{"id":"fa0d8354-4e09-421c-a83a-4ce3c7bbe306","arxiv_id":"2411.10439","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces the L^p-Legendre transform and proves a functional L^p-Santaló inequality: after translation by the L^p-Santaló point, the L^p-Mahler integral of any convex function is bounded by that of the Gaussian |x|^2/2.","lead":"This paper defines an L^p version of the Legendre transform for convex functions and proves that, after an optimal translation, the associated Mahler integral is no larger than the Gaussian's. This functional Santaló inequality generalizes a classical convex-geometry result and links it to a Fokker-Planck heat-flow proof.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.11 applies Proposition 1.15 to translated functions that do not satisfy the Fokker–Planck equation; the missing drift term invalidates the monotonicity step as written.","rationale":"The reader's weakest_assumption focuses on the superlinear growth condition (Lemma A.1) needed for integration by parts. That condition is guaranteed for every φ∈Cvx(R^n), so it is not the most fragile point of the proof. The central theorem is proved via monotonicity along the Fokker–Planck heat flow, and the decisive move in §6.3.3 is differentiating a Mahler integral after a time-dependent translation by the Santaló point. The presentation applies Proposition 1.15 to T_{s_p(φ)}φ as if it were a solution of (6.1), but translations are not symmetries of the Ornstein–Uhlenbeck drift. This is a genuine logical gap in the proof as written. It is likely fixable by a direct evolution computation for the tilted measure e^{-φ^{*p}+⟨s,y⟩}dy, where the extra drift term produces M_p⟨s,b⟩ and vanishes at b=0; however, the paper does not contain this computation. Since the gap is real but plausibly repairable, the appropriate outcome is the same conditional verdict the reader gave, rather than rejection or unconditional acceptance. My main concern therefore differs from the reader's, so agreement is 'disagree' while the final verdict remains unchanged.","tokens_in":39095,"tokens_out":11426,"duration_ms":103504,"concrete_test":"Recompute the proof of Theorem 1.11 with a time-dependent translation. For fixed s, set ψ_s(t,x)=φ(t,x+s) and derive ∂_tV(ψ_s^{*p}) directly from Lemma 6.5, keeping the extra −⟨s,y⟩ term in ∂_tψ_s^{*p} that arises from the non-commutation of translation with the Fokker–Planck drift. Then impose s=s_p(t) and b(ψ_s^{*p})=0 and check whether ∂_tM_p(ψ_s)≥0 follows from the same Cramér–Rao and Brascamp–Lieb estimates; in particular, verify that the extra term M_p⟨s,b⟩ vanishes exactly at the Santaló point and does not upset the inequality. If the lower bound fails or cannot be established, the monotonicity argument does not prove Theorem 1.11.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §6.3.3, the proof sets g(t)=M_p(T_{s_p(φ)}φ) and writes g'(t)=∂_tM_p(T_{s_p(φ)}φ)+⟨∂_t s_p(φ),(D_xM_p(T_xφ))(s_p(φ))⟩. The second term vanishes because s_p minimizes x↦M_p(T_xφ). However, the first term is a time derivative with the translation s_p held fixed, and the function ψ_s(t,x)=φ(t,x+s) is not a solution of the Fokker–Planck equation (6.1). The drift term ⟨x,Dφ⟩ breaks translation invariance: ∂_tψ_s satisfies (6.1) with an additional +⟨s,Dψ_s⟩. Consequently Proposition 1.15 cannot be invoked for T_{s_p(φ)}φ. A direct computation shows this extra drift adds a term M_p(ψ_s)⟨s,b(ψ_s^{*p})⟩ to the evolution equation for M_p(ψ_s); at s=s_p, the barycenter condition b(ψ_s^{*p})=0 makes this term vanish, but this cancellation is not present in the paper. Without this missing computation, the inequality ∂_tM_p(T_sφ) ≥ −(p/(p+1))M_p|b|^2 is unproved, and the monotonicity g'(t)≥0 does not follow. The reader's identified concern about superlinear growth is actually satisfied for Cvx by Lemma A.1; the translation non-invariance is the more load-bearing gap.