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On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$

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Pith's one-line read For unconditional convex bodies, the dual Brunn-Minkowski inequality for q-th dual quermassintegrals holds for every $0<q\le n+1$, with equality for smooth strictly convex bodies only when the two bodies are dilates; outside the…

desk verdict Sharp dual Brunn-Minkowski thresholds are likely right, but two proof gaps (a false L^p calculation and a wrong exponent estimate in the singular Reilly step) need fixing. read the letter →

arxiv 2608.03949 v1 pith:ZQYIHAXF submitted 2026-08-04 math.DG math.APmath.MG

Haizhong Li , Yao Wan , Yijia Zhang This is my paper · ORCID
classification math.DGmath.APmath.MG MSC 52A4052A2026D15
keywords dualquermassintegralBrunn-MinkowskiinequalityunconditionalconvexbodysingularweightedReillyformulaHardycurvaturemeasurepolarmomentofinertiaorigin-symmetric
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 asks whether the Brunn-Minkowski inequality for q-th dual quermassintegrals can extend beyond $q=n$, and it answers the question with a near-complete map of where the inequality lives. For arbitrary convex bodies the inequality fails for every $q>n$, and for origin-symmetric bodies it fails for every $q>n+2$; at the endpoint $q=n+2$ it holds for origin-symmetric bodies. The main positive result is that for unconditional convex bodies the inequality holds for every $0

What carries the argument

The carrying object is the singular weighted Reilly formula (Proposition 6.5), an integration-by-parts identity for the operator $L_\alpha u=|x|^{-\alpha}\,\mathrm{div}(|x|^\alpha \nabla u)$ with $\alpha=q-n$, obtained by excising a small ball around the origin and letting its radius tend to zero. The inner-sphere boundary terms vanish because the Neumann solution has a controlled expansion near the singularity, and the Hessian remains square-integrable against the weight $|x|^\alpha$. Two coordinate-slice Hardy inequalities with an offset parameter $c=(|x|^2-x_i^2)^{1/2}$, one for $0<\alpha<3$ with constant $3-\alpha$ and one for negative $\alpha$ with constant $-\alpha$, dominate the tangential-gradient terms, giving nonnegativity of the Hessian functional and strict log-concavity of the weighted volume along unconditional perturbations.

What would settle it

Take the planar unconditional rectangles $K=[-2,2]\times[-1,1]$ and $L=[-1,1]\times[-2,2]$ with $q=3=n+1$; the theorem asserts $eV_3(K+L)^{1/3} \ge eV_3(K)^{1/3}+eV_3(L)^{1/3}$, so a direct numerical evaluation of these radial integrals giving the reverse inequality would refute Theorem 1.6.

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Extended reading notes

Core claim

The central claim is a sharp threshold picture for the dual Brunn-Minkowski inequality $eV_q(K+L)^{1/q} \ge eV_q(K)^{1/q} + eV_q(L)^{1/q}$. When $q>n$ the inequality is shown to fail for arbitrary convex bodies by testing the local second variation at the Euclidean ball. In the origin-symmetric class it fails for all $q>n+2$ through a dimension-reduction construction that collapses an $n$-dimensional body to a product of a planar rectangle and a thin cube; at the endpoint $q=n+2$ the inequality holds for all origin-symmetric bodies via a classical inequality for the polar moment of inertia, with equality only for dilates. The main theorem establishes the inequality in the full range $0<q\le n+1$ for unconditional convex bodies and characterizes equality for smooth strictly convex unconditional bodies as equality exactly when the bodies are dilates. This extends the known $0<q\le n$ range and settles the equality question for the unconditional class.

Load-bearing premise

The proof depends on the singular weighted Reilly formula: after cutting a small ball around the origin out of the body and shrinking it, the boundary terms on that inner sphere must contribute nothing and the second derivatives of the auxiliary Neumann solution must stay integrable against $|x|^{q-n}$; if that failed, an uncontrolled contribution from the origin would break the strict log-concavity that produces the inequality.

