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Illumination bodies on Riemannian manifolds

T0 review · 1 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The volume asymptotic for illumination bodies of convex sets extends to Riemannian manifolds with Ricci curvature bounded from below.

desk verdict The paper generalizes the illumination-body volume asymptotic to manifolds with a Ricci lower bound, but the stress-test concern about uncontrolled sectional-curvature errors in the Jacobian looks like it needs direct checking in the proof. read the letter →

arxiv 2606.21112 v1 pith:Z6HU5NUC submitted 2026-06-19 math.DG math.MG

Rotem Assouline , Carsten Schütt , Elisabeth M. Werner This is my paper · ORCID
classification math.DGmath.MG
keywords illuminationbodiesRiemannianmanifoldsRiccicurvatureboundconvexasymptoticvolumeformulasminimizinggeodesicsdistortion
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 generalizes a known Euclidean-space result on illumination bodies to curved settings. An illumination body collects all points whose connecting minimizing geodesics to a given convex set sweep out total volume at most delta. The authors show that the volume of this body obeys the same leading asymptotic in delta as in flat space, once the manifold's Ricci curvature is bounded from below. This matters because the lower curvature bound controls how volumes stretch or shrink along geodesics, letting the original volume-counting argument carry over without new error terms.

What carries the argument

The δ-illumination body, defined via the union of minimizing geodesic segments to the convex set having volume at most δ, with the Ricci lower bound supplying the volume-comparison control needed for the asymptotic.

What would settle it

Compute the volume of the δ-illumination body for successively smaller δ on a manifold whose Ricci curvature is unbounded from below and check whether the leading term deviates from the Euclidean power of δ.

Watch

Extended reading notes

Core claim

We prove a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body, which holds on Riemannian manifolds with Ricci curvature bounded from below. The δ-illumination body of a subset of a Riemannian manifold is defined to be the set of all points such that the union of all minimizing geodesic segments joining the point to the set has volume at most δ.

Load-bearing premise

A lower bound on Ricci curvature is enough to control the volume distortion of geodesic segments so the Euclidean asymptotic argument transfers without invalidating extra error terms.

Editorial extensions

If this is right

  • The same volume asymptotic applies on any manifold satisfying a uniform Ricci lower bound, including spheres and hyperbolic spaces of appropriate curvature.
  • Volume estimates for illumination bodies become available for convex sets in any geometry where Ricci curvature can be bounded from below.
  • The construction remains intrinsic because it uses only minimizing geodesics and the manifold's own volume measure.
  • The proof strategy adapts the original Euclidean comparison by inserting the Ricci bound to absorb distortion effects along short geodesics.

Reading between the lines

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

  • The result opens the door to replacing Euclidean illumination bodies by their manifold versions in any application that previously relied on the asymptotic volume count.
  • Similar generalizations may be possible for other volume-based bodies once an analogous curvature condition is identified that controls geodesic volume distortion.
  • Numerical checks on model spaces such as the sphere could verify the rate at which the asymptotic is approached for concrete convex sets.
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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

1 major / 2 minor

Summary. The manuscript generalizes Werner's asymptotic formula for the volume of the illumination body of a convex body to Riemannian manifolds with Ricci curvature bounded from below. The δ-illumination body of a subset K is defined as the set of points p such that the volume of the union of all minimizing geodesic segments from p to K is at most δ; the paper claims an asymptotic formula for the volume of this set as δ → 0.

Significance. If the central claim holds, the result would extend a Euclidean convex-geometry asymptotic to the Riemannian setting under a Ricci lower bound alone, using volume comparison. This could be useful for comparison geometry, though the significance is tempered by the need to verify that curvature-dependent error terms in the geodesic Jacobian do not alter the leading coefficient or exponent.

major comments (1)
  1. [§4–5, main theorem] Main theorem (proof in §4–5): The claimed asymptotic for vol(I_δ(K)) as δ → 0 is asserted to carry over verbatim from the Euclidean case using only Ric ≥ −(n−1)κ. However, the volume of the union of minimizing geodesics depends on the Jacobian determinant J(r,θ) along the geodesic flow; its expansion contains sectional-curvature terms at order r^3 that are not controlled by a Ricci lower bound (Bishop–Gromov gives only an upper volume bound on cones). Without an accompanying upper curvature bound or injectivity-radius control near K, the o(δ^α) remainder may acquire manifold-dependent corrections that change the constant or the power in the formula. An explicit expansion or error estimate addressing these terms is required.
minor comments (2)
  1. [Definition 1.2] Definition 1.2: the precise meaning of 'union of all minimizing geodesic segments' when multiple geodesics exist should be clarified with respect to the cut locus.
  2. [Theorem 1.1] The statement of the asymptotic should explicitly record the dependence (or independence) of the leading constant on the Ricci bound constant κ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

  1. Referee: [§4–5, main theorem] Main theorem (proof in §4–5): The claimed asymptotic for vol(I_δ(K)) as δ → 0 is asserted to carry over verbatim from the Euclidean case using only Ric ≥ −(n−1)κ. However, the volume of the union of minimizing geodesics depends on the Jacobian determinant J(r,θ) along the geodesic flow; its expansion contains sectional-curvature terms at order r^3 that are not controlled by a Ricci lower bound (Bishop–Gromov gives only an upper volume bound on cones). Without an accompanying upper curvature bound or injectivity-radius control near K, the o(δ^α) remainder may acquire manifold-dependent corrections that change the constant or the power in the formula. An explicit expansion or error estimate addressing these terms is required.

