{"id":"d75cefd4-9e36-4bb6-a379-edad9115d526","arxiv_id":"2608.02098","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The optimal constant in the isomorphic Busemann-Petty problem for arbitrary even densities has the sharp order √n: a new lower bound C_n ≥ c√n matches the known upper bound C_n ≤ √n.","lead":"This mathematics paper pins down the exact size of a constant in the Busemann-Petty comparison problem for arbitrary densities: it grows like the square root of the dimension, and the new lower bound matches the known upper bound. The result closes a question the authors opened in 2015 and offers a reusable gluing technique for turning one-body slicing examples into two-body comparison theorems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharp lower bound depends entirely on Proposition 6.1, an unproved-in-paper black box from Klartag–Livshyts; if that construction fails, only the c√(n/log n) bound survives. Internal proof is otherwise correct.","rationale":"The paper's internal argument for Theorem 1.2 is mathematically sound: the one-scale construction is self-contained, the spherical-averaging identity (31) and Lemma 6.2 correctly yield the mass-concentration estimate (30) (giving (34)), and the gluing and annular-density lemmas are valid. The only typo we found—Lemma 2.4's statement claims c√n but its proof gives c/√n—is non-load-bearing because Proposition 2.7 uses the proof's correct bound c0/√n ≤ p_n ≤ C0/√n. The genuinely load-bearing premise is Proposition 6.1, the Klartag–Livshyts black box. The authors disclose their reliance and point to precise steps in [14], but they do not reproduce the construction; this is the same weakness the reader identified. Given that the sharp lower bound collapses to the one-scale bound if Proposition 6.1 fails, a conditional verdict is appropriate until the external construction is independently verified. No internal error changes this assessment, so the reader's CONDITIONAL verdict stands.","tokens_in":14229,"tokens_out":17748,"duration_ms":117273,"concrete_test":"Obtain the published/corrected Klartag–Livshyts paper (arXiv:1810.06189v4) and verify Steps 1–3 of its Theorem 1.1 produce: (i) T = C6K origin-symmetric with √n B ⊂ K (hence r0√n B ⊂ T) and |T|^{1/n} ≤ C0; (ii) an even C∞ density g, convolution of standard Gaussian with an even discrete measure, satisfying ∫_T g ≥ 1/2 and ∫_{ξ⊥} g ≤ a0/√n for all ξ∈S^{n-1} (or at least the central sections needed here). If any property fails as stated, check whether symmetrization or a different scaling of C6 repairs it; if not, the sharp lower bound is unproven and the verdict should be reduced to the one-scale theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2's lower bound C_n ≥ c√n is a direct corollary of Proposition 6.1, which asserts existence of an origin-symmetric body T with r0√n B ⊂ T and |T|^{1/n} ≤ C0, and an even C∞ probability density g with ∫_T g ≥ c0 and ∫_{ξ⊥} g ≤ a0/√n. The paper does not prove Proposition 6.1; it cites Steps 1–3 of [14, Theorem 1.1] and gives a brief translation. If the Klartag–Livshyts construction does not actually deliver a density with these exact properties (e.g., if the density is not even, or the volume-radius bound requires a different scaling, or the section bound is only for a different measure), the sharp result would not follow, though the one-scale c√(n/log n) theorem would remain. The paper's internal derivation from Proposition 6.1 to Theorem 1.2 is correct; we checked the spherical-averaging lemma (31), the mass-concentration bound (30) giving (34), the annular density construction, and the gluing argument. (Note: Lemma 2.4's statement has a typo—the correct bounds are c/√n ≤ Eφ(n⟨Θ,ξ⟩) ≤ C/√n, as its proof shows; this does not affect Proposition 2.7.) The only real soft spot is the black-box reliance.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the optimal constant C_n in the isomorphic Busemann–Petty problem for arbitrary even, continuous, strictly positive densities on R^n. The main result, Theorem 1.2, asserts c√n ≤ C_n ≤ √n, thereby determining the sharp order. The upper bound was previously proved by the authors. For the lower bound, the paper first gives a complete one-scale construction (Theorem 1.3) yielding C_n ≥ c√(n/log n), using Gluskin polytopes, Gaussian mixtures, and a support-separation/gluing principle. It then upgrades this to the sharp c√n lower bound by importing the Klartag–Livshyts random-rounding construction as a black box (Proposition 6.1) and combining it with a spherical-averaging support-separation lemma (Lemma 6.2, Proposition 6.3). All in-paper estimates are proved with explicit absolute constants; the sharp part is logically downstream of Proposition 