{"id":"11a39bbc-28ab-4f58-8600-ac4a5405f542","arxiv_id":"1908.06426","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sharp generalized Hermite-Hadamard inequality is proved and applied to obtain the optimal volume-to-central-section constant 2^n/n for symmetric projections.","lead":"A new inequality bounds the average of a convex function of a concave function over a symmetric convex body by an integral over a single line through the origin. It gives the optimal constant for estimating a convex body's volume from its central section, answering a question posed by Francisco Santos.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equality case in Theorems 1.2 and 1.3 fails for constant functions because the proof divides by δ and cannot treat δ=0; e.g., f≡0 on any 0-symmetric body gives equality without the cylinder structure.","rationale":"Read in good faith, the paper's main inequality is plausible and the geometric application to Santos's question is supported; the proof of the inequality itself can be repaired by a δ=0 case using convexity, since φ(f)≤φ(f(0)) and the average of φ(f(0)(1+t)) over [−1,1] is at least φ(f(0)). The reader's weakest assumption is correct and is the same as the principal concern found here: the equality characterizations in Theorems 1.2 and 1.3 are false as stated for constant functions, and the proof excludes δ=0 by division. This does not overturn the inequality or the optimal constant, so a revision that adds the non-constant/positive qualification and treats δ=0 separately should suffice. This is a technical gap in the equality case, not an attack on the authors or on the main quantitative result.","tokens_in":11834,"tokens_out":13465,"duration_ms":139051,"concrete_test":"Instantiate Theorem 1.2 with n=2, C=B_2^2, f≡0, and φ(t)=t^2: the left side is 0 and the right side is (1/2)∫_{−1}^{1}0 dt=0, so equality holds although B_2^2 is not of the form [−x0,x0]+({0}×C0). Instantiate Theorem 1.3 with the same C and f≡1: both sides equal 1 (the right side is the limiting value lim_{a→1}(a^2−1)/log(a^2)=1), again equality without the cylinder structure. Either computation directly refutes the stated equality characterization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2's equality proof is not valid when the chosen supporting affine function g has δ=0: immediately after (9) it sets γ=g(0)/δ. If δ=0 and f(0)=0, concavity plus 0-symmetry forces f≡0, and then both sides of Theorem 1.2 are 0 for every C∈K^n_0, not only for the generalized cylinders listed in the equality characterization. The claim that equality in (11) forces |M**_t|=0 relies on strict convexity via the function F(t)=φ(a(1+t/t0))+φ(a(1−t/t0)) being strictly increasing; when a=f(0)=0 this F is identically 0, so no such conclusion follows. The same degeneracy propagates to Theorem 1.3: for f constant positive, u−u0≡0, equality in the auxiliary application of Theorem 1.2 is attained, and Theorem 1.3 gives equality on every C, contradicting its 'only if' characterization. For constant positive f in Theorem 1.2 itself equality actually fails (the average of φ(c(1+t)) exceeds φ(c)), so the missing case is precisely f≡0 (and constant positive f in Theorem 1.3). This is a genuine but local flaw in the equality statements, not in the main inequality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a functional inequality of Orlicz-Hermite-Hadamard type: for an n-dimensional 0-symmetric convex body C, a concave function f:C->[0,∞), and a convex function φ:[0,∞)->[0,∞) with φ(0)=0, one has (1/|C|)∫_C φ(f(x))dx ≤ (1/2)∫_{-1}^{1} φ(f(0)(1+t))dt. It also states equality characterizations for strictly convex φ, derives a corresponding estimate for log-concave functions (Theorem 1.3), and applies the power case to obtain the optimal constant in Francisco Santos's question on bounding |K| by |P_H K||K∩H^⊥| (Theorem 1.1). The inequality proof combines an affine majorant of f, Fubini's theorem, Schwarz symmetrization, and Brunn-Minkowski equality conditions.","tokens_in":12081,"tokens_out":7243,"duration_ms":74543,"significance":"If the inequality itself is correct, it is a clean extension of the classical Hermite-Hadamard inequality with an explicit and optimal constant for the Santos volume-section question, and the log-concave version is a useful companion estimate. The argument is explicit and does not rely on circular reasoning or fitted parameters. However, the equality characterizations, which are a prominent part of the abstract and of Theorems 1.2 and 1.3, are false as stated for constant functions, and the proof contains an unhandled degenerate case. These are genuine defects in load-bearing statements, though they appear local and repairable.","major_comments":[{"comment":"The equality characterization in Theorem 1.2 is false as stated because f≡0 gives equality for every C∈K^n_0: both sides of the inequality are 0, regardless