Editorial extensions

If this is right

  • The dual Brunn-Minkowski inequality now holds for all $0<q\le n+1$ in the unconditional class, extending the previously known $0<q\le n$ range for symmetric bodies.
  • At the endpoint $q=n+2$ the inequality holds for all origin-symmetric bodies, so the only remaining open range for the symmetric problem is $n<q<n+2$.
  • The equality characterization for smooth strictly convex unconditional bodies settles the equality case of the earlier symmetric-range inequality within this class.
  • The Brunn-Minkowski inequalities imply corresponding Minkowski inequalities for dual curvature measures, yielding uniqueness of the $(1,q)$-th dual curvature measure for $0<q\le n+1$ in the unconditional class and for $q=n+2$ in the origin-symmetric class.
  • The counterexample thresholds show that the range $q\le n$ is sharp without symmetry and $q\le n+2$ is the natural boundary with origin symmetry, since the inequality fails beyond those ranges.

Reading between the lines

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

  • The dimension-reduction counterexample for symmetric bodies makes it plausible that any positive range for general origin-symmetric bodies cannot exceed $q=n+2$; the open interval $n<q<n+2$ is exactly where a strengthening of the Hardy estimates would have to be found.
  • The excision-and-limit treatment of the singular weight is likely to work for other radially singular densities of the form $|x|^{q-n}$ with different symmetry classes, so the same machinery could test Brunn-Minkowski behavior for partial symmetries beyond unconditional bodies.
  • At $q=1$ the derived uniqueness statement must allow dilates because the dual curvature measure in question is dilation-invariant; this suggests that the dilation ambiguity at $q=1$ is a structural feature of $(1,q)$-dual curvature measures rather than a defect of the proof.
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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

2 major / 3 minor

Summary. The paper investigates the Brunn-Minkowski inequality for q-th dual quermassintegrals, focusing on the range q>n posed by Sadovsky and Zhang. It establishes negative results: the inequality fails for arbitrary convex bodies when q>n, and fails for origin-symmetric convex bodies when q>n+2, via a second-variation argument and a dimension reduction to planar rectangles. It also proves the endpoint case q=n+2 for origin-symmetric bodies using Hadwiger's inequality for the polar moment of inertia, and for unconditional convex bodies it establishes the full range 0<q≤n+1 using a singular weighted Reilly formula and coordinate-slice Hardy inequalities. Applications to uniqueness of dual curvature measures are derived as consequences.

Significance. If the main results are correct, the paper makes a substantial contribution to the dual Brunn-Minkowski theory. Theorem 1.6 extends the known range for unconditional bodies up to q≤n+1 and gives an equality case that answers a question of Sadovsky and Zhang in that symmetry class. The counterexamples sharpen previous negative results, and the endpoint q=n+2 via Hadwiger's inequality is an elegant and nontrivial application. The paper is also careful in deriving many auxiliary estimates from external benchmarks such as the Kolesnikov-Milman Reilly formula and A2 weighted Poincaré theory, and in the parts I checked the technical estimates are coherent. However, two technical issues, one in the central proof of the unconditional theorem and one in a secondary theorem, need to be addressed before the results are fully established as written.

major comments (2)
  1. [Section 6.1, Lemma 6.3] The exclusion of the singular branch r^{β_m^-}Y_m is not justified as written for 0<q<1. The displayed chain '2β_m^-+q-3 = -(q-1)-2β_m ≤ -(q-1)<-1 since q>0' is false in that range, because -(q-1)=1-q>0. This step is load-bearing: it is used to obtain the expansion (6.7), the integrability (6.8), the vanishing of inner-sphere boundary terms in Lemma 6.4, and hence the singular Reilly formula (6.13) in Proposition 6.5, which underlies Theorem 1.6. The gap appears repairable: from the indicial equation (6.9), for m≥1 and q<2 one has β_m>2-q, so 2β_m^-+q-3 = 1-q-2β_m < q-3 < -1, while for q≥2, β_m>0 gives the same divergence. Since neither bound is stated or proved in the manuscript, the proof of Lemma 6.3 must be corrected before the full range 0<q≤n+1 in Theorem 1.6 is established.
  2. [Section 4.3, Proposition 4.3] The identity '1/2·R_t +_p 1/2·R_-t = r_{p,t}R_0' used at the start of the proof is not correct for p<1 (and for p=0). For n=2, along u=(1,0) the support function of the L_p combination equals r_{p,t}, while along u=(1,1)/√2 it equals √2 independently of t. Since h_{R_0}(u) equals 1 on the first direction and √2 on the diagonal, equality would force r_{p,t}=1 for every t, which holds only for p=1. Consequently the formula for F''_{p,Q}(0), the threshold q_p^{(n)}, and the claimed L_p Brunn-Minkowski failure do not follow from the given argument. This does not affect Theorems 1.3-1.6, but Proposition 4.3 is a stated theorem and needs either a corrected proof or a corrected statement.
minor comments (3)
  1. [Section 4.3] The monotonicity of P(Q) is asserted from the integrand being increasing in Q, but P is a quotient of two increasing integrals; a direct proof of monotonicity would improve the presentation if Proposition 4.3 is retained.
  2. [Section 6.1, Lemma 6.3] In the convergence statement, the phrase 'together with all its derivatives' should be qualified: the local uniform convergence is on compact subsets of B_{R_0}\{0\}, and near the origin the derivatives are not uniformly controlled for negative α.
  3. [Section 5, Step 2 of Theorem 5.1] The assertion that the ratios T_E^+(K_τ)/T_E^-(K_τ) and T_E^+(L_τ)/T_E^-(L_τ) vary continuously from 0 to ∞ and from ∞ to 0 is plausible but not proved; a short justification would make the intermediate value step fully transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main results are derived from external benchmarks and from lemmas proved in the paper, with no fitted input, no load-bearing self-citation, and no self-definitional reduction.