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the L^p-Legendre transform and L^p-Mahler integral for convex functions, establishes their basic properties, computes several explicit examples, and proves existence and uniqueness of L^p-Santaló points. Its main result, Theorem 1.11, asserts that for every p>0 and φ in Cvx(R^n), the infimum over translations of M_p(T_x φ) is bounded above by M_p(|x|^2/2). The proof is via monotonicity of the Mahler integral along the Fokker-Planck heat flow, after centering at the L^p-Santaló point. The paper also gives asymptotics of M_p(B_2^n) for p=1 and discusses connections to Nakamura-Tsuji and Tao's Laplace transform inequality.","tokens_in":39460,"tokens_out":7989,"duration_ms":66074,"significance":"The functional L^p-Santaló inequality (Theorem 1.11) is a significant new result if the proof can be completed. It generalizes known body-level inequalities and matches the sharp Gaussian extremizer, with connections to Bourgain's conjecture via the L^p-polarity program. The paper contains useful explicit computations of L^p-Legendre transforms and Mahler integrals, and it clearly discloses the relationship with the Nakamura-Tsuji inequality. The p=1 asymptotic computation of M_1(B_2^n) is a valuable technical contribution, though it is currently restricted to odd n. The main proof relies on a Fokker-Planck flow; as written, it contains a gap involving the translation of the evolving function that must be repaired.","major_comments":[{"comment":"The proof of Theorem 1.11 invokes Proposition 1.15 for the translated function T_{s_p(φ)}φ, but translation does not preserve the Fokker-Planck equation (6.1). Specifically, if φ solves (6.1), then ψ(t,x)=φ(t,x+s) satisfies ∂_t ψ = Δψ - |Dψ|^2 + ⟨x+s,Dψ⟩ - n, which contains an extra ⟨s,Dψ⟩ drift term. Proposition 1.15 therefore cannot be applied directly to T_s φ. The missing computation is to derive the evolution of M_p(T_s φ) for fixed s, which yields an additional term M_p(T_s φ)⟨s,b((T_s φ)^{*p})⟩; at s=s_p(φ(t)) this term vanishes by the barycenter characterization. Until this computation is included, the inequality g'(t) ≥ 0 is not justified.","section":"§6.3.3"},{"comment":"The proof of Conjecture 5.1 for p=1 uses Corollary 5.6, which is proved only for odd n (the formula for the integral of t^{2m+n/2+1} K_{n/2+1}(t) relies on (n+1)/2 being an integer). The subsequent conclusion M_1(B_2^n)=(4π)^n e^{o(n)} is therefore not established for even n. The statement 'We prove Conjectures 5.1 and 5.2 for p=1' overstates what is shown.","section":"§5.2"}],"minor_comments":[{"comment":"The notation s_p(φ) is used for the Santaló point of the time-evolving function without making the time dependence explicit; this makes the chain rule computation hard to follow. Use s_p(t) or s_p(φ(t,·)).","section":"§6.3.3"},{"comment":"The proof of Theorem 1.11 uses the chain rule requiring ∂_t s_p(φ); differentiability of the Santaló point along the flow is not proved. The implicit function theorem applied to the smooth strictly convex map x↦M_p(T_x φ(t)) should yield it, but a statement is needed.","section":"§6.3.3"},{"comment":"Typos: 'B/suppress locki' appears in the abstract and in reference [4]; 'Lemma 6.11' in the proof of Corollary 1.12 should read 'Corollary 6.11'.","section":"§1.2 / References"},{"comment":"The measure in (6.7) is denoted dφ_{p,y} but later occurrences use dφ^{p,y}; unify the notation for clarity.","section":"§6.2.2"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely true and the gap in §6.3.3 appears repairable by the computation indicated by the referee. The paper should also address the odd-n restriction in the p=1 asymptotics or clearly state the result for odd n only. The overlap with Cordero-Erausquin-Fradelizi-Langharst and with Nakamura-Tsuji is disclosed; the author should ensure proper citation and attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it introduces a genuinely new object, the Lp-Legendre transform, and proves a functional Lp-Santaló inequality that goes beyond the Nakamura–Tsuji even case. Second, the proof of the main theorem as written has a gap: translated functions do not satisfy the Fokker–Planck equation, so Proposition 1.15 cannot be applied directly. The gap looks fixable, but it is real and needs to be addressed.