6.1.","tokens_in":14463,"tokens_out":11213,"duration_ms":99803,"significance":"If the black-box input is valid, the paper settles the sharp order of the isomorphic Busemann–Petty constant for arbitrary measures, matching the known upper bound up to an absolute factor. The proof introduces clean, reusable support-separation and gluing tools, and the one-scale bound is fully self-contained and independent of [14]. The paper is transparent about the black-box nature of Proposition 6.1. The main caveat is that the c√n lower bound rests entirely on an unproved consequence of the proof of [14, Theorem 1.1]; if that consequence fails, only the c√(n/log n) result would survive.","major_comments":[{"comment":"Proposition 6.1 is the sole external input for the sharp lower bound, yet its proof is only a reference to Steps 1–3 of [14, proof of Theorem 1.1]. The manuscript does not reproduce the construction or verify all the listed properties. Since Theorem 1.2 is entirely downstream of this proposition, please either include a complete proof of Proposition 6.1 in the paper, or state it as a formally quoted theorem of [14] with explicit page/equation references and demonstrate that the constants behave as asserted. In particular, check (27)–(29) and the evenness, smoothness, and probability normalization of g. If any of these properties fails or requires a different scaling, the c√n lower bound collapses, although the one-scale Theorem 1.3 would remain valid.","section":"Section 6, Proposition 6.1"},{"comment":"The sentence 'Their density is the convolution of the standard Gaussian density with an even discrete probability measure' is asserted without derivation. This evenness is load-bearing: Proposition 6.3 and the gluing principle require the densities p, h, and f to be even. Please either define the discrete measure explicitly or cite the precise location in [14] where its evenness is established. Similarly, the notation 'T = C6K' is ambiguous and should be written as C_6 K with the meaning of C_6 specified.","section":"Section 6, Proof of Proposition 6.1"}],"minor_comments":[{"comment":"Statement typo: the correct bounds are c/√n ≤ Eφ(n⟨Θ,ξ⟩) ≤ C/√n, as the proof establishes. The printed statement c√n ≤ Eφ(...) ≤ C√n is inconsistent with φ ≤ 1 and could mislead a reader. The later use in Proposition 2.7 correctly uses the c/√n lower bound.","section":"Lemma 2.4"},{"comment":"Identity (31) is derived only in prose. A short display-level derivation, or a reference to a standard spherical-Radon-transform formula, would improve verifiability.","section":"Section 6, Lemma 6.2"},{"comment":"Reference [14] lists the corrected arXiv version. If the published version in the GAFA 2020 lecture notes is the primary source, cite it with the page range and any corrigendum.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The internal argument is sound and the one-scale theorem is a solid standalone contribution. The only substantive concern is the unproved, load-bearing Proposition 6.1. If the authors can either prove that proposition or give a fully precise statement from [14] with step-by-step verification, I would be willing to support acceptance. The paper is within the journal's scope and contains no evident circularity or fitted parameters."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper settles the order of C_n: the two-body isomorphic Busemann–Petty constant for arbitrary even densities is exactly sqrt(n). The upper bound was already in [21]; the new lower bound C_n ≥ c sqrt(n) is the content, and it is genuine. The proof has two layers. The one-scale construction, yielding c sqrt(n/log n), is complete and self-contained: the random direction selection, the polytope volume estimate, the Gaussian marginal calculation, and the annular comparison density all check out. The sharp step is a clean reduction to Klartag–Livshyts: a spherical-averaging lemma forces most mass of a density with small hyperplane sections to lie outside a large Euclidean ball, and a gluing lemma converts that into the two-body comparison. Those support-separation and gluing lemmas are the real new geometric content, and they are well done.\n\nThere are two small mechanical issues, both noted by the reader. Proposition 6.3's printed bound is a factor n too large and as written cannot imply (34); Lemma 6.2's correct estimate repairs it, so this is a typo. Lemma 2.4 states the lower bound incorrectly, though its proof gives the right one. Neither affects the argument once fixed.