{"id":"f383b374-2606-4cc5-a060-99e539aec41a","arxiv_id":"1907.06122","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For positive subharmonic f on convex Omega in R^n the volume average is at most c_n times the surface average with n-1 <= c_n <= 2 n^{3/2}, plus the sharp geometric inequality |partial Omega1|/|Omega1| * |Omega2|/|partial Omega2| <= n for nested convex domains.","lead":"The paper proves an improved bound on the constant relating the volume average of a positive subharmonic function to its surface average over the boundary of a convex domain in n dimensions. A smart generalist might read it for the sharpened classical inequality and the accompanying geometric relation between nested convex bodies.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the convexity and subharmonicity hypotheses; because the full proof is unavailable, no tighter or softer spot inside the argument can be isolated. The verdict therefore remains UNVERDICTED.","tokens_in":1736,"tokens_out":228,"duration_ms":49048,"concrete_test":"Recompute the surface-to-volume ratio product for the standard n-simplex and its inscribed ball (or a scaled copy) in dimension n=3; if the product exceeds 3 the geometric claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Full manuscript text was referenced but not supplied, preventing line-by-line inspection of the proof. The abstract states a uniform bound c_n ≤ 2n^{3/2} for positive subharmonic functions on arbitrary convex domains together with the matching lower bound n-1 and the geometric consequence; these statements are internally consistent with the maximum principle for subharmonic functions and with John's theorem scaling, and no contradiction with the listed assumptions is visible from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that for convex Ω ⊂ R^n and positive subharmonic f (Δf ≥ 0), the volume average satisfies (1/|Ω|) ∫_Ω f dx ≤ (c_n / |∂Ω|) ∫_∂Ω f dσ with explicit uniform bound c_n ≤ 2n^{3/2}; the optimal constant satisfies c_n ≥ n-1. It also derives the geometric inequality |∂Ω1|/|Ω1| ⋅ |Ω2|/|∂Ω2| ≤ n whenever Ω2 ⊂ Ω1 are convex. The claims extend Hermite-Hadamard inequalities from convex to subharmonic functions with improved constants.","tokens_in":1812,"tokens_out":351,"duration_ms":12873,"significance":"If the stated bounds hold, the work supplies explicit, dimension-dependent constants that improve on prior results limited to convex functions, together with a sharp geometric consequence obtained as a byproduct. The approach via the maximum principle for subharmonic functions is consistent with standard tools in the field and yields falsifiable predictions (the lower bound n-1 and the geometric inequality).","major_comments":[],"minor_comments":[{"comment":"The abstract states that the inequality 'was previously only known for convex functions with a much larger constant' but supplies neither the prior constant nor a citation; adding this reference would clarify the improvement.","section":"Abstract"},{"comment":"Notation for surface measure (dσ) and volume measure (dx) is standard but should be defined explicitly on first use for readers outside convex geometry.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive report, accurate summary of our results on improved Hermite-Hadamard bounds for subharmonic functions and the geometric inequality, and the recommendation of minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1269,"tokens_out":67,"duration_ms":10639,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this work improves the Hermite-Hadamard-type inequality from convex functions (with a larger constant) to positive subharmonic ones, reaching c_n ≤ 2n^{3/2}, while also proving the lower bound c_n ≥ n-1 and the nested-domain geometric fact that |∂Ω1|/|Ω1| ⋅ |Ω2|/|∂Ω2| ≤ n for Ω2 ⊂ Ω1 convex. Both the extension and the geometric byproduct are new on the evidence of the abstract. The bound is consistent with the maximum principle for subharmonics and with John's ellipsoid scaling, so the statements line up internally without obvious circularity or free parameters. The geometric inequality in particular looks like it could be useful on its own for volume-surface estimates. The main soft spot is that the abstract gives no proof outline, so the derivation of the 2n^{3/2} factor remains opaque; if it relies on crude ellipsoid approximation then the gap to the n-1 lower bound is unsurprising but still leaves room for later sharpening. The positivity assumption on f is stated explicitly and seems necessary. This is a targeted improvement rather than a broad new theory, so it is mainly for people already working on convex-domain inequalities or subharmonic estimates in PDE. The claims are precise enough and the extension is substantive, so the paper deserves a serious referee rather than a desk rejection.","headline":"The paper sharpens the constant to 2n^{3/2} for positive subharmonic functions on convex domains and adds a clean geometric inequality for nested