of whether C is a generalized cylinder. The proof excludes this case implicitly when it defines γ=g(0)/δ immediately after Eq. (9), since for f≡0 the affine majorant g has g(0)=0 and δ=0, so γ is undefined. The theorem should either assume f is not identically zero (or f(0)>0) and add a separate trivial case, or correctly describe the full equality class for f≡0.","section":"Theorem 1.2 and proof immediately after Eq. (9)"},{"comment":"The same degeneracy invalidates the equality case of Theorem 1.3. If f≡c>0 on C, then fmin=f(0)=c, the right-hand side is c (the quotient tends to 1 as f(0)/fmin tends to 1), and the left-hand side is c, so equality holds for every C∈K^n_0. The proof applies Theorem 1.2 to u-u0≡0, which is exactly the unhandled constant-zero case identified above. The equality statement must exclude constant f or include the full set of equality cases.","section":"Theorem 1.3 and its proof"},{"comment":"The proof of the inequality itself is incomplete for affine majorants with δ=0. For a constant positive f, one may take g=f and then δ=0, so the quantity γ=g(0)/δ used in Eq. (9) is undefined. While the inequality for constant functions is elementary and could be recovered by approximation, the current proof does not provide any argument for this case. The same gap affects the equality proof, which later asserts that equality in (10) forces δ=f(0) without first establishing δ>0.","section":"Proof of Theorem 1.2, use of Eq. (9)"}],"minor_comments":[{"comment":"The abstract states f:C->[0,1) and g:[0,1)->[0,1), whereas Theorem 1.2 and the introduction use f:C->[0,∞) and φ:[0,∞)->[0,∞); these codomains should be corrected.","section":"Abstract"},{"comment":"In the displayed equation before (10), the integrand is written as (g(0)+tδ/t0)|M_t^{**}|dt, but it should be φ(g(0)+tδ/t0)|M_t^{**}|dt; an application of φ is missing.","section":"Equation (10)"},{"comment":"The text says M_t'=(te1+L)∩C and (-te1+L)∩C=-2te1+M_t', but M_t' was defined as the sections of C'=σ_{e1}(C), so the intersections should be with C' rather than C.","section":"Proof of Theorem 1.2, paragraph after Eq. (9)"},{"comment":"The example in Remark 2.1 takes f(x)=⟨x,e1⟩-1, which is non-positive on C=[-e1,e1]×C0 and hence not a function with values in [0,∞); it also does not vanish on the basis -e1+C0. The intended example is presumably a positive affine function such as f(x)=c(1+⟨x,e1⟩) with c>0, vanishing on the opposite basis.","section":"Remark 2.1"}],"recommendation":"major_revision","confidential_remarks":"The main inequality and the Santos constant appear sound, but the equality statements printed in Theorems 1.2 and 1.3 are false as stated because of the constant-function degeneracy, and the proof has a corresponding δ=0 gap. This is repairable within the manuscript's scope, but it affects the central claims in the abstract and theorems, so the revision should be more than a copyedit. The incorrect Remark 2.1 and the missing φ in Eq. (10) also suggest that the author should carefully re-read the displayed formulae."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main inequality is genuinely new and the application is worth having. The paper answers Santos's question with the optimal constant, and the Hermite-Hadamard-type bound with the center fixed at the origin is a real extension of the Milman–Pajor and Borell versions. The proof idea—compare the original body to a Schwarz-symmetrized cylinder with the same slices, then use convexity to push the integral to a two-point boundary term—is clean and, as far as I can tell, correct for the inequality itself.\n\nThe soft spot is the equality characterisation, and it is real but local. In Theorem 1.2, if f is identically zero then both sides are zero for every 0-symmetric C, so the “only if” part cannot require the generalized cylinder structure. The proof breaks at the point where γ = g(0)/δ is introduced: δ = 0 is not handled, and δ = 0 is exactly the constant-zero case. The fix is straightforward: either exclude f ≡ 0 from the equality statement, or add a sentence that if f ≡ 0 equality is trivial and otherwise f(0) > 0 and one can choose the supporting affine function with δ > 0. Theorem 1.3 inherits the same issue: constant positive log-concave functions give equality for every C, so the equality case there is false as stated. That is also easily fixed by excluding constant functions or stating the degenerate case separately.\n\nThere is also a sign typo in Remark 2.1: f(x) = ⟨x, e1⟩ − 1 is nonpositive on the cylinder, so it should be f(x) = 1 − ⟨x, e1⟩. Minor, but it should be corrected.\n\nThe citation pattern looks fine; the paper builds on standard results and does not fit parameters. The application to the volume question is solid because the critical part of Theorem 1.1 only needs the inequality, not the flawed equality case.