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Centro-affine Poincar\'e inequality: Unconditional convex bodies

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Pith's one-line read This paper proves that the centro-affine Poincaré inequality holds with the sharp constant n on every smooth unconditional convex body, with equality only for the zero function.

desk verdict Sharp centro-affine Poincaré inequality for unconditional bodies, with an elegant parity argument; the main caveat is heavy reliance on the authors' own unpublished preprint for the equality cases. read the letter →

arxiv 2607.20223 v1 pith:TYSRWT3L submitted 2026-07-22 math.AP

Yingxiang Hu , Mohammad N. Ivaki This is my paper · ORCID
classification math.AP MSC 52A4035P1553A15
keywords centro-affinePoincaréinequalityunconditionalconvexbodyspectralgapcone-volumemeasureevenlog-MinkowskiL_pMinkowskiuniquenessparitydecompositionLaplacian
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

This paper proves a sharp spectral-gap estimate for the centro-affine Laplacian on every smooth unconditional convex body: any smooth function orthogonal to constants and to the n coordinate functions l_i = x_i/h satisfies n∫F² dV ≤ ∫|∇F|² dV, with equality only for F ≡ 0. Previously this was known for unconditional functions; the paper removes the symmetry restriction on F by decomposing F into 2^n parity components and handling each type of component. The one-coordinate-odd components are reduced to the unconditional case by adding a suitable multiple of l_i, while the two-coordinate-odd components are treated by slicing the body into two-dimensional sections and applying planar inequalities fiberwise. What makes this worth caring about is that the strict spectral gap n directly yields the even log-Minkowski inequality when one body is unconditional, and the uniqueness of solutions to the L_p Minkowski problem in the supercritical range p ∈ [-n-1, -n) for n ≥ 3.

What carries the argument

The central device is the parity decomposition of F into components even or odd under each coordinate reflection. For the component odd in exactly one coordinate i, the paper adds a carefully chosen multiple of the eigenfunction l_i = x_i/h (which satisfies ∆l_i = -(n-1)l_i) to make it mean-zero on the positive orthant, then applies the known unconditional-function Poincaré inequality; the orthogonality condition ∫ F l_i dV_K = 0 ensures the added term does not change the quadratic form. For components odd in two coordinates i and j, the paper slices K into two-dimensional sections parallel to the (i,j)-plane. On each section the induced speed function is odd in both coordinate reflections,

What would settle it

Find a smooth unconditional convex body and a nonzero smooth function F satisfying both orthogonality conditions with n∫F² dV_K = ∫|∇F|² dV_K; this would disprove the equality characterization in Theorem 1.1. A concrete place to look is the equality case of the companion unconditional-function theorem: if any nonconstant unconditional function attains equality there, the main theorem's equality claim collapses.

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Extended reading notes

Core claim

The paper establishes Theorem 1.1: for a C^∞_+ unconditional convex body K in R^n, equipped with the centro-affine metric g = (1/h)(∇̄²h + hḡ) and cone-volume measure dV_K, every smooth function F on the sphere with ∫ F dV_K = 0 and ∫ F (x_i/h) dV_K = 0 for each i satisfies n∫ F² dV_K ≤ ∫ |∇F|²_g dV_K, with equality if and only if F ≡ 0. This is the sharp Poincaré constant n on the subspace orthogonal to the constant function and the n coordinate functions l_i = x_i/h. The proof is structural rather than analytic: it decomposes F into 2^n pieces by parity with respect to the coordinate reflections, proves the inequality separately for pieces whose parity pattern has zero, one, and at least t

Load-bearing premise

The argument relies on a companion paper's equality characterization for unconditional functions and a strict-positivity lemma; if either of those is wrong, the strictness and equality conclusions of the main theorem do not follow.

