$L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow

Vlassis Mastrantonis

REVIEW 2 major objections 4 minor 34 references

$L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.10439 v2 pith:UEUXFATN submitted 2024-11-15 math.FA

Vlassis Mastrantonis This is my paper · ORCID

classification math.FA MSC 52A4152A4026B2546E3060J60

keywords Lp-LegendretransformLp-MahlerintegralLp-SantalópointfunctionalSantalóinequalityFokker–Planckheatflowconvexfunctionslog-concaveMahlerconjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

This 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[§6.3.3] 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.
[§5.2] 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.

minor comments (4)

[§6.3.3] 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,·)).
[§6.3.3] 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.
[§1.2 / References] 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'.
[§6.2.2] The measure in (6.7) is denoted dφ_{p,y} but later occurrences use dφ^{p,y}; unify the notation for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the functional Lp-Santaló inequality is derived from the Fokker–Planck flow, not from its target.

full rationale

The central claim (Theorem 1.11) is proved by deriving evolution equations for the Lp-Legendre transform and the Lp-Mahler integral under the Fokker–Planck heat flow (Lemmas 6.5 and 6.8), then using monotonicity after centering by the Lp-Santaló point. These evolution equations are computed from the PDE (6.1) by direct differentiation, integration by parts, and standard inequalities (Cramér–Rao and Brascamp–Lieb); no parameter is fitted to Mahler-integral data and no target-dependent assumption is used. The extremal value M_p(|x|^2/2) enters only as the explicit large-time limit of the flow (Lemma 6.3) and is computed independently (Lemma 3.6). The paper explicitly discloses that Theorem 1.11 for even functions is equivalent to Nakamura–Tsuji's Theorem 3.10 via Lemma 1.13; this is an acknowledged equivalence with an external result, not a hidden import of the conclusion. Self-citations to the author's prior work [3,22,24] provide background definitions and the convex-body analog, but the functional proof does not reduce to those citations. A possible concern in §6.3.3 is that the translated function T_{s_p(φ)}φ does not obviously satisfy the Fokker–Planck equation (6.1), so applying Proposition 1.15 to it may require an extra drift-term computation; however, this is a correctness issue, not circularity, since no assertion is being assumed in place of its proof. No self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chain can be exhibited from the text.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard inequalities (Cramér-Rao, Brascamp-Lieb, Prékopa-Leindler, Gaunt), the definitional class Cvx, and the superlinear growth needed for boundary-free integration by parts. There are no fitted free parameters. No new physical entities are postulated; the L^p-Legendre transform and L^p-Mahler integral are new mathematical definitions but do not carry independent-evidence obligations.

assumptions (7)

standard math Standard measure-theoretic tools: Hölder, Jensen, Fatou, dominated convergence, integration by parts.
Used throughout Sections 2, 4 and 6 without separate proofs.
standard math Prékopa-Leindler inequality
Invoked in the proof of Lemma 2.14 to control the volume of the infimal convolution.
standard math Cramér-Rao inequality for probability measures
Invoked as Theorem 6.9 to lower-bound Fisher information by the inverse covariance.
standard math Brascamp-Lieb variance inequality
Invoked as Lemma 6.12 to bound the trace of the covariance of φ^{*,p} by an integral of Fisher information.
domain assumption Gaunt's uniform bound on modified Bessel functions
Lemma 5.3, cited from the literature, is used to replace I_{n/2}(r)^{-1/p} by an equivalent expression with K_{n/2+1}(r)^{1/p} in the asymptotics.
domain assumption Membership in Cvx(R^n): proper, convex, lower semi-continuous, with 0 < V(φ) < ∞
This class is the domain of the theorem. It ensures finite volume and, by Lemma A.1, superlinear growth φ(x) ≥ a|x|+b.
domain assumption Superlinear growth (or convexity) for vanishing boundary terms in integration by parts
Stated explicitly in the proof of Corollary 1.12: 'Super-linearity is essential to ensure the validity of the integration by parts used throughout.'

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Pith. "Pith review of $L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow." pith.science (2026). https://pith.science/paper/UEUXFATN

@misc{pith2026241110439,
  author       = {Pith},
  title        = {Pith review of: $L^p$-Legendre Transforms and Mahler integrals: Asymptotics and the Fokker--Planck heat flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEUXFATN}},
  note         = {Machine review of arXiv:2411.10439}
}

abstract

$L^p$-polarity and $L^p$-Mahler volumes were recently introduced by Berndtsson, Rubinstein, and the author as a new approach, inspired by complex geometry, to the Mahler, Bourgain, and Blocki conjectures. This paper serves two purposes. First, it introduces functional analogues of these notions and establishes functional versions of key theorems previously formulated in the setting of convex bodies. This involves introducing the $L^p$-Legendre transform and analyzing the associated Santal\'o points and the dimensional asymptotics of the Mahler volumes of conjectured extremizers. Second, the paper investigates the connection between the $L^p$-Legendre transform and the recent work of Nakamura--Tsuji on the Fokker--Planck heat flow. As a byproduct, a functional $L^p$-Santal\'o inequality is established. The proof is based on deriving the evolution equations for the $L^p$-Legendre transform and Mahler integral under the Fokker--Planck heat flow. A second approach, using the geometric method of Artstein--Klartag--Milman is also presented, for which the necessary asymptotics are derived in the $L^1$-case.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [7]

On a Santal\'o point for Nakamura-Tsuji's Laplace transform inequality

D. Cordero-Erausquin, M. Fradelizi, D. Langharst, On a Santal´ o point for Nakamura– Tsuji’s Laplace transform inequality, preprint, 2024, arXiv:2409.05541

work page Pith review arXiv 2024

[2] [1]

Artstein, B

S. Artstein, B. Klartag, V. Milman, The Santal´ o point of a function, and a functional form of Santa´ o inequality, Mathematika 51 (2004), pp. 33–48

work page 2004

[3] [2]

Ball, Logarithmically concave functions and sections of convex s ets in Rn, Studia Math

K. Ball, Logarithmically concave functions and sections of convex s ets in Rn, Studia Math. 88 (1988), pp. 69–84

work page 1988

[4] [3]

Berndtsson, V

B. Berndtsson, V. Mastrantonis, Y.A. Rubinstein, Lp-polarity, Mahler volumes, and the isotropic constant, Anal. PDE 17-6 (2024), pp. 2179–2245

work page 2024

[5] [4]

B/suppress locki,On Nazarov’s complex analytic approach to the Mahler conjec ture and the Bourgain-Milman inequality, in: Complex Analysis and Geometry (F

Z. B/suppress locki,On Nazarov’s complex analytic approach to the Mahler conjec ture and the Bourgain-Milman inequality, in: Complex Analysis and Geometry (F. Bracci et al., Eds.), Springer, 2015, pp. 89–98

work page 2015

[6] [5]

Brascamp, E

H. Brascamp, E. H. Lieb, On extensions of the Brunn–Minkowski and Pr´ ekopa–Leindler theorems, including inequalities for log concave function s, and with an application to the diﬀusion equation, J. Funct. Anal. 22 (1976), pp. 366–389

work page 1976

[7] [6]

Brazitikos, A

S. Brazitikos, A. Giannopoulos, P. Valletas, B-H. Vrits iou, Geometry of isotropic convex bodies, Amer. Math. Soc., 2014

work page 2014

[8] [8]

Duality and Heat flow

D. Cordero-Erausquin, N. Gozlan, S. Nakamura, H. Tsuji, Duality and Heat ﬂow, preprint, 2024, arXiv:2403.15357

work page Pith review arXiv 2024