full rationale

I find no circularity in the derivation. Theorem 1.6 is obtained by proving log-concavity of mu-tilde_alpha along unconditional segments (Proposition 7.7), whose engine is the singular weighted Reilly formula (Proposition 6.5). That formula is derived from Kolesnikov-Milman's weighted Reilly identity (an external benchmark) together with the paper's own asymptotic expansion (Lemma 6.3) and vanishing inner-boundary estimate (Lemma 6.4). The Hardy inequalities used in Section 7 (Lemmas 7.1 and 7.3) are proved inside the paper by explicit changes of variable and integration by parts, with sharp constants computed directly, not imported from the theorem being proved. The endpoint q=n+2 rests on Hadwiger's independent inequality for polar moments of inertia, and the counterexamples in Sections 3-4 are self-contained second-variation and dimension-reduction calculations. No parameter is fitted to the target inequality, no normalization encodes the conclusion, and the authors do not invoke their own prior work to force the result. The only concern I see is a possible internal gap in Lemma 6.3 for 0<q<1, where the displayed exponent bound '-(q-1)<-1' is false; that is a correctness issue in the proof as written, not a circular dependency, since the missing bound would need only a further estimate from the indicial equation, not the Brunn-Minkowski inequality itself. Accordingly, the honest finding is a clean circularity score of 0.

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

The central results rest on standard theorems (Reilly formula, Hadwiger, A2 theory) plus in-paper lemmas. The only nonstandard analytic content (singular Reilly formula and Hardy inequalities) is proved in the paper itself, not assumed. There are no fitted parameters and no new postulated entities.

assumptions (6)
  • standard math Kolesnikov-Milman weighted Reilly formula (2018) for smooth potentials
    Used as the starting identity (6.10) in Section 6.2; the paper extends it to the singular weight |x|^alpha.
  • standard math Hadwiger's Brunn-Minkowski inequality for the polar moment of inertia (1956)
    Used to prove the endpoint case q=n+2 for origin-symmetric bodies in Section 5 via I(K)=n/(n+2) eV_{n+2}(K).
  • standard math Muckenhoupt A2 property and weighted Poincare inequality for |x|^alpha with |alpha|<n
    Used in Lemma 6.1 to guarantee the weighted Neumann problem has a unique solution via Lax-Milgram.
  • standard math Spectral gap facts of the spherical Laplacian: first eigenvalue n-1 and first even eigenvalue 2n
    Used in Section 3 for the q>n counterexample at the ball and in Remark 3.1 for the local symmetric threshold.
  • standard math Existence of an orthonormal basis diagonalizing the quadratic form of a traceless symmetric matrix to zeros
    Used in Step 3 of the proof of Hadwiger's theorem (Theorem 5.1) to find directions u_i with T_{u_i⊥}(K)=T_{u_i⊥}(L). Not stated in the paper, implicit.
  • domain assumption Unconditional bodies have coordinate slices that are symmetric intervals, and unconditional functions restrict to odd functions on those slices
    Used throughout Section 7 to apply the one-dimensional Hardy inequalities on [0,R_i(x')] with f(0)=0.

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Pith. "Pith review of On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$." pith.science (2026). https://pith.science/paper/ZQYIHAXF

@misc{pith2026260803949,
  author       = {Pith},
  title        = {Pith review of: On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQYIHAXF}},
  note         = {Machine review of arXiv:2608.03949}
}
abstract

In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.

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