Editorial extensions

If this is right

  • The even log-Minkowski inequality holds for every origin-symmetric convex body L when K is a C^∞_+ unconditional convex body, with equality only if L = cK (Corollary 1.2).
  • For n ≥ 3 and -n-1 ≤ p < -n, the only smooth unconditional convex bodies with h^{1-p}_K = 1 are unit balls, settling uniqueness in that supercritical range (Corollary 1.3).
  • The first nonzero even eigenvalue of the centro-affine Laplacian on an unconditional body is strictly greater than n, because equality in Theorem 1.1 forces the function to vanish.
  • The strict inequality with equality characterization upgrades the earlier inequality-only results for unconditional bodies to a full spectral-gap statement with no extremal functions.

Reading between the lines

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

  • The slicing proof suggests an inductive strategy for the same inequality on arbitrary convex bodies: if every two-dimensional coordinate section satisfied the planar inequality uniformly, Fubini would give the n-dimensional result; this may point toward a proof of the full centro-affine Poincaré inequality by dimension induction.
  • The equality characterization in the companion unconditional-function theorem is load-bearing; if it were replaced by a non-sharp version, the strictness in Theorem 1.1 and the uniqueness in Corollary 1.3 would fail, but the inequality itself might still hold.
  • A quantitative version of the strict gap (λ > n) could yield stability estimates for the even log-Minkowski inequality for unconditional bodies, measuring how close a body is to being a homothetic copy in terms of the deficit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proves a sharp centro-affine Poincaré inequality on S^{n-1} for C∞_+ unconditional convex bodies: if F has zero mean with respect to the cone-volume measure dV_K and is orthogonal to the functions l_i = x_i/h, then n∫F² dV ≤ ∫|∇F|²_g dV, with equality iff F ≡ 0. The proof decomposes F into parity components under coordinate reflections; the fully even component is handled by a result quoted from the authors' preprint [HI26], the components odd in exactly one coordinate by a new orthogonality argument (Corollary 2.6), and the components odd in at least two coordinates by a new section-wise estimate (Lemma 2.4). The paper also derives applications: a local-to-global log-Minkowski inequality with one unconditional body (Corollary 1.2) and a uniqueness result in the supercritical L_p Minkowski problem (Corollary 1.3).

Significance. If the main theorem holds, it establishes the expected spectral gap n for the centro-affine Laplacian on every smooth unconditional convex body, extending earlier results for symmetric bodies and for unconditional functions. The proof is structurally interesting: the parity decomposition is clean, and the reduction to planar sections via Lemma 2.2 and Lemma 2.3 is elegant. The new Lemma 2.4 is a substantial technical contribution. The applications to the log-Minkowski inequality and to supercritical uniqueness are nontrivial and would be of interest to the community. However, the equality characterization and strictness arguments depend on two results quoted without proof from the authors' own unpublished preprint [HI26], so the central claim is not fully self-contained. A related proof in Corollary 1.3 also contains an erroneous factor 1/n, although it appears to be a fixable typo.

