{"id":"ea3a845a-b5c3-4802-a597-68f2b6ec5ea2","arxiv_id":"2606.21112","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalizes Werner's asymptotic volume formula for δ-illumination bodies to Riemannian manifolds with Ricci curvature bounded below.","lead":"The paper proves a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body to Riemannian manifolds with Ricci curvature bounded from below. A smart generalist might read it to see how volume estimates from flat-space convex geometry extend to curved spaces.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Ricci lower bound controls volume growth but may not eliminate curvature-dependent errors in the precise asymptotic for vol(illumination body)","rationale":"The reader's weakest assumption directly identifies the comparison step that must be checked; the concrete test isolates whether the Ricci bound is quantitatively sufficient for the error term. Because the full manuscript is available, the test can be performed directly on the existing proof rather than on an abstract claim.","tokens_in":1543,"tokens_out":375,"duration_ms":41277,"concrete_test":"In the proof of the asymptotic (likely §3 or §4), isolate the volume-comparison step for the geodesic cone; replace the model space by a space form of constant sectional curvature −κ and recompute the leading coefficient of vol(cone(p)) as dist(p,∂K)→0. If the coefficient differs from the Euclidean case by more than the o(1) term allowed in the argument, the claimed generalization fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the Euclidean-style asymptotic for vol({p : vol(union of min. geodesics from p to K) ≤ δ}) as δ → 0 carries over verbatim on a manifold with only Ric ≥ −(n−1)κ. Bishop–Gromov supplies an upper bound on the volume of geodesic cones, yet the leading coefficient and the o(δ^α) error in the sublevel-set volume depend on the expansion of the Jacobian determinant along the geodesics; this expansion contains sectional-curvature contributions at order r^3 and higher that are not controlled by a Ricci lower bound alone. Without an accompanying upper curvature bound or injectivity-radius control near K, the error term may acquire manifold-dependent corrections that alter the constant or the exponent in the claimed formula.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1706,"tokens_out":480,"duration_ms":19113,"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":[{"comment":"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.","section":"§4–5, main theorem"}],"minor_comments":[{"comment":"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.","section":"Definition 1.2"},{"comment":"The statement of the asymptotic should explicitly record the dependence (or independence) of the leading constant on the Ricci bound constant κ.","section":"Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[§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."}],"tokens_in":1198,"tokens_out":385,"duration_ms":24113,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this work extends Werner's Euclidean asymptotic for the volume of the illumination body to Riemannian manifolds under a lower Ricci bound. The definition carries over directly: the delta-illumination body consists of points p where the volume of the union of minimizing geodesics from p to the convex set is at most delta, and the claim is that its volume behaves asymptotically like a fixed power of delta as delta goes to zero.\n\nThey handle the setup cleanly by using the manifold exponential map and invoking standard volume comparison. That part is straightforward and shows the idea adapts without major new machinery.\n\nThe soft spot is the error term. Bishop-Gromov supplies volume upper bounds from the Ricci assumption, yet the volume of the geodesic union depends on the Jacobian determinant along each ray. That determinant expands with sectional-curvature contributions starting at order r cubed. A Ricci lower bound alone does not cap those sectional terms, so the leading coefficient or the little-o remainder could pick up manifold-dependent corrections unless the proof adds an upper curvature bound or shows the extra terms integrate to something negligible. The stress-test note flags exactly this gap, and nothing in the abstract rules it out.\n\nThis is aimed at people who already work on asymptotic invariants of convex bodies and want to see how they behave under curvature. A reader who cares about precise volume asymptotics in comparison geometry would find the statement useful to test. It is worth sending to a serious referee because the claim is specific enough to be checked and the approach is standard, even if revisions on the curvature hypotheses are likely needed.","headline":"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.","tokens_in":2194,"tokens_out":404,"would_cite":false,"duration_ms":28583,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The volume asymptotic for illumination bodies of convex sets extends to Riemannian manifolds with Ricci curvature bounded from below.","keywords":["illumination bodies","Riemannian manifolds","Ricci curvature bound","convex bodies","asymptotic volume formulas","minimizing geodesics","volume distortion"],"falsifier":"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.","tokens_in":2439,"feed_emoji":"","tokens_out":658,"duration_ms":22353,"temperature":0.7,"pith_summary":"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.","feed_headline":"Illumination body volume asymptotics extend to Ricci-bounded manifolds","feed_subtitle":"The Euclidean power-law in delta persists when a lower Ricci bound controls geodesic volume distortion.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Werner's formula for illumination body volume generalized to Ricci manifolds","Asymptotic volume formula for illumination bodies holds on Ricci manifolds","Illumination body volumes asymptotics proven for Ricci bounded manifolds","Generalized asymptotics of delta illumination bodies under Ricci bound"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Werner's formula for illumination body volume generalized to Ricci manifolds","Asymptotic volume formula for illumination bodies holds on Ricci manifolds","Illumination body volumes asymptotics proven for Ricci bounded manifolds","Generalized asymptotics of delta illumination bodies under Ricci bound"]},"model":"grok-4.3","cost_usd":0.010702,"raw_usage":{"total_tokens":4634,"prompt_tokens":491,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":107024500,"prompt_tokens_details":{"text_tokens":491,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4085,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":491,"tokens_out":58,"duration_ms":32020,"temperature":1.0,"reasoning_tokens":4085,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:46:07.978756+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}