Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

Gaoyong Zhang; Shay Sadovsky

REVIEW 4 major objections 6 minor 26 references

Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves two inequalities in the dual Brunn-Minkowski theory: a Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies, and a reverse isoperimetric inequality showing the cube maximizes dual…

desk verdict Two genuinely new dual Brunn-Minkowski/reverse-isoperimetric inequalities with a sound strategy, but the proof of Theorem 2 needs patching around evenness, the approximating weight, and a misstated lemma. read the letter →

arxiv 2505.23748 v1 pith:TVAXJC43 submitted 2025-05-29 math.MG math.FA

Shay Sadovsky , Gaoyong Zhang This is my paper

classification math.MGmath.FA MSC 52A40

keywords convexbodydualquermassintegralBrunn-MinkowskiinequalityreverseisoperimetricJohn'spositioncuberadialfunctionlog-concavity

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 establishes two inequalities in the dual Brunn-Minkowski theory, which studies convex bodies through radial functions rather than support functions. First, for origin-symmetric convex bodies in any dimension, the q-th dual quermassintegral satisfies a Brunn-Minkowski inequality for all 0

What carries the argument

The main object is the dual quermassintegral Ṽ_q(K) = (1/n)∫_{$S^{{n-1}}$} ρ_K(u)^q du, equivalently (q/n)∫_K |x|^{q-n} dx for q>0. For Theorem 2, the carrying mechanism is log-concavity of the singular measure |x|^{q-n}dx on origin-symmetric convex bodies, established via a second-variation criterion and a Poincaré-type inequality for the weighted Laplacian, followed by a homogeneity normalization trick. For Theorem 3, the machinery is a distributional inequality comparing the superlevel sets of radial functions under the Gaussian measure, derived from a continuous Brascamp-Lieb inequality; layer-cake decomposition then converts this distributional dominance into integral inequalities for all q<0, q=0, and 0<q<n, with q=n supplied by the classical volume ratio inequality.

What would settle it

Compute the dual quermassintegrals of two explicit origin-symmetric convex bodies, such as two orthogonally oriented elongated ellipsoids, at a value q strictly between 1 and n, and check whether (Ṽ_q(K+L))^{1/q} is at least (Ṽ_q(K))^{1/q}+(Ṽ_q(L))^{1/q}; a violation would disprove Theorem 2. For Theorem 3, numerically maximize the normalized dual quermassintegral over a small family of origin-symmetric perturbations of the cube that remain in John's position; any value exceeding that of the cube for some q in (−∞,n) would contradict the claimed equality case.

Watch

Extended reading notes

Core claim

The central claim is that two sharp inequalities hold. Theorem 2 states that for origin-symmetric convex bodies K and L in R^n and any 0<q≤n, the q-th root of the dual quermassintegral of the Minkowski sum dominates the sum of the q-th roots of the two summands: (Ṽ_q(K+L))^{1/q} ≥ (Ṽ_q(K))^{1/q} + (Ṽ_q(L))^{1/q}, with equality at least when K and L are dilates. Theorem 3 states that if an origin-symmetric convex body is in John's position, meaning the Euclidean ball is its maximal volume ellipsoid, then its normalized dual quermassintegral is at most that of the cube B^n_∞ for every q in (−∞,n], with equality only for rotations of the cube. The paper proves Theorem 2 by showing log-concavity of the measure |x|^{q-n}dx on origin-symmetric convex bodies through a limiting argument from smooth radial approximations, and Theorem 3 by integrating a distributional comparison of radial functions under Gaussian measure.

Load-bearing premise

The proof of Theorem 2 assumes that the log-concavity criterion proved for the smooth approximating measures $e^{{-(n-q)log(|x|+ε)}}$ passes to the limit and gives log-concavity for the singular measure |x|^{q-n}dx, and that the auxiliary Neumann function used in the criterion can be taken even.

Editorial extensions

If this is right

Lutwak's Brunn-Minkowski conjecture for dual quermassintegrals is now a theorem in the range 0<q≤n, extending the previously known case q≤1.
Every dual quermassintegral of an origin-symmetric body in John's position is bounded by that of the cube, giving a family of sharp volume-ratio-type inequalities interpolating between the classical volume ratio (q=n) and the dual entropy (q=0).
The layer-cake proof shows the cube domination holds simultaneously for all q in (−∞,n), revealing a distributional extremal property rather than an exponent-by-exponent one.
The equality case of the Brunn-Minkowski inequality remains open for 1<q≤n, so a complete characterization of when equality occurs is still unresolved.
The Gaussian formulation of dual quermassintegrals connects these geometric inequalities to probabilistic integrals, potentially opening them to rearrangement and concentration techniques.