major comments (3)
  1. [Section 2, Theorem 2.5; Corollary 2.6; Theorem 1.1] Theorem 2.5 is quoted from the authors' preprint [HI26, Thm. 8.12] without proof; the manuscript says only 'This is part of [HI26, Thm. 8.12]'. This result supplies both the inequality and the equality characterization used in Corollary 2.6 and in the F_0 term of Theorem 1.1. The statement in Theorem 2.5 applies to arbitrary u∈C^1(S^{n-1}), while the introduction describes [HI26, Thm. 8.12] as an unconditional-function inequality. Thus the equality characterization 'only when u is constant on Ω_+' is not established in this manuscript. If that characterization fails, the equality statement of Theorem 1.1 (and hence Corollaries 1.2 and 1.3) does not follow. The authors should either include a full proof of Theorem 2.5 and the relevant part of [HI26], or clearly state this as an external result and verify that the hypotheses of [HI26, Thm. 8.12] are satisfied in the needed cases.
  2. [Lemma 2.4, second proof; also Lemma 2.2] The strictness arguments in the second proof of Lemma 2.4 rely on [HI26, Lem. 2.3] to conclude ⟨b,ξ⟩ > 0 whenever b≠0 (or P_F x≠0). Without this positivity lemma, the proof only gives a non-strict inequality, and the equality cases of Theorem 1.1 would collapse. Similarly, Lemma 2.2 uses [HI26, Lem. 2.3] for the strict sign of X_1 X_2. These positivity facts are load-bearing and are not proved in the manuscript. They should be stated as a lemma and proved, or the dependence on [HI26] should be made explicit with a precise statement and the status (published/unpublished) of [HI26] clarified.
  3. [Corollary 1.3, proof] The displayed definition of F_i contains an erroneous factor 1/n: F_i = X_i - (V/n)∑_j (A^{-1})_{ij} l_j yields ∫ F_i l_k dV = V(1 - 1/n)δ_{ik} ≠ 0, so Theorem 1.1 cannot be applied. The subsequent algebra — in particular ∑_i ∫ F_i² dV = ∫|X|² dV - V² tr(A^{-1}) — corresponds to the definition F_i = X_i - V∑_j (A^{-1})_{ij} l_j, without the factor 1/n. Please correct this typo; the proof is sound after this correction.
minor comments (6)
  1. [Section 1, capillary example] The paper states a half-space capillary inequality and then says 'We omit the details'. Since this result is not used in the main theorem and is presented without proof, it should be framed as a remark or conjecture rather than as a proved assertion.
  2. [Section 2, Theorem 2.5] The one-line proof is only a reference to [HI26, Thm. 8.12]. If the journal permits citing a preprint, the exact statement from [HI26] should be quoted and the hypotheses checked, especially because the statement here is broader than the unconditional-function description in the introduction.
  3. [Lemma 2.3] The term dS_{D0} in the final integral is not defined. Please define the surface measure used, or use dV_{D0} if that is intended.
  4. [First proof of Lemma 2.4, after (2.8)] The assertion that w_ξ(a) depends smoothly on (ξ,a) is stated without details. Please spell out the regularity of the minimizer in (2.4) and the dependence on the section parameter ξ.
  5. [Throughout] The paper uses notation and conventions from [HI26] (connections, Laplacian, dV_K, etc.) that are not fully defined in the present text. Adding a short list of conventions would improve readability.
  6. [References] Several references are to unpublished preprints ([HI26], [CHI26], [Iff26], [Du25]). If these are publicly available, please include arXiv identifiers and submission dates so that the reader can verify the cited results.

Circularity Check

2 steps flagged · score 4.0 of 10

Equality characterization in Theorem 1.1 rests on two unproved results from the authors' own preprint [HI26]; the inequality component has independent support, but the strict/uniqueness conclusions are load-bearing self-citations.

  1. self citation load bearing [Theorem 2.5, Section 2; used in Corollary 2.6 and in the proof of Theorem 1.1]
    "Theorem 2.5. Let u ∈ C^1(S^{n-1}). Then n ∫_{Ω_+} (u − ū_+)^2 dV_K ≤ ∫_{Ω_+} |∇u|^2_g dV_K, ū_+ := ... . Equality holds only when u is constant on Ω_+. Proof. This is part of [HI26, Thm. 8.12]; the unconditionality assumption on u was used there only for the second inequality (8.7)."

    The main equality analysis of Theorem 1.1 reduces the unconditional component F_0 to this quoted theorem, and Corollary 2.6 reduces every odd-in-one-coordinate component to it by adding t l_i. The equality characterization 'only when u is constant on Ω_+' is not proved or independently cited here; it is imported verbatim from the authors' own preprint [HI26]. Without that equality characterization, the equality cases of Theorem 1.1 and Corollaries 1.2/1.3 do not follow. The inequality part has independent published support from [KM22], but the equality part is a load-bearing self-citation.

  2. self citation load bearing [Second proof of Lemma 2.4, Section 2]
    "Due to [HI26, Lem. 2.3], ⟨b, ξ⟩ = Σ_{k ∉ {i,j}} x_k X_k(x) > 0, and in view of (2.11) we have s̃_{ξ,0}(a) s̈̃_{ξ,0}(a) ≤ −h(x)⟨b, ξ⟩ u(x)^2 < 0."

