On the conjectured capillary Blaschke-Santal\'o inequality

Carlos Cabezas-Moreno; Mohammad N. Ivaki; Yingxiang Hu

REVIEW 3 major objections 4 minor 19 references

On the conjectured capillary Blaschke-Santal\'o inequality

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves the conjectured capillary Blaschke–Santaló inequality for unconditional strictly convex capillary hypersurfaces when the contact angle is acute, and shows the volume product is unbounded for obtuse angles.

desk verdict The main theorem is right but the printed constant is wrong: it should be 2n, not 2^n; with that fix the paper is a solid proof of the unconditional capillary Blaschke–Santaló case. read the letter →

arxiv 2509.20257 v2 pith:NK2SH7QX submitted 2025-09-24 math.DG math.FAmath.MG

Carlos Cabezas-Moreno , Yingxiang Hu , Mohammad N. Ivaki This is my paper · ORCID

classification math.DGmath.FAmath.MG MSC 52A4053A15

keywords capillaryhypersurfaceBlaschke–SantalóinequalityunconditionalconvexbodyvolumeproductPrékopa–LeindlerLegendretransformcentro-affinecontactangle

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 proves a half-space analog of the classical Blaschke–Santaló inequality: among convex bodies that meet the boundary plane at a fixed acute angle and are symmetric in each coordinate direction, the spherical cap maximizes the product of the body's volume and the volume of its capillary polar body. This confirms a conjecture for the unconditional case. For obtuse contact angles, the paper constructs families of convex bodies for which this volume product grows without bound, so no such inequality can hold there. If correct, the result extends a cornerstone of convex geometry to a capillary setting and provides a sharp equality characterization.

What carries the argument

The proof reduces the geometric inequality to an analytic one via the Legendre-transform pair V = ½p_C² and V* = ½h_C², where C is the convex body obtained by reflecting the capillary cap across the boundary hyperplane. The core identity det D²V(x) · det D²V*(DV(x)) = 1, together with the concavity of V(√x) and the Prékopa–Leindler inequality, yields the integral inequality (2.3). The capillary polar volume is expressed through the capillary Gauss map and support function, bridging the analytic bound to the geometric volume product.

What would settle it

Construct an unconditional, strictly convex capillary hypersurface with θ ∈ (0, π/2) (e.g., a small diagonal perturbation of the spherical cap) and compute vol(bΣ) vol(cΣ*) numerically; if any such product exceeds vol(bC_θ)², the central claim collapses. Alternatively, verify the paper's obtuse-angle example by evaluating the support-function bound h_{bΣ}(ψ) for the ellipse family and checking that the integral grows linearly in b.

Watch

Extended reading notes

Core claim

For θ ∈ (0, π/2), the authors establish that for any unconditional convex body K in R^n, the weighted integral inequality vol(K)/2^n ∫_{S^{n-1}_θ} (1 − cosθ x_n)^{n+1} / h_K^n dx ≤ vol( bC_θ)^2 holds, where bC_θ is the capillary spherical cap. When K is the double of an unconditional, strictly convex capillary hypersurface Σ, this directly gives vol(bΣ) vol(cΣ*) ≤ vol(bC_θ)^2, with equality if and only if Σ is a homothetic copy of C_θ. In contrast, for θ ∈ (π/2, π), they exhibit two explicit one-parameter families of convex bodies showing that vol(bΣ) vol(cΣ*) → ∞.

Load-bearing premise

The proof assumes that for every unconditional, strictly convex capillary hypersurface, the capillary polar volume can be represented by the same integral formula as for the model cap, specifically that the capillary support function s_Σ equals the standard support function h_K at shifted arguments; if that identification fails for some admissible surface, the geometric inequality does not follow from the analytic bound.