Reading between the lines

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

Because the density |x|^{q-n} is q-homogeneous, the same log-concavity strategy might extend to arbitrary convex bodies after a suitable translation; the paper explicitly leaves this as an open question.
The distributional dominance used for Theorem 3 likely implies extremality of the cube for any monotone integral of the radial function, not just the power integrals defining dual quermassintegrals; testing radial moments of arbitrary order would be a direct extension.
The case q>n, where the Gaussian representation diverges but the radial integral still makes sense, is a natural stress test; if the inequalities persist there they would cover the full real range of q.

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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

Dear colleague,

The headline: this paper proves two substantial new inequalities in the dual Brunn-Minkowski theory, and the proofs are mostly right, but the write-up has three analytic loose ends that need to be tied before Theorem 2 is fully certified.

What is new: Theorem 2 resolves Lutwak's conjecture for origin-symmetric convex bodies for all 0

The soft spots are all in Section 2. First, Lemma 5 needs the key integral nonnegative for every smooth u solving the Neumann problem, but Lemma 6 only supplies it for even u. The paper never justifies that the solution can be taken even when W is radial and the boundary data is even. That is very likely true via symmetrization, but it is not written. Second, the approximating W_epsilon(t)=(n-q)log(t+epsilon) has a cusp at the origin and is not C^2 there, so identity (11) from [13] may not apply directly; a smoothing or limiting argument is missing. Third, Lemma 6 as printed claims the integral is at least 1/n independent of the body. That cannot be true for small K (take W=0, u=|x|^2/(2n): the integral is |K|/n). The proof only needs nonnegativity, so this looks like a typo, but it makes the external result hard to check. None of these look fatal; they are fixable in a revision.

The paper is honest about its limitations: equality cases for Theorem 2 are not resolved and are stated as an open problem. Citation practice looks fine; the Zhang self-citations are to related volume inequalities and do not create a circularity.

For whom: convex geometers working in Brunn-Minkowski theory and dual mixed volumes. This deserves a serious referee. I would send it to peer review and ask the referee to focus the authors on the three analytic gaps above.

Referee Report

4 major / 6 minor

Summary. The manuscript proves two inequalities in the dual Brunn-Minkowski theory. Theorem 2 states that for origin-symmetric convex bodies K and L in R^n and 0<q≤n, the q-th dual quermassintegral satisfies a dimension-free Brunn-Minkowski inequality, (\tilde V_q(K+L))^{1/q} ≥ (\tilde V_q(K))^{1/q} + (\tilde V_q(L))^{1/q}. The proof passes to the measure dμ=|x|^{q-n}dx, approximates it by dμε=(|x|+ε)^{q-n}dx, and invokes results of Kolesnikov-Milman and Cordero-Erausquin-Rotem to conclude log-concavity of με on origin-symmetric convex bodies; the limit ε→0 and homogeneity then yield the inequality. Theorem 3 states that if K is an origin-symmetric convex body in John's position, then the normalized dual quermassintegral satisfies \bar V_q(K) ≤ \bar V_q(B_∞^n) for all q∈(-∞,n], with equality only for rotations of the cube. The proof uses a Gaussian stochastic domination result of Schechtman and Schmuckenschläger, proved via Barthe's continuous Brascamp-Lieb inequality, followed by layer-cake integration.

Significance. These are substantial results if the proofs are made fully rigorous. Theorem 2 resolves Lutwak's conjecture for all 0<q≤n, extending the previously known case q≤1, and Theorem 3 generalizes Ball's volume ratio inequality to dual quermassintegrals. The paper is conceptually clear, is parameter-free, and reduces the main inequalities to known analytic machinery without circularity. The authors also honestly state the remaining equality problem for Theorem 2. The main weaknesses are in the precision of the analytic arguments in Section 2: the evenness hypothesis in Lemma 6, the singularity of the weight Wε at the origin, and an incorrect quantitative statement in Lemma 6. These appear fixable, but as submitted the proof of Theorem 2 is not complete.

major comments (4)