    The strict inequality needed to conclude strict positivity in Lemma 2.4 (and hence in Theorem 1.1) depends entirely on [HI26, Lem. 2.3] guaranteeing ⟨b, ξ⟩ > 0. That lemma is from the same authors' preprint and is not proved in this paper. If [HI26, Lem. 2.3] only gives nonnegativity or requires extra hypotheses, the displayed < 0 becomes ≤ 0 and the strictness conclusion collapses. This is another load-bearing import for the equality characterization, not an independently established step.

full rationale

The central inequality is not a disguised restatement of its inputs: Lemma 2.4 is new and largely self-contained, the odd-in-one-coordinate case is obtained from Theorem 2.5 by the t l_i projection, and the decomposition into parity components is standard. Moreover, the inequality component has independent published support in [KM22]. However, the equality characterization claimed in Theorem 1.1 and Corollaries 1.2 and 1.3 is not independently established in this paper. It rests on two results from the authors' own preprint [HI26]: Theorem 8.12 (quoted as Theorem 2.5) supplies the equality characterization for the unconditional component, and Lemma 2.3 supplies the strict positivity ⟨b,ξ⟩>0 used in the second proof of Lemma 2.4. Both are invoked without proof, and the paper explicitly labels Theorem 2.5 'part of [HI26, Thm. 8.12]'. The paper also explicitly omits details in the capillary aside ('We omit the details'), but that aside is not used in the main chain. Because the inequality content is supported independently while the strict/uniqueness content reduces to same-author citations, the appropriate circularity score is moderate, not maximal.

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

No empirical parameters are fitted; the ledger is about imported theorems. The main caveat is that Theorem 2.5 and Lemma 2.3 come from the same authors' preprint [HI26]; the inequality component has external support in [KM22].

assumptions (7)
  • domain assumption K is a C^∞_+ unconditional convex body in R^n with n≥2.
    Assumed throughout; the proof uses invariance under all coordinate reflections and smoothness/strict convexity.
  • domain assumption [HI26, Thm. 8.12]: the centro-affine Poincaré inequality for unconditional functions on Ω+, with equality only for constants.
    Quoted verbatim as Theorem 2.5; proof omitted and delegated to the authors' previous preprint. The inequality part has published support in [KM22], but equality characterization is not independently derived here.
  • domain assumption [HI26, Lem. 2.3] and the fact that unconditionality gives X_k = x_k B_k with B_k smooth and positive.
    Used to obtain the strict inequality <b,ξ> > 0 in the second proof of Lemma 2.4; asserted rather than proved in this paper.
  • domain assumption [HI25, Lem. 4.2]: ∫ x_i x_j dV_K = V δ_ij when dV_K = h^p dx and p ≠ -n.
    Used in the proof of Corollary 1.3; taken from the authors' published IMRN paper without proof.
  • standard math [BLYZ12, Thm. 1.3]: planar log-Brunn-Minkowski inequality for origin-symmetric planar convex bodies.
    Used in the second proof of Lemma 2.4 to compare the geometric-mean body with the section; accepted published theorem.
  • standard math [IM23a, Thm. 2.6]: local-to-global principle for the Lp-Minkowski problem via a spectral gap of a family.
    Used in Corollary 1.2 to convert λ_{1,e} > n into the global even log-Minkowski inequality; published theorem.
  • standard math Standard spectral theory and elliptic regularity give a smooth even first eigenfunction.
    Invoked in Corollary 1.2.

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Pith. "Pith review of Centro-affine Poincar\'e inequality: Unconditional convex bodies." pith.science (2026). https://pith.science/paper/TYSRWT3L

@misc{pith2026260720223,
  author       = {Pith},
  title        = {Pith review of: Centro-affine Poincar\'e inequality: Unconditional convex bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYSRWT3L}},
  note         = {Machine review of arXiv:2607.20223}
}
abstract

We prove that the centro-affine Poincar\'e inequality holds with constant $n$ for every $C^{\infty}_+$ unconditional convex body and every smooth function with natural orthogonality conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 3 linked inside Pith

  1. [1]

    Andrews, Classification of limiting shapes for isotropic curve flows, J

    B. Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), no. 2, 443--459

    2003

  2. [2]

    K. J. B\"or\"oczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn--Minkowski inequality, Adv. Math. 231 (2012), no. 3--4, 1974--1997