\n\nWhat is new and good: the Lp-Legendre transform is a clean interpolation between the classical support function and the Legendre transform, and the paper establishes the basic properties (convexity, l.s.c., smoothness, tensoriality, reverse inequalities). The examples—L1 norm, quadratic, simplex, ball—are carefully and usefully computed. The evolution equations under the Fokker–Planck flow are derived transparently, and the author is honest that the even case is equivalent to Nakamura–Tsuji and that the preprint [7] substantially overlaps. The p=1 ball asymptotics for odd dimensions is a real computation, even if limited.\n\nSoft spots: the main one is in §6.3.3. The proof sets g(t)=M_p(T_{s_p(φ)}φ) and then applies Proposition 1.15 to T_s φ, but T_s φ does not solve (6.1) because the drift term ⟨x,Dφ⟩ is not translation-invariant. The stress-test note is right that the extra drift contributes a term proportional to ⟨s,b(ψ_s^{*p})⟩, which vanishes at the Santaló point by the barycenter condition. That computation is absent, so the monotonicity step is unproved as written. I do not see this as fatal—it is a missing computation plus a slightly misleading use of the proposition—but a referee should ask for it. Also, the claim that Conjectures 5.1 and 5.2 are proved for p=1 is overstated: the Bessel-function argument explicitly assumes odd n. The even case is not done. Finally, the geometric approximation route is openly deferred to a \"subsequent paper,\" which is fine but means the first approach advertised in the introduction is incomplete.\n\nThis paper is for people working in convex geometry, functional Santaló inequalities, and log-concave functions. It deserves a serious referee and, I think, publication after a revision that closes the translation gap and corrects the p=1 statement to odd dimensions only.","headline":"A genuinely new Lp-Legendre transform and a promising functional Santaló inequality, but the main proof has a translation-invariance gap that is likely repairable.","tokens_in":39963,"tokens_out":3776,"would_cite":true,"duration_ms":36773,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A41","52A40","26B25","46E30","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a functional $L^p$-Santaló inequality: for every $p\\in(0,\\infty)$, every finite-volume convex function has $L^p$-Mahler integral at most $M_p(|x|^2/2)$ after optimal translation.","keywords":["Lp-Legendre transform","Lp-Mahler integral","Lp-Santaló point","functional Santaló inequality","Fokker–Planck heat flow","convex functions","log-concave functions","Mahler conjecture"],"falsifier":"Evaluate the explicit formulas in Lemmas 3.4 and 3.6 for any $p$ and $n$: if $M_p$ of the $L^1$ norm, suitably translated, ever exceeded $M_p(|x|^2/2)$, Theorem 1.11 would be false; for $p=1$, $n=1$ this check is $(32/3)$ versus $4\\pi$. A direct computational check of the flow would also suffice: run the Fokker–Planck evolution on a centered non-quadratic convex function and test whether its $L^p$-Mahler integral ever decreases; a single decrease would contradict Proposition 1.15.","tokens_in":38922,"feed_emoji":"📐","tokens_out":11769,"duration_ms":101618,"temperature":0.7,"pith_summary":"The paper builds functional analogues of $L^p$-polarity and the Mahler volume: for a convex function $\\varphi$ it defines an $L^p$-Legendre transform $\\varphi^{*,\\kern0.4pt p}$ and an $L^p$-Mahler integral $M_p(\\varphi)=V(\\varphi)V(\\varphi^{*,\\kern0.4pt p})$, where $V(\\varphi)=\\int e^{-\\varphi}$. Its main theorem is a functional $L^p$-Santaló inequality: for any $p\\in(0,\\infty)$ and any proper, lower semi-continuous, convex function $\\varphi$ with $0<V(\\varphi)<\\infty$, one has $\\inf_{x\\in\\mathbb{R}^n} M_p(T_x\\varphi) \\le M_p(|x|^2/2)$. Since $|x|^2/2$ is the potential of the standard Gaussian density, this identifies the Gaussian, after optimal centering, as the maximizer among all such functions. The proof shows that the $L^p$-Mahler integral is monotone nondecreasing along the Fokker–Planck heat flow once the evolving function is recentered at its $L^p$-Santaló point, and that the flow converges to the quadratic potential. The paper also computes explicit $L^p$-Legendre transforms and Mahler integrals for basic examples and derives the $p=1$ dimensional asymptotics of the Euclidean ball needed for a second, geometric proof approach.","feed_headline":"The Gaussian maximizes the Lp-Mahler