\n\nThe substantive soft spot is Proposition 6.1. The sharp theorem is entirely downstream of that black box: an even C-infinity density with section integrals of order 1/sqrt(n) and positive mass on a body of bounded volume radius. The authors do not reproduce the proof; they say it follows from Steps 1–3 of [14, Theorem 1.1] and point to a corrected version. If that construction does not deliver these exact properties, only the one-scale bound survives. This is not a manufactured flaw—it is a real dependency, and the referee should verify it. It is also the natural division of labor: the paper is transparent about the black box and contributes the comparison machinery.\n\nThe citation pattern is fine. The paper uses Gluskin's classical estimate and cites the relevant slicing literature; no circularity, no fitted constants.\n\nWho should read this: anyone working on slicing inequalities for arbitrary measures or on Busemann–Petty-type comparisons. It completes the two-body analogue of the one-body slicing result and gives a reusable support-separation technique. It deserves a serious referee. I would send it to review, expecting the referee to check Proposition 6.1 and the typo fixes, not to redo the whole proof.","headline":"The paper proves the sharp C_n ~ sqrt(n) lower bound for the two-body Busemann–Petty problem; the new geometric lemmas are solid, and the only real soft spot is the black-box reliance on Klartag–Livshyts.","tokens_in":15100,"tokens_out":2017,"would_cite":true,"duration_ms":17110,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","52A40","52A23","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The optimal constant in the isomorphic Busemann–Petty problem for arbitrary even densities grows exactly like the square root of the dimension: c√n ≤ C_n ≤ √n.","keywords":["Busemann-Petty problem","arbitrary measures","hyperplane sections","convex bodies","isomorphic constant","random rounding","Gaussian mixtures","slicing inequality"],"falsifier":"A reader could falsify the sharp lower bound by checking the constants in the quoted random-rounding construction: if the body $T$ has volume radius growing faster than a constant or the density $g$ has central sections larger than $a_0/\\sqrt{n}$, then Proposition 6.1 collapses and only the one-scale $\\sqrt{n/\\log n}$ bound would follow. More directly, exhibiting any even continuous strictly positive density and two origin-symmetric convex bodies satisfying the hyperplane-section inequalities with total-mass ratio smaller than $c\\sqrt{n}$ for every absolute $c>0$ would refute the theorem.","tokens_in":13986,"feed_emoji":"📐","tokens_out":4570,"duration_ms":32503,"temperature":0.7,"texified_at":"2026-08-05T21:59:13.886792+00:00","pith_summary":"The paper establishes that the optimal constant in the isomorphic Busemann–Petty problem for arbitrary even densities grows exactly like the square root of the dimension. Earlier work had shown the constant is at most $\\sqrt{n}$; this paper proves a matching lower bound $c\\sqrt{n}$, so the order is sharp. The proof combines a one-scale construction that gives $\\sqrt{n/\\log n}$ with a sharpening using a random-rounding construction as a black box and a spherical-averaging support-separation argument. If correct, it settles the dimensional dependence of this two-body comparison problem, in contrast with the volume case where the analogous constant is dimension-free.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5627,"prompt_tokens":768,"completion_tokens":4859,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":768,"completion_tokens_details":{"reasoning_tokens":4121}},"feed_headline":"Busemann-Petty constant for arbitrary densities is exactly order √n","feed_subtitle":"For every even density, comparing hyperplane sections only controls total mass up to a √n factor—and that factor is unavoidable.","key_machinery":"The argument rests on three mechanisms. A support-separation lemma converts a density with uniformly small central hyperplane integrals into a compactly supported probability density whose support lies in an outer body $T$ but outside a Euclidean ball $L$, using spherical averaging of the central Radon transform. An annular comparison density $h$, supported in $L \\setminus T$, has central sections bounded below by an absolute constant (or by $1/v$ in the one-scale version). A gluing principle combines these two densities with a small Gaussian perturbation to enforce strict positivity, producing a two-body comparison with section inequalities in one direction and total-mass inequality in the other. The sharp l","core_discovery":"The central claim is Theorem 1.2: there is an absolute constant $c>0$ such that for every $n\\ge 2$, $c\\sqrt{n} \\le C_n \\le \\sqrt{n}$, where $C_n$ is the smallest factor such that for every even, continuous, strictly positive density $f$ and every pair of origin-symmetric convex bodies $K,L$, hyperplane-section inequalities $\\int_{K\\cap \\xi^\\perp} f \\le \\int_{L\\cap \\xi^\\perp} f$ for all $\\xi$ imply $\\int_K f \\le C_n \\int_L f$. The paper proves the lower bound by constructing explicit witness bodies and densities. The sharp example uses a body $T$ with bounded volume radius and an even probability density $g$ whose central hyperplane integrals are $O(1/\\sqrt{n})$; a new lemma shows that small central sections force most of the mass of such a density to lie away from any ball of ra","pith_inferences":["The support-separation lemma suggests a general principle: a density with all central hyperplane sections O(n^{-1/2}) must be spread out, so any convex body with bounded volume radius that carries such a density must have most of its mass at distance ≳√n from the origin—this may transfer to other geometric tomography questions.","The gap between the one-scale bound (√(n/log n)) and the sharp bound (√n) is exactly the volume-radius gap: replacing the one-scale body's √log n volume radius by an absolute constant yields the sharp constant, so any improvement in the volume-radius of such random constructions would directly sharpen such two-body comparisons.","A natural testable extension is whether the same sharp order holds for even densities that are not strictly positive or for non-symmetric densities; the gluing step uses strict positivity only via a small Gaussian term, so a limiting argument may extend the result to the closure of this class.","The non-log-concavity of the witness density clarifies that the log-concave Busemann–Petty constant remains dimension-free; any attempt to improve the arbitrary-measure constant must leave the log-concave class."],"forward_implications":["The optimal constant for arbitrary even densities is dimensionally order √n; no dimension-free bound is possible in this setting.","The witness bodies and densities show that continuous strictly positive even densities cannot be replaced by log-concave ones in an order-√n result, as the paper notes explicitly.","The one-scale lower bound √(n/log n) follows from a complete, self-contained geometric construction and identifies the core mechanism; the sharp bound needs the stronger random-rounding input.","The spherical-averaging identity that converts small central sections into mass decay away from the origin has independent uses in comparing full-dimensional and lower-dimensional integrals of densities.","The sharp example produces a density that is not log-concave, delineating the boundary beyond which the isomorphic constant must grow with dimension."],"fun_headline_variants":["Busemann-Petty constant for arbitrary measures scales as √n","Sharp order: hyperplane sections control mass up to √n factor","For arbitrary densities, section comparison constant is √n","Matching lower bound: arbitrary-measure BP constant is √n","√n is the exact order for arbitrary-measure Busemann-Petty"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The sharp lower bound depends on the existence, for every large dimension, of an origin-symmetric body with volume radius bounded by an absolute constant and of an even probability density that gives a fixed positive mass to the body while having all central hyperplane integrals at most a constant times $1/\\sqrt{n}$; this is asserted from a cited random-rounding construction and not proved in this paper.","fun_headline_variants_meta":{"raw":{"variants":["Busemann-Petty constant for arbitrary measures scales as √n","Sharp order: hyperplane sections control mass up to √n factor","For arbitrary densities, section comparison constant is √n","Matching lower bound: arbitrary-measure BP constant is √n","√n is the exact order for arbitrary-measure Busemann-Petty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":1000,"prompt_tokens":754,"completion_tokens":246,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":170}},"tokens_in":498,"tokens_out":246,"duration_ms":13739,"temperature":1.0,"reasoning_tokens":170,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:10:54.355173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A reader could falsify the sharp lower bound by checking the constants in the quoted random-rounding construction: if the body $T$ has volume radius growing faster than a constant or the density $g$ has central sections larger than $a_0/\\sqrt{n}$, then Proposition 6.1 collapses and only the one-scale $\\sqrt{n/\\log n}$ bound would follow. More directly, exhibiting any even continuous strictly positive density and two origin-symmetric convex bodies satisfying the hyperplane-section inequalities with total-mass ratio smaller than $c\\sqrt{n}$ for every absolute $c>0$ would refute the theorem.","supporting_citations":[],"review_version":1}