    2012

  3. [3]

    Brendle, K

    S. Brendle, K. Choi, P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math. 219 (2017), no. 1, 1--16

    2017

  4. [4]

    Cabezas-Moreno, Y

    C. Cabezas-Moreno, Y. Hu, M. N. Ivaki, On the conjectured capillary Blaschke--Santal\'o inequality, Mathematika 72 (2026), no. 3, Paper No. e70109

    2026

  5. [5]

    Colesanti, A

    A. Colesanti, A. Kolesnikov, G. Livshyts, L. Rotem, On weighted Blaschke--Santal\'o and strong Brascamp--Lieb inequalities, arXiv:2409.11503 (2024)

    arXiv 2024

  6. [6]

    S. Chen, Y. Huang, Q.-R. Li, J. Liu, The L_p -Brunn--Minkowski inequality for p<1 , Adv. Math. 368 (2020), Paper No. 107166

    2020

  7. [7]

    Du, Uniqueness of L_p Minkowski problem in the supercritical range, arXiv:2504.06946 (2025)

    S.-Z. Du, Uniqueness of L_p Minkowski problem in the supercritical range, arXiv:2504.06946 (2025)

    arXiv 2025

  8. [8]

    van Handel, The local logarithmic Brunn--Minkowski inequality for zonoids, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2020--2022, Lecture Notes in Math

    R. van Handel, The local logarithmic Brunn--Minkowski inequality for zonoids, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2020--2022, Lecture Notes in Math. 2327, Springer, Cham, 2023, 355--379

    2020

Show all 19 references
  1. [9]

    Y. Hu, M. N. Ivaki, Stability of the cone-volume measure with near constant density, Int. Math. Res. Not. IMRN 2025, no. 6, Paper No. rnaf062, 9 pp

    2025

  2. [10]

    Y. Hu, M. N. Ivaki, Weighted centro-affine Poincar\'e inequalities, arXiv:2606.04774 (2026)

    2026 arXiv

  3. [11]

    Iffland, The local logarithmic Brunn--Minkowski inequality for bodies of revolution, arXiv:2602.17912 (2026)

    L. Iffland, The local logarithmic Brunn--Minkowski inequality for bodies of revolution, arXiv:2602.17912 (2026)

    2026

  4. [12]

    M. N. Ivaki, E. Milman, L^p -Minkowski problem under curvature pinching, Int. Math. Res. Not. IMRN 2024, no. 10, 8638--8652

    2024

  5. [13]

    M. N. Ivaki, E. Milman, Uniqueness of solutions to a class of isotropic curvature problems, Adv. Math. 435 (2023), part A, Paper No. 109350

    2023

  6. [14]

    A. V. Kolesnikov, E. Milman, Local L^p -Brunn--Minkowski inequalities for p<1 , Mem. Amer. Math. Soc. 277 (2022), no. 1360, v+78 pp

    2022

  7. [15]

    A. V. Kolesnikov, G. Livshyts, L. Rotem, On p -Brunn--Minkowski and Brascamp--Lieb inequalities, arXiv:2507.12099 (2025)

    2025

  8. [16]

    Milman, Centro-affine differential geometry and the log-Minkowski problem, J

    E. Milman, Centro-affine differential geometry and the log-Minkowski problem, J. Eur. Math. Soc. 27 (2025), no. 2, 709--772

    2025

  9. [17]

    Putterman, Equivalence of the local and global versions of the L^p -Brunn--Minkowski inequality, J

    E. Putterman, Equivalence of the local and global versions of the L^p -Brunn--Minkowski inequality, J. Funct. Anal. 280 (2021), no. 9, Paper No. 108956, 20 pp

    2021

  10. [18]

    R. T. Rockafellar, Convex analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, NJ, 1970

    1970

  11. [19]

    Schneider, Convex bodies: The Brunn--Minkowski theory, 2nd expanded ed., Encyclopedia Math

    R. Schneider, Convex bodies: The Brunn--Minkowski theory, 2nd expanded ed., Encyclopedia Math. Appl. 151, Cambridge University Press, Cambridge, 2014

    2014

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