    Authors: We thank the referee for this observation. The proof in §§4–5 relies on the Bishop–Gromov volume comparison theorem under the sole assumption Ric ≥ −(n−1)κ to control the volume of the union of minimizing geodesics. For the leading asymptotic as δ → 0 the relevant geodesics have lengths tending to zero, so that the integrated effect of the order-r^3 sectional-curvature terms in the Jacobian expansion remains o(δ^α) and does not alter the Euclidean leading coefficient or exponent. Nevertheless, to make the error control fully explicit we will add a detailed expansion of J(r,θ) together with the resulting remainder estimate in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation to base Euclidean result; manifold extension derived independently via volume comparison

full rationale

The paper defines the δ-illumination body directly from the volume of geodesic unions and claims an asymptotic generalization of a prior Euclidean formula by Werner (one co-author). The derivation adapts standard Bishop-Gromov volume comparison under Ric ≥ −(n−1)κ, which is an external theorem independent of the target asymptotic. No equations reduce the claimed volume formula to a fitted parameter, self-definition, or unverified self-citation chain; the Ricci bound supplies the necessary control without importing the result itself. This is the typical non-circular extension of an external theorem.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the Ricci curvature bound is treated as a standard domain assumption rather than an ad-hoc postulate.

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Cite this review

Pith. "Pith review of Illumination bodies on Riemannian manifolds." pith.science (2026). https://pith.science/paper/Z6HU5NUC

@misc{pith2026260621112,
  author       = {Pith},
  title        = {Pith review of: Illumination bodies on Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z6HU5NUC}},
  note         = {Machine review of arXiv:2606.21112}
}
abstract

We prove a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body, which holds on Riemannian manifolds with Ricci curvature bounded from below. The $\delta$-illumination body of a subset of a Riemannian manifold is defined to be the set of all points such that the union of all minimizing geodesic segments joining the point to the set has volume at most $\delta$.

Figures

Figures reproduced from arXiv: 2606.21112 by the authors.

Figure 1
Figure 1. The set C(x, K). Denote by Vol the Riemannian volume measure on M. Definition 1.1 (Illumination body). Let δ > 0. The δ-illumination body of K is the set Kδ := {x ∈ M \ K : Vol(C(x, K)) ≤ δ} ∪ K. (2) Our main result generalizes to the Riemannian setting the asymptotic formula established in [28] for the volume of the illumination body of a convex body. Since there are several notions of convexity in Riemannian geome… view at source ↗
Figure 2
Figure 2. Lemma 3.6 asserts that only points lying in the shaded region can be mapped to [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. The ball Bℓ(t) (xt) has twice the radius of the smallest ball containing C(xt , K). By compactness of K, we may assume that zj → z ∈ ∂K. Suppose that z ̸= x0. Upon passing to a subsequence, the geodesics γj converge to a nonconstant geodesic γ joining x0 to z. Since x0 and z belong to K, assumption (⋆) implies that the geodesic γ must be contained in K. But the relative interiors of the geodesics γj are disjoint fro… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The set Ps. The expression in parentheses therefore vanishes. But sj → 0 since zj → x0, whence necessarily v n j M2 j tj → −q(u) < 0. (27) By Lemma 3.5 and by (25), (26) and (27), r [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: A right circular cone. Lemma 3.13. There exists c1 > 0 such that for every h, b ∈ (0, R), a right circular cone with height h and radius b which is contained in the set U has volume ≥ c1 · h · b n−1 . Proof. Let Y be a right circular cone contained in the set U, with x…
Figure 6
Figure 6. Figure 6: Proof of Lemma 3.13. Lemma 3.14. Let x ∈ Σ. If ρ(x) > 0 then x ∈ Σ ′ . Proof. If ρ(x) > 0 then the tangent cone TxK contains the tangent cone of some ball at x, which is a half-space. Hence TxK is a half-space and x ∈ Σ ′ . Lemma 3.15. Let x ∈ Σ ′ and let ρ ∈ (0, R]. I…
Figure 7
Figure 7. Figure 7: Proof of Lemma 3.16. (II) the cone Y is disjoint from the interior of K. By Lemma 3.14, x0 ∈ Σ ′ . By the definition of ρ, the set K contains a closed ball of radius ρ whose boundary contains x0, and by Lemma 3.15, its center is p := γx0 (−ρ). Let γ be a unit-speed geo…
Figure 8
Figure 8. Figure 8: Proof of Lemma A.6 Since dψ|0 ̸= 0, for sufficiently small δ the Euclidean Hausdorff distance between ∂Pt ∩ Bδ(0) and ∂P0 ∩ Bδ(0) is O(t). Hence if we denote by rt the Euclidean signed distance function to ∂Pt then rt = r0 + O(t) on Bδ(0). (57) We claim that r = r0 + o…

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