integral","feed_subtitle":"Heat-flow proof: every convex function's Lp-Mahler integral sits below |x|^2/2 after centering.","key_machinery":"The central mechanism is the Fokker–Planck heat flow on convex functions, written as $\\partial_t \\varphi = \\Delta\\varphi - |D\\varphi|^2 + \\langle x,D\\varphi\\rangle - n$, which is the transport of the log-concave density $e^{-\\varphi}$ toward a Gaussian and which keeps the volume $V(\\varphi)$ constant. Along this flow the paper computes the evolution of $\\varphi^{*,\\kern0.4pt p}$ and of $M_p(\\varphi)$; the formula involves the Fischer information of the tilted probability measures $d\\varphi_{p,y}$ proportional to $e^{p\\langle x,y\\rangle-(p+1)\\varphi(x)}\\,dx$. Two standard inequalities close the loop: the Cramér–Rao inequality, bounding Fischer information by the inverse covariance, and a Brascamp–Lieb variance bound for log-concave measures. Together they give $\\partial_t M_p(\\varphi) \\ge -\\frac{p}{p+1} M_p(\\varphi)\\,|b(\\varphi^{*,\\kern0.4pt p})|^2$, so whenever $\\varphi$ is centered at its $L^p$-Santaló point (where $b(\\varphi^{*,\\kern0.4pt p})=0$) the Mahler integral is monotone nondecreasing and, because the flow ends at $|x|^2/2$ up to constants, the Gaussian value is an upper bound.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.11: for every $p\\in(0,\\infty)$ and every $\\varphi\\in\\mathrm{Cvx}(\\mathbb{R}^n)$, $\\inf_{x\\in\\mathbb{R}^n} M_p(T_x\\varphi) \\le M_p(|x|^2/2)$. The $L^p$-Mahler integral factors as the product of the volume of $\\varphi$ and the volume of its $L^p$-Legendre transform, mirroring the classical Mahler volume of a convex body, and the optimal translation is the unique $L^p$-Santaló point, characterized by the vanishing of the barycenter of the transformed function. The theorem says that among finite-volume convex functions, the quadratic potential $|x|^2/2$ — equivalently the standard Gaussian density $e^{-|x|^2/2}$ — has the largest $L^p$-Mahler integral after centering. The author proves this by running $\\varphi$ through the Fokker–Planck heat flow, deriving the evolution equations for the $L^p$-Legendre transform and the Mahler integral, and using them to show the centered Mahler integral is monotone increasing in time and converges to the Gaussian value.","pith_inferences":["The monotonicity statement suggests that the centered $L^p$-Mahler integral is a Lyapunov functional for the Fokker–Planck flow, so the proof may also yield quantitative convergence rates or stability bounds around the Gaussian maximizer.","A natural equality-case conjecture, not stated as a theorem in the paper, is that maximizers are exactly convex quadratics (affine images of $|x|^2/2$); this could be tested by examining strictness in the Cramér–Rao and Brascamp–Lieb steps.","The heat-flow proof indicates that the same monotonicity should hold for the classical Mahler functional at $p=\\infty$, offering a possible new proof of the functional Santaló inequality that bypasses the usual geometric limit arguments.","The explicit formulas for the $L^1$ norm and the functional simplex provide ready-made numerical checks of Conjectures 1.8 and 1.9 in finite dimension, which could guide the search for counterexamples or sharpen the conjectured constants."],"forward_implications":["The $L^p$-Mahler integral of any finite-volume convex function, optimally translated, is at most the Gaussian value; this is a sharp functional Santaló-type bound holding for every $p\\in(0,\\infty)$.","Taking the limit $p\\to\\infty$ recovers the classical functional Santaló inequality and connects the new conjectures to the $p=\\infty$ functional Mahler conjectures for convex functions.","For even functions the theorem is equivalent to a sharp bound for the $L^p$ norms of Laplace transforms of log-concave functions, so it transfers a known heat-flow inequality from the even to the general convex case.","The $p=1$ computation of the Mahler volume of the Euclidean ball, $M_1(B_2^n)=(4\\pi)^n e^{o(n)}$, matches the Gaussian asymptotic and supplies the missing ingredient for a geometric approximation proof of the same inequality.","Because convexity is used only for the superlinear growth it guarantees, the same