\n\nBottom line: the main inequality and the constant for Santos's question are correct and worth publishing. The equality statements need revision before the paper is final. I would send it to a serious referee; a competent referee will find these issues quickly, but they are fixable and do not undermine the main results.","headline":"New inequality with a clean proof and the optimal constant for Santos's question; the equality characterisation is false for constant functions and needs a small fix.","tokens_in":12575,"tokens_out":7166,"would_cite":true,"duration_ms":69916,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","52A38","52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a generalized Hermite-Hadamard inequality for convex functions composed with concave functions on 0-symmetric convex bodies, and uses it to determine the optimal constant in a volume-versus-central-section bound with full…","keywords":["Hermite-Hadamard inequality","convex geometry","0-symmetric convex bodies","Schwarz symmetrization","Brunn-Minkowski inequality","log-concave functions","volume of central sections","optimal constants"],"falsifier":"Take any 0-symmetric convex body $C$ in dimension $n\\ge2$ that is not a generalized cylinder, for example a Euclidean ball, set $f(x)=0$ for all $x$, and set $\\varphi(t)=t^2$, which is strictly convex with $\\varphi(0)=0$. Then the left and right sides of Theorem 1.2 are both 0, so equality holds, even though the equality characterization would require $C$ to be a generalized cylinder. Checking this example directly would settle that the 'only if' part of the characterization is false without an extra nonconstant assumption on $f$.","tokens_in":11627,"feed_emoji":"📐","tokens_out":10913,"duration_ms":97737,"temperature":0.7,"pith_summary":"This paper proves a sharp, higher-dimensional extension of the Hermite-Hadamard inequality. For any 0-symmetric convex body $C$, any concave nonnegative function $f$ on it, and any convex nondecreasing $\\varphi$ with $\\varphi(0)=0$, the average of $\\varphi(f(x))$ over $C$ is no larger than the one-dimensional average of $\\varphi(f(0)(1+t))$ over $t\\in[-1,1]$. When $\\varphi$ is strictly convex, equality forces $C$ to be a generalized cylinder and $f$ to be affine and vanish on one base. The inequality yields an optimal constant $c_n=2^n/n$ for bounding the volume of a convex body by the volume of a central hyperplane section when its projection on the normal line is symmetric, and it gives a sharp Hermite-Hadamard bound for log-concave functions. The proof runs through Schwarz symmetrization and a volume-preserving comparison with a cylinder, so the same mechanism carries both the inequality and its equality cases.","feed_headline":"Sharp bound ties symmetric convex bodies to a central slice","feed_subtitle":"A generalized Hermite-Hadamard inequality yields the optimal constant and equality cases for volume-section bounds.","key_machinery":"The argument is carried by a one-dimensional comparison lemma for convex functions: for $\\varphi$ convex, nondecreasing, with $\\varphi(0)=0$, the symmetric sum $\\varphi(a-r)+\\varphi(a+r)$ is no larger than $\\varphi(a-\\gamma r)+\\varphi(a+\\gamma r)$ whenever $a\\ge \\gamma r\\ge0$ and $\\gamma\\ge1$ (inequality (9)). This is applied after symmetrizing $C$ with respect to the direction $e_1$ where an affine majorant $g$ of $f$ has its steepest descent. The symmetrized body $C'$ is sandwiched between two cylinders, $R_{t_0}\\subset C'\\subset R_0$, built from the section $M_0'$ and the extreme section $M_{t_0}'$; a volume-preserving intermediate cylinder $R$ with the same volume as $C'$ is selected. The slicewise comparison then shifts the $\\delta$-slope in $g(0)+(t/t_0)\\delta$ up to the full slope $g(0)(1+t/t_0)$, turning the integral over $C$ into the one-dimensional integral on the right. The equality case is decided by Brunn-Minkowski: equal section volumes force all sections of $C$ to be translates of one $(n-1)$-dimensional body, which is exactly a generalized cylinder.","core_discovery":"The central claim is Theorem 1.2: if $C$ is an $n$-dimensional 0-symmetric compact convex set, $f:C\\to[0,\\infty)$ is concave, and $\\varphi:[0,\\infty)\\to[0,\\infty)$ is convex, not identically zero, with $\\varphi(0)=0$, then\n$$\\frac{1}{|C|}\\int_C \\varphi(f(x))\\,dx \\le \\frac12\\int_{-1}^{1}\\varphi(f(0)(1+t))\\,dt.