[Section 2, proof of Theorem 2, normalization step] The line defining K1 and L1 reads "K1 = μ(K)^{-1/q}K, L1 = μ(K)^{-1/q}L". The definition of L1 should use μ(L)^{-1/q}, not μ(K)^{-1/q}. As printed, μ(L1)=μ(K)^{-1}μ(L), which is not equal to 1 in general, so the final equalities μ(K1)=μ(L1)=1 and μ(K1)^{1-λ}μ(L1)^λ=1 do not hold. This is presumably a typographical error, but it occurs in a load-bearing step of the proof.
[Section 2, Lemmas 5 and 6 and their use] Lemma 5 requires the nonnegativity of the integral for every smooth solution u of the Neumann problem, while Lemma 6 is stated only for smooth even u. In the application to Wε(x)=(n-q)log(|x|+ε) on an origin-symmetric K with even boundary data φ=h_L(ν)-h_K(ν), the manuscript never proves that the Neumann solution can be taken to be even. This is a genuine gap in the chain "By Lemmas 5 and 6, με is log-concave". A symmetrization argument, for instance taking u_e(x)=(u(x)+u(-x))/2, would likely repair the gap, but it is absent from the text.
[Section 2, proof of Theorem 2, smoothness of Wε] The function Wε(x)=(n-q)log(|x|+ε) is not C^2 at the origin, and in fact is not C^1 there, while the integration-by-parts identity (11) and the cited Kolesnikov-Milman results require smoothness of W on the domain. The paper only approximates the measure by με; it does not approximate Wε by smooth functions before invoking (11), nor does it give a limiting argument that first establishes the log-concavity of με. Without such an argument, the derivation of log-concavity of με from Lemmas 5 and 6 is not justified.
[Section 2, Lemma 6] Lemma 6 as stated claims the integral is at least 1/n, independent of K. This is false: for W=0 and u=|x|^2/(2n), one has Δu=1 and the left side equals μ(K)/n, which can be smaller than 1/n when μ(K)<1. The proof of Theorem 2 only needs nonnegativity, and the intended result from [7] likely gives the stronger lower bound μ(K)/n. The lemma and its quotation of the external result should be corrected.

minor comments (6)

[Section 2, proof of Theorem 2] In the sentence "This measure nearly satisfies the condition of Theorem 6", the reference should be to Lemma 6, not Theorem 6.
[Section 2] The phrase "Newmann boundary problem" is a typo and should read "Neumann boundary problem".
[Section 3, Theorem 7] The notation ρ_K^{-1}(x) is ambiguous; it should be written as 1/ρ_K(x) or defined explicitly in the statement of Theorem 7.
[Section 3, proof of Theorem 3] The proof treats 0<q<n, q<0, and q=0, but not q=n. The introduction notes that q=n is Ball's volume ratio theorem, but the proof section should state this explicitly to cover the full range claimed in Theorem 3.
[Section 3, proof of Theorem 3] Several layer-cake displays have mismatched braces, for example the final display in the proof of Theorem 3; these should be corrected for readability.
[Section 3, Theorem 8] The statement says F is a positive continuous function, but then sets f=1_{[a,b]}F, which is not continuous. The equality condition for this discontinuous class should be stated precisely, or an approximation argument should be indicated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: both main theorems are derived from external inequalities (Kolesnikov-Milman/Livshyts and Schechtman-Schmuckenschläger/Barthe) that do not presuppose the target results.

full rationale

The derivation chain for Theorem 2 starts from the equivalent measure formulation dµ = |x|^{q-n} dx, approximates the singular weight by w_epsilon(t) = (n-q) log(t+epsilon), and invokes Lemmas 5 and 6 to conclude log-concavity of µ_epsilon on origin-symmetric convex bodies. Lemmas 5 and 6 are imported from Kolesnikov-Livshyts and Cordero-Erausquin-Rotem; neither of these external results assumes Lutwak's conjecture or the desired dual quermassintegral Brunn-Minkowski inequality. The homogeneity trick then converts log-concavity into the claimed inequality, which is a standard and non-circular reduction. The proof of Theorem 3 is likewise a direct corollary of Schechtman-Schmuckenschläger's radial level-set dominance, integrated via the layer-cake formula; the equality cases are proved using Barthe's continuous Brascamp-Lieb inequality. Zhang's self-citations (references [9], [10], [11], [18], [19], [20]) appear in context-setting remarks or as alternative routes and are not load-bearing for either theorem. The analytic concerns raised about the proof of Theorem 2, such as the evenness of the Neumann solution and the smoothness of w_epsilon at the origin, are possible correctness gaps in the application of external lemmas, not instances where a conclusion is assumed as an input; they therefore do not constitute circularity. The paper contains no fitted parameters presented as predictions, no self-referential uniqueness theorem, and no renaming of a known result as a new derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The two theorems rest on a chain of established external results: John's theorem, Barthe's Brascamp-Lieb, Kolesnikov-Milman formulas, and Cordero-Erausquin-Rotem's log-concavity estimate. No free parameters are fitted. The singular radial measure is handled by an approximation that is standard but not fully detailed. Invented entities: none.