Editorial extensions

If this is right

The conjectured capillary Blaschke–Santaló inequality is true for all unconditional, strictly convex capillary hypersurfaces with θ ∈ (0, π/2), with spheres as the unique maximizers up to scaling.
A linearization (Theorem 1.2) yields a sharp second-order inequality involving the centro-affine Laplacian on the cap, giving a necessary condition for local maximality.
Corollary 2.5 provides entropy-type and L^p-type inequalities for unconditional convex bodies, extending the volume product bound to other functionals.
For θ ∈ (π/2, π), the absence of a finite upper bound shows the acute-angle condition is essential for any capillary Blaschke–Santaló-type statement.

Reading between the lines

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

The technique may extend to non-unconditional bodies if the symmetry assumption can be relaxed, though the Prékopa–Leindler argument relies on coordinate-wise convexity; this is left open by the paper.
The unboundedness constructions for obtuse angles suggest that for θ > π/2 the capillary polar volume can be made arbitrarily large by 'flattening' the body near the boundary plane, hinting at a possible phase transition in the geometry.
A quantitative stability version of the equality case might follow from known stability results for Prékopa–Leindler, though the paper does not address it explicitly.

Share X Bluesky LinkedIn Reddit HN

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. First, the paper actually proves the conjecture it claims in the unconditional class, and the obtuse-angle unboundedness result is new. Second, Theorem 1.1 as printed has a wrong denominator: the proof gives 2n, not 2^n, and the stated 'in particular' and equality case only follow after that correction.

What is genuinely new: the unconditional capillary Blaschke–Santaló inequality, the equality characterization, and the unboundedness for θ > π/2. The argument is clean: build a symmetric body C from the cap, show V(√x) is concave, use a functional form of the Blaschke–Santaló inequality from CKLR24, and convert the Prékopa–Leindler step into the capillary volume product using the capillary polar volume formula from MWW25. Theorem 1.2, a linearization, is a reasonable bonus. Section 3 gives two explicit families showing the product is unbounded for obtuse angles; the construction is a bit ad hoc but it checks out.

The constant bug is real and not cosmetic. Combining (2.2) with the three identities before (2.3) gives vol(K)/(2n) I ≤ vol(bCθ)², not vol(K)/2^n I ≤ vol(bCθ)². Since I = n vol(cΣ*) and vol(K ∪ R(K)) = 2 vol(bΣ), the 2n denominator is exactly what produces vol(bΣ)vol(cΣ*) ≤ V(bCθ)². The printed 2^n leaves a factor 2n/2^n on the left, so for n ≥ 3 the capillary implication and the equality claim do not follow as written; e.g., n=3, θ→π/2, K=B³ gives strict inequality with the printed constant. This is a one-character fix in Theorem 1.1 and (2.3), and the proof already contains the right constants, so I do not treat it as fatal—but it must be corrected before publication. The reader's note about Lemma 2.2 omitting a homogeneity normalization is also worth addressing: add one line explaining the identity is for the normalized support function or state the homogeneous version explicitly. The equality-case argument is terse but credible.

Who is this for: convex geometers working on capillary problems or functional Blaschke–Santaló inequalities. It deserves a serious referee; I would send it out with the expectation that the authors fix the constant and tighten the cited identity. After that, it is a solid advance.

Referee Report

3 major / 4 minor

Summary. The paper addresses the conjectured capillary Blaschke–Santaló inequality for even, strictly convex capillary hypersurfaces in the upper half-space. For θ∈(0,π/2), the authors claim the inequality holds for all unconditional such hypersurfaces, with equality if and only if the hypersurface is a dilate of the capillary spherical cap Cθ. The proof strategy is functional-analytic: they introduce the convex body C = bCθ ∪ R(bCθ), prove concavity of V(√x) for V=(1/2)p_C², import a gradient inequality from [CKLR24], apply the Prékopa–Leindler inequality, and reduce the desired capillary inequality to a weighted integral inequality for unconditional convex bodies K. The paper also sketches two examples showing that for θ∈(π/2,π) the capillary volume product is unbounded, and it derives a linearized spectral inequality.