$$\nWhen $\\varphi$ is strictly convex, equality occurs exactly when, up to rotation, $C=[-x_0,x_0]+\\{0\\}\\times C_0$ for some $C_0\\in\\mathcal{K}^{n-1}_0$ and $f$ is affine with $f(-x_0+x)=0$ for every $x$ in the opposite base. From this the paper derives a volumetric inequality: if $K$ is convex and $H$ is a subspace whose projection $P_H K$ is symmetric, then $|K|\\le \\frac{2^{n-i}}{n-i+1}|P_H K||K\\cap H^\\perp|$, with equality characterized by a generalized cylinder structure. The same functional inequality gives a Hermite-Hadamard estimate for log-concave functions in terms of $\\log(f(0)/f_{\\min})$. The proof reduces the problem to one dimension by orienting along a support line, replacing the body by its Schwarz symmetrization, and sandwiching that symmetric body between two cylinders whose slices are concentric balls.","pith_inferences":["One extension not pursued in the paper: because the core comparison only needs convexity, monotonicity, and $\\varphi(0)=0$, the same proof should yield exponential-moment bounds $\\frac{1}{|C|}\\int_C e^{\\lambda f(x)}dx \\le \\frac12\\int_{-1}^1 e^{\\lambda f(0)(1+t)}dt$ for $\\lambda>0$, giving Laplacian-type estimates for concave functions on symmetric bodies.","The sandwich-by-cylinders argument is robust enough to suggest that the optimal constant extends to higher moments of section volumes: for any even moment of $|K\\cap(x+H^\\perp)|$, the extremal body should be the same generalized cylinder.","A natural testable extension is to non-symmetric bodies by replacing the role of 0 with the centroid or a related center; the inequality would then compare the average of $\\varphi(f)$ to a one-dimensional integral centred at that point, possibly with a dimension-dependent constant."],"forward_implications":["For $\\varphi(t)=t^\\alpha$, $\\alpha\\ge1$, Corollary 2.2 gives $\\frac{1}{|C|}\\int_C f^\\alpha \\le \\frac{2^\\alpha}{\\alpha+1} f(0)^\\alpha$, with equality if and only if $C$ is a generalized cylinder and $f$ is affine and vanishes on one base.","The volume inequality with the optimal constant $2^n/n$ answers the 2017 question: when a convex body's projection on a line is $[-e_1,e_1]$, the body's volume is at most $(2^n/n)$ times the volume of its intersection with the hyperplane $e_1^\\perp$; equality cases are explicitly described.","For log-concave $f$, Theorem 1.3 gives $\\frac{1}{|C|}\\int_C f \\le f_{\\min}\\frac{(f(0)/f_{\\min})^2-1}{\\log((f(0)/f_{\\min})^2)}$, with equality only for generalized cylinders and log-affine functions attaining the minimum on one base.","Theorem 3.2 removes the symmetry assumption in the hyperplane case: for any $K\\in\\mathcal{K}^n$ and $H\\in\\mathcal{L}^n_{n-1}$, $|K|\\le |P_H K||K\\cap(x_{P_H K}+H^\\perp)|$, with equality characterized. This choice can improve on the earlier centroid-section bound of (2)."],"supporting_citations":[{"why":"Supplies Jensen's inequality, the base of the Hermite-Hadamard bound used for hyperplane-section volume estimates.","marker":"[J]"},{"why":"Supplies Schwarz symmetrization and the fact that it preserves convexity and volume, the core geometric reduction.","marker":"[Gru, Section 9.3]"},{"why":"Supplies the Brunn-Minkowski inequality and its equality cases, used to identify generalized cylinders in the equality analysis.","marker":"[Ga]"},{"why":"Supplies Brunn's concavity principle, used to make the section-volume function concave in the volume estimate.","marker":"[Giann, Prop. 1.2.1]"},{"why":"Provides the earlier section/projection inequality that the new volumetric bound extends and improves in the symmetric-projection case.","marker":"[MP]"}],"fun_headline_variants":["Generalized Hermite-Hadamard sharpens convex volume bounds","Optimal volume-section inequality for symmetric convex bodies","Central slice bounds convex body volume with sharp equality","New Hermite-Hadamard type answer to Santos's volume question"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the equality case assumes the supporting line used to bound f actually tilts along the chosen direction; when f is constant, the key comparison step divides by this tilt and the equality characterization as stated misses those cases.","fun_headline_variants_meta":{"raw":{"variants":["Generalized Hermite-Hadamard sharpens convex volume bounds","Optimal volume-section inequality for symmetric convex bodies","Central slice bounds convex body volume with sharp equality","New Hermite-Hadamard type answer to Santos's volume question"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1730,"prompt_tokens":1043,"completion_tokens":687,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":621}},"tokens_in":659,"tokens_out":687,"duration_ms":7135,"temperature":1.0,"reasoning_tokens":621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:47:56.016653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any 0-symmetric convex body $C$ in dimension $n\\ge2$ that is not a generalized cylinder, for example a Euclidean ball, set $f(x)=0$ for all $x$, and set $\\varphi(t)=t^2$, which is strictly convex with $\\varphi(0)=0$. Then the left and right sides of Theorem 1.2 are both 0, so equality holds, even though the equality characterization would require $C$ to be a generalized cylinder. Checking this example directly would settle that the 'only if' part of the characterization is false without an extra nonconstant assumption on $f$.","supporting_citations":[],"review_version":1}