assumptions (6)

standard math Barthe's continuous Brascamp-Lieb inequality (Theorem 8)
Quoted and used to prove the Gaussian slab comparison in Theorem 7; no proof given in this paper.
domain assumption John's theorem for origin-symmetric convex bodies
Invoked in the proof of Theorem 7 to obtain an isotropic contact measure ν with B_2^n ⊂ K ⊂ Z∞^*.
standard math Kolesnikov-Milman boundary integral formulas (10) and (11)
Used in Lemma 5 for the second variation of log μ(K_s); taken from [14,13] without proof.
standard math Cordero-Erausquin-Rotem log-concavity estimate (Lemma 6)
The central analytic estimate ∫(||∇^2u||^2_2 + ⟨∇^2W∇u,∇u⟩)dμ ≥ 1/n for radially symmetric convex potentials; quoted from [7].
domain assumption Ball's volume ratio inequality for q=n
Used as the endpoint of Theorem 3; the paper cites [2] and does not derive or prove the equality condition for this case.
ad hoc to paper Approximation of the singular measure |x|^{q-n}dx by w_ε and limit ε→0
The proof of Theorem 2 assumes log-concavity passes through the limit w_ε(t)=(n-q)log(t+ε)→(n-q)log t without a detailed convergence argument.

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Cite this review

Pith. "Pith review of Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals." pith.science (2026). https://pith.science/paper/TVAXJC43

@misc{pith2026250523748,
  author       = {Pith},
  title        = {Pith review of: Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVAXJC43}},
  note         = {Machine review of arXiv:2505.23748}
}

read the original abstract

This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The second, generalizing Ball's volume ratio inequality, is a reverse isoperimetric inequality: among all origin-symmetric convex bodies in John's position, the cube maximizes the dual quermassintegrals.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [13]

Kolesnikov and Emanuel Milman,Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J

Alexander V. Kolesnikov and Emanuel Milman,Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Funct. Anal.27(2017), no. 2, 1680–1702

work page 2017

[2] [7]

3, 987–1003

Dario Cordero-Erausquin and Liran Rotem,Improved log-concavity for rotationally invariant measures of symmetric convex sets, The Annals of Probability51(2023), no. 3, 987–1003

work page 2023

[3] [1]

Milman,Asymptotic Geometric Analysis, Part I, 06 2015

Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman,Asymptotic Geometric Analysis, Part I, 06 2015

work page 2015

[4] [2]

Ball,Some remarks on the geometry of convex sets, In Geometric Aspects of Functional Analysis, edited by V

Keith M. Ball,Some remarks on the geometry of convex sets, In Geometric Aspects of Functional Analysis, edited by V. D. Milman and G. Schechtman, Lecture Notes in Math1317(1988), 224–231

work page 1988

[5] [3]

2, 351–359

,Volume ratios and a reverse isoperimetric inequality, Journal of the London Mathematical Society2(1991), no. 2, 351–359

work page 1991

[6] [4]

Franck Barthe,An extremal property of the mean width of the simplex, Mathematische Annalen310 (1998), 685–693

work page 1998

[7] [5]

Milman and G

Frank Barthe,A continuous version of the Brascamp-Lieb inequalities, Geometric Aspects of Functional Analysis, V.D. Milman and G. Schechtman (Eds.), Lecture Notes in Math.1850(2004), 53–63

work page 2004

[8] [6]

Functional Analysis273(2017), no

Andrea Colesanti, Galyna Livshyts, and Arnaud Marsiglietti,On the stability of Brunn-Minkowski type inequalities, J. Functional Analysis273(2017), no. 3, 1120–1139

work page 2017