Significance. If the constant issue described below is corrected, this is a significant and clean result: it proves the conjectured capillary Blaschke–Santaló inequality in the unconditional class, with a sharp equality statement. The proof is genuinely non-circular: it imports the capillary polar-volume formulas from [MWW25] and a gradient inequality from [CKLR24], uses external results (Prékopa–Leindler and Dubuc's equality case) but introduces no free parameters or fitting. The unboundedness for obtuse contact angles is also an interesting complement. The main theorem is of clear interest to convex geometry and capillary hypersurface theory.

major comments (3)

[§2.1, Eq. (2.3) and Theorem 1.1] The displayed simplification leading to (2.3) has the wrong constant and the equality statement is inconsistent with it. With I = ∫_{Sθ} h_C^{n+1}/h_K^n dσ, the three preceding identities give A = (nc/2^n)V(K), B = (c/2^{n-1})I, C = (nc/2^{n-1})V(bCθ). Substituting AB ≤ C² yields V(K)/(2n) I ≤ V(bCθ)², not V(K)/2^n I ≤ V(bCθ)². The printed 2^n inequality is weaker; it does not imply the capillary inequality, and its equality case is false (for n≥3, K=C gives strict inequality). Please correct (2.3) and Theorem 1.1 to the 2n form and re-verify the equality-case argument under the corrected statement.
[§2.1, passage from Σ to K] The proof uses the identification h_K(ζ+cosθE_n)=sΣ(ζ) for K=bΣ∪R(bΣ) to convert the capillary polar volume into the integral over Sθ. This is not proved or referenced. It is essential: without it the geometric inequality for Σ does not follow from the analytic inequality for K. Please supply a short supporting-hyperplane argument (the point on Σ with Euclidean unit normal u=ζ+cosθE_n supports bΣ, hence K, in direction u) or a precise reference.
[§3, unboundedness] The two examples are constructed in R². The abstract claims no finite upper bound, presumably in every dimension, but the text only says the planar bodies may be rotated about the x_n-axis. To make the claim rigorous for general n, the rotation reduction and the corresponding estimates for the n-dimensional body and its capillary polar need to be stated explicitly; otherwise the result is proved only for n=2.

minor comments (4)

[§2.1, display before spherical coordinates] In the first integral after 'By Lemma 2.2 and Lemma 2.3', the integrand should be e^{-1/2 h_K²(y)} (equivalently e^{-1/2 p_{K°}²(y)}), not e^{-1/2 p_K²(y)}. Since Φ*=1/2 h_K², the printed p_K is a typo that would make the subsequent h_K formula unjustified.
[§2.1, equality-case paragraph] In the equality case, from p_K(e^t)=p_C(e^{t+w/2}) the correct diagonal matrix is diag(e^{-w_i/2}), not diag(e^{w_i/2}), to obtain p_{AK}=p_C. The final conclusion is unaffected, but the sign should be fixed.
[§2.2, Theorem 1.2] The notation ΔCθ is introduced only via the phrase 'centro-affine Laplacian' and a reference; a one-line definition or formula would improve readability.
[§3, Example 1] In the construction of Kλ, the displayed inequality for s_{Σλ}(ζ) is terse; a short explanation of why the other parts of the union do not affect the capillary support function on Dθ would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is an analytic derivation from external lemmas with no fitted parameters and no self-citation of the target result.