sets.","tokens_in":2288,"tokens_out":362,"would_cite":false,"duration_ms":9803,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Classical convex-analysis bounds on subharmonic integrals; no RS structures","alignment":"orthogonal","rationale":"The paper derives dimension-dependent constants for volume-to-surface integral inequalities for positive subharmonic functions on convex domains in R^n, using the torsion function, maximum principle, Steinhagen width bounds, coarea formula, and mixed volumes. None of its machinery (Hermite-Hadamard extensions, Cheeger constants, or the geometric nesting inequality) invokes J-cost, φ-ladders, 8-tick periodicity, or distinction-forced emergence. RS theorems (reality_from_one_distinction, J-uniqueness via Aczél, AlexanderDuality for D=3, etc.) are silent on this domain.","tokens_in":48049,"confidence":"high","tokens_out":167,"duration_ms":5015,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For positive subharmonic functions on convex domains in R^n the volume average is at most 2n^{3/2} times the surface average.","keywords":["subharmonic functions","convex domains","Hermite-Hadamard inequality","higher dimensions","surface-to-volume ratios","boundary integrals"],"falsifier":"A single convex domain together with one positive subharmonic function on it whose volume-to-surface average ratio exceeds 2n^{3/2}.","tokens_in":2655,"feed_emoji":"📐","tokens_out":641,"duration_ms":14067,"temperature":0.7,"pith_summary":"The paper proves an inequality that bounds the average value of a positive subharmonic function inside a convex domain by a multiple of its average on the boundary. The multiple c_n satisfies c_n less than or equal to 2n to the power 3/2 and is at least n minus 1. This extends earlier results that required the stronger assumption that the function itself is convex and used larger constants. The same argument produces a geometric bound: when one convex domain sits inside another the product of their surface-to-volume ratios is at most n.","feed_headline":"Subharmonic volume average bounded by 2n^{3/2} times surface average","feed_subtitle":"New constant works for any positive subharmonic function on a convex domain and yields a sharp bound on nested surface-to-volume ratios.","key_machinery":"The inequality comparing the normalized volume integral of a positive subharmonic function to its normalized surface integral, controlled by a dimension-dependent constant c_n.","core_discovery":"Let Omega subset R^n be convex and let f be positive and subharmonic on Omega. Then the volume average of f is bounded above by c_n times the surface average of f, where c_n is at most 2n^{3/2}. The optimal constant is at least n-1. As a consequence, any two nested convex domains Omega2 subset Omega1 satisfy |partial Omega1|/|Omega1| times |Omega2|/|partial Omega2| less than or equal to n.","pith_inferences":["The bound may become sharper when the function is harmonic rather than merely subharmonic.","The nested-domain inequality could be used to compare isoperimetric ratios across a chain of convex sets.","Explicit computation on the Euclidean ball would give a concrete numerical check on how close 2n^{3/2} is to the true optimum."],"forward_implications":["The stated bound on c_n holds for all positive subharmonic functions, not merely convex ones.","The optimal constant c_n is at least n-1 in every dimension.","Nested convex domains obey the product bound on their surface-to-volume ratios.","The geometric inequality is sharp in the sense that equality is attained in limiting cases."],"fun_headline_variants":["Subharmonic volume avg <= 2n^{3/2} surface avg on convex domains","Volume to surface avg ratio <= 2n^{3/2} for positive subharmonic f","2n^{3/2} upper bounds subharmonic volume-surface average ratio","Nested convex domains satisfy surface-volume ratio product <= n"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The domain must be convex and the function must be positive with nonnegative Laplacian.","fun_headline_variants_meta":{"raw":{"variants":["Subharmonic volume avg <= 2n^{3/2} surface avg on convex domains","Volume to surface avg ratio <= 2n^{3/2} for positive subharmonic f","2n^{3/2} upper bounds subharmonic volume-surface average ratio","Nested convex domains satisfy surface-volume ratio product <= n"]},"model":"grok-4.3","cost_usd":0.014864,"raw_usage":{"total_tokens":6388,"prompt_tokens":670,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":148637000,"prompt_tokens_details":{"text_tokens":670,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5635,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":670,"tokens_out":83,"duration_ms":31594,"temperature":1.0,"reasoning_tokens":5635,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T21:42:15.966697+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A single convex domain together with one positive subharmonic function on it whose volume-to-surface average ratio exceeds 2n^{3/2}.","supporting_citations":[],"review_version":1}