature.
  • The same differentiability holds in projective Finsler geometries.
  • The derivative of the volume yields a well-defined surface area measure on the boundary of the convex body.
  • The Euclidean weighted-volume result is the technical foundation that carries the argument to the non-Euclidean settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same volume-derivative construction could be examined in Finsler geometries that are not projective if suitable regularity can be established.
  • The explicit examples in the appendix furnish concrete test cases for comparing the new surface area with other curvature-dependent notions.
  • The method supplies a uniform way to associate surface area measures to convex bodies across several families of geometries that admit a projective structure.
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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

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Summary. The manuscript extends the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. It proves that the derivative of their volume defines a surface area measure for convex bodies in these settings, generalizing the affine surface area from Euclidean space. The argument reduces the claim to a general differentiability result for weighted volumes of weighted illumination bodies already established in Euclidean space, with explicit non-Euclidean examples supplied in the appendix.

Significance. If the reduction is valid and the regularity conditions hold, the work supplies a natural generalization of affine surface area to these non-Euclidean geometries. The reduction to a prior Euclidean theorem is efficient, and the appendix examples constitute a concrete strength by furnishing explicit instances that can be checked directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

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No significant circularity

full rationale

The derivation reduces the non-Euclidean differentiability claim to a general result on weighted-volume derivatives already established in Euclidean space, as stated in the abstract. This external reduction, together with the appendix examples of explicit non-Euclidean instances, keeps the argument self-contained against independent benchmarks rather than self-referential. No load-bearing step collapses by definition, fitted-parameter renaming, or self-citation chain within the provided structure.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

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Cite this review

Pith. "Pith review of Illumination Bodies in Projective Geometries." pith.science (2026). https://pith.science/paper/HY7RDUPD

@misc{pith2026260525122,
  author       = {Pith},
  title        = {Pith review of: Illumination Bodies in Projective Geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HY7RDUPD}},
  note         = {Machine review of arXiv:2605.25122}
}
read the original abstract

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space. The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space. In the appendix, we give some explicit examples for non-Euclidean illumination bodies.

Figures

Figures reproduced from arXiv: 2605.25122 by the authors.

Figure 2.1
Figure 2.1. In a Hilbert geometry (X, H) the Finsler norm H(p, Xp) = ∥Xp∥p for Xp ∈ Tp(X) is defined as the harmonic mean of the distances t+(p, Xp) and t−(p, Xp) to the boundary ∂X on the line through p in direction Xp. where D[K] = 1 2 {x − y : x, y ∈ K} denotes the difference body of a convex body, K∗ is the dual body that is K∗ = {y ∗ ∈ (R n ) ∗ : y ∗ (x) ≤ 1 for all x ∈ K}, and we identify (R n ) ∗ with (TpX) ∗ ; see [43].… view at source ↗
Figure 3.1
Figure 3.1. The frontside boundary ∂K+(z) (in red) and backside boundary ∂K−(z) (in blue) as seen from z ̸∈ K partition the boundary of K. We may parametrize y ∈ (int[z, K])\K by y(x, s) = [z, x]s = (1−s)z+sx for s ∈ (0, 1) and x ∈ K+(z). The Jacobian is Jy(x, s) = s n−1 [(z−x)·nK(x)] for almost all x ∈ K+(z). Using this we find V φ K (z) = volφ n ([z, K] \ K) = Z ([z,K]\K)∩U φ dλn = Z ∂K+(z) |(z − x) · nK(x)| Z 1 0 1U ([z, x]… view at source ↗
Figure 4.1
Figure 4.1. Sketch for the proof of Proposition 4.2. Given a geodesic segment γ that is outside of K we consider the point γ(t0) that determines the boundary points p and q. H+ is the closed half-space that is determined by the points p, q ∈ ∂K and for |t − t0| small enough γ(t) is contained in H+. Then the convex hull [γ(0), K] is the union of K (in blue) with the triangle △(t0) = conv{p, q, γ(t0)} (in orange). The triangle △(… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Illumination bodies on Riemannian manifolds

    math.DG 2026-06 unverdicted novelty 6.0 of 10

    Generalizes Werner's asymptotic volume formula for δ-illumination bodies to Riemannian manifolds with Ricci curvature bounded below.

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