full rationale

The main inequality (2.3) is derived from the Prékopa–Leindler inequality applied to unconditional convex functions Φ = 1/2 p_K^2 and V = 1/2 p_C^2, using changes of variables, the Legendre transform, and the auxiliary function V^* = 1/2 h_C^2. No parameter is fitted and no form of the capillary Blaschke–Santaló inequality is assumed. The passage from the analytic inequality to the capillary volume product uses the representation vol(cΣ*) = (1/n)∫_{S^{n-1}_θ} (h_C/h_K)^n h_C dσ, which is quoted from the prior work [MWW25, Prop. 2.9] by Mei–Wang–Weng, not by the present authors. The equality case is handled through Dubuc's equality characterization in the Prékopa–Leindler theorem and the paper's own Lemma 2.4. Although the skeptic's note identifies a possible constant-factor arithmetic error in the printed inequality, that is a correctness concern, not a circularity: the claimed result is not equivalent to any of its inputs by construction, and the cited external results are independent support rather than self-citations of the conjecture being proved. Therefore, no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof is parameter-free; no constants are fitted to data. It relies on standard convex geometry, the Prekopa-Leindler inequality, and capillary-specific volume formulas imported from prior work. No new entities are postulated.

assumptions (5)

standard math Prekopa-Leindler inequality and Dubuc equality characterization
Used in Section 2.1 to derive inequality (2.2) and the equality case analysis.
domain assumption Capillary Gauss map is a diffeomorphism and capillary polar volume formulas from [MWW25, Prop. 2.9] and [MWWX25, Lem. 2.2]
Establishes vol(bSigma) = integral of sigma/G dV and vol(cSigma*) = integral of sigma^{-n} dV, and the identification s_Sigma = h_K on S^{n-1}_theta; load-bearing for converting the geometric product into the analytic inequality.
domain assumption Weighted gradient inequality (2.1) from [CKLR24, Lem. 5.17]: <a, DV(b)> >= 2V(sqrt(ab)) for unconditional 2-homogeneous V with V(sqrt(x)) concave
Key step in the Prekopa-Leindler argument; cited rather than re-derived.
standard math Support function Hessian identity det D^2(1/2 h^2) = h^{n+1}/K, used via [MP14, Thm. 3.1] in normalized form
Computes the Monge-Ampere measure of V* in Lemma 2.2; the paper quotes the formula without an explicit homogeneity normalization but applies the correct normalized version.
standard math Classical Blaschke-Santalo inequality and standard convex body facts from Schneider
Background comparison and standard tools such as the volume formula in terms of the gauge function.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the conjectured capillary Blaschke-Santal\'o inequality." pith.science (2026). https://pith.science/paper/NK2SH7QX

@misc{pith2026250920257,
  author       = {Pith},
  title        = {Pith review of: On the conjectured capillary Blaschke-Santal\'o inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NK2SH7QX}},
  note         = {Machine review of arXiv:2509.20257}
}

abstract

We prove that the conjectured capillary Blaschke-Santal\'o inequality holds for any unconditional, strictly convex capillary hypersurface when $\theta \in \left(0, \tfrac{\pi}{2}\right)$. Moreover, for $\theta \in \left(\tfrac{\pi}{2}, \pi\right)$, we show that the capillary volume product has no finite upper bound.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 2 linked inside Pith

[1]

Artstein-Avidan, B

S. Artstein-Avidan, B. Klartag, V. Milman, The Santal\'o point of a function, and a functional form of the Santal\'o inequality, Mathematika 51 (2004), 33--48

2004
[2]

Blaschke, Vorlesungen \"uber Differentialgeometrie

W. Blaschke, Vorlesungen \"uber Differentialgeometrie. Band I: Elementare Differentialgeometrie, 3rd ed., Springer, Berlin, 1945

1945
[3]

K. M. Ball and K. J. B\"or\"oczky, Stability of the Pr\'ekopa--Leindler inequality, Mathematika 56 (2010), 339--356

2010
[4]

Borell, Convex set functions in d -space, Period

C. Borell, Convex set functions in d -space, Period. Math. Hung. 6 (1975), 111--136

1975
[5]

K. J. B\"or\"oczky, A. De, Stability of the Pr\'ekopa--Leindler inequality for log-concave functions, Adv. Math. 386 (2021), 107810

2021
[6]

K. J. B\"or\"oczky, A. Figalli, J. P. G. Ramos, A quantitative stability result for the Pr\'ekopa--Leindler inequality for arbitrary measurable functions, Ann. Inst. H. Poincar\'e C 41 (2023), 565--614