proof gives the inequality for non-convex measurable functions with linear growth at infinity (Corollary 1.12)."],"supporting_citations":[{"why":"Introduces $L^p$-polarity, $L^p$-Mahler volumes, and $L^p$-Santaló points for convex bodies, the construction this paper extends to functions.","marker":"[3]"},{"why":"Provides the sharp heat-flow bound for even functions that the paper shows is equivalent to its Theorem 1.11 in the even case.","marker":"[26]"},{"why":"Supplies the Fokker–Planck heat-flow method used to obtain monotonicity of the Mahler integral.","marker":"[25]"},{"why":"The Brascamp–Lieb variance inequality is the key input in bounding the trace of the covariance in Corollary 6.13.","marker":"[5]"},{"why":"Defines the Santaló point of a function and provides the geometric approximation method used as the paper's second route toward Theorem 1.11.","marker":"[1]"},{"why":"States the $L^p$-Santaló inequality for convex bodies that the paper generalizes to the functional setting.","marker":"[22]"}],"fun_headline_variants":["Heat flow proves Gaussian maximizes Lp-Mahler integral","Gaussian extremal for Lp-Mahler after centering","Fokker-Planck flow shows Lp-Mahler max at Gaussian","Lp-Mahler integral: every convex function loses to Gaussian","Centered Lp-Mahler maximizes at Gaussian potential"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the function $\\varphi$ grows at least linearly at infinity, $\\varphi(x) \\ge a|x| + b$ with $a>0$, which makes the integrations by parts in the evolution equations free of boundary terms; for convex functions with finite positive volume the paper proves this growth is automatic.","fun_headline_variants_meta":{"raw":{"variants":["Heat flow proves Gaussian maximizes Lp-Mahler integral","Gaussian extremal for Lp-Mahler after centering","Fokker-Planck flow shows Lp-Mahler max at Gaussian","Lp-Mahler integral: every convex function loses to Gaussian","Centered Lp-Mahler maximizes at Gaussian potential"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1663,"prompt_tokens":1045,"completion_tokens":618,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":661,"tokens_out":618,"duration_ms":5985,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:37:38.158928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the explicit formulas in Lemmas 3.4 and 3.6 for any $p$ and $n$: if $M_p$ of the $L^1$ norm, suitably translated, ever exceeded $M_p(|x|^2/2)$, Theorem 1.11 would be false; for $p=1$, $n=1$ this check is $(32/3)$ versus $4\\pi$. A direct computational check of the flow would also suffice: run the Fokker–Planck evolution on a centered non-quadratic convex function and test whether its $L^p$-Mahler integral ever decreases; a single decrease would contradict Proposition 1.15.","supporting_citations":[{"cited_title":"Artstein, B","cited_arxiv_id":null,"evidence_quote":"Defines the Santaló point of a function and provides the geometric approximation method used as the paper's second route toward Theorem 1.11."},{"cited_title":"Berndtsson, V","cited_arxiv_id":null,"evidence_quote":"Introduces $L^p$-polarity, $L^p$-Mahler volumes, and $L^p$-Santaló points for convex bodies, the construction this paper extends to functions."},{"cited_title":"The functional volume product under heat flow","cited_arxiv_id":"2401.00427","evidence_quote":"Provides the sharp heat-flow bound for even functions that the paper shows is equivalent to its Theorem 1.11 in the even case."},{"cited_title":"Hypercontractivity beyond Nelson's time and its applications to Blaschke--Santal\\'{o} inequality and inverse Santal\\'{o} inequality","cited_arxiv_id":"2212.02866","evidence_quote":"Supplies the Fokker–Planck heat-flow method used to obtain monotonicity of the Mahler integral."},{"cited_title":"Brascamp, E","cited_arxiv_id":null,"evidence_quote":"The Brascamp–Lieb variance inequality is the key input in bounding the trace of the covariance in Corollary 6.13."},{"cited_title":"Mastrantonis, A Santal´ o inequality for theLp-polar body, preprint, 2024, to appear in Contemp","cited_arxiv_id":null,"evidence_quote":"States the $L^p$-Santaló inequality for convex bodies that the paper generalizes to the functional setting."}],"review_version":1}