2023
[7]

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
[8]

Dubuc, Crit\`eres de convexit\'e et in\'egalit\'es int\'egrales, Ann

S. Dubuc, Crit\`eres de convexit\'e et in\'egalit\'es int\'egrales, Ann. Inst. Fourier (Grenoble) 27 (1977), 135--165

1977

Show all 19 references

[9]

Fradelizi, M

M. Fradelizi, M. Meyer, Some functional forms of Blaschke-Santal\'o inequality, Math. Z. 256 (2007), 379--395

2007
[10]

Marini, G

M. Marini, G. De Philippis, A note on Petty's problem, Kodai Math. J. 37 (2014), 586--594

2014
[11]

Meyer, A

M. Meyer, A. Pajor, On the Blaschke-Santal\'o inequality, Arch. Math. (Basel) 55 (1990), 82--93

1990
[12]

Leindler, On a certain converse of H\"older's inequality

L. Leindler, On a certain converse of H\"older's inequality. II, Acta Sci. Math. (Szeged) 33 (1972), 217--223

1972
[13]

X. Mei, G. Wang, L. Weng, C. Xia, Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary, II, Math. Z. 310 (2025), 71

2025
[14]

X. Mei, G. Wang, L. Weng, The capillary Gauss curvature flow, arXiv:2506.09840 (2025)

2025 arXiv
[15]

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. (2023), DOI: 10.4171/JEMS/1386

2023 doi
[16]

Pr\'ekopa, Logarithmic concave measures with application to stochastic programming, Acta Sci

A. Pr\'ekopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301--316

1971
[17]

Saint-Raymond, Sur le volume des corps convexes sym\'etriques, S\'eminaire d'Initiation \`a l'Analyse, 20\`eme ann\'ee (1980/81), Expos\'e no

J. Saint-Raymond, Sur le volume des corps convexes sym\'etriques, S\'eminaire d'Initiation \`a l'Analyse, 20\`eme ann\'ee (1980/81), Expos\'e no. 11, Universit\'e Pierre et Marie Curie, Paris, 1--25

1980
[18]

L. A. Santal\'o, An affine invariant for convex bodies of n -dimensional space, Portugaliae Math. 8 (1949), 155--161

1949
[19]

Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 2013

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 2013

2013

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Artstein-Avidan, B

S. Artstein-Avidan, B. Klartag, V. Milman, The Santal\'o point of a function, and a functional form of the Santal\'o inequality, Mathematika 51 (2004), 33--48

2004

[2] [2]

Blaschke, Vorlesungen \"uber Differentialgeometrie

W. Blaschke, Vorlesungen \"uber Differentialgeometrie. Band I: Elementare Differentialgeometrie, 3rd ed., Springer, Berlin, 1945

1945

[3] [3]

K. M. Ball and K. J. B\"or\"oczky, Stability of the Pr\'ekopa--Leindler inequality, Mathematika 56 (2010), 339--356

2010

[4] [4]

Borell, Convex set functions in d -space, Period

C. Borell, Convex set functions in d -space, Period. Math. Hung. 6 (1975), 111--136

1975

[5] [5]

K. J. B\"or\"oczky, A. De, Stability of the Pr\'ekopa--Leindler inequality for log-concave functions, Adv. Math. 386 (2021), 107810

2021

[6] [6]

K. J. B\"or\"oczky, A. Figalli, J. P. G. Ramos, A quantitative stability result for the Pr\'ekopa--Leindler inequality for arbitrary measurable functions, Ann. Inst. H. Poincar\'e C 41 (2023), 565--614

2023

[7] [7]

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

[8] [8]

Dubuc, Crit\`eres de convexit\'e et in\'egalit\'es int\'egrales, Ann

S. Dubuc, Crit\`eres de convexit\'e et in\'egalit\'es int\'egrales, Ann. Inst. Fourier (Grenoble) 27 (1977), 135--165

1977