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Concavity principles for weighted marginals

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Pith's one-line read For rotationally invariant measures, weighted marginals of concave functions with symmetric sections are concave after the $1/(\beta+n)$-power, for every $\beta>0$.

desk verdict A genuine functional Brunn-Minkowski advance for rotationally invariant measures, with a few fixable rough edges that should be addressed before publication. read the letter →

arxiv 2506.16941 v1 pith:6OLMCFDB submitted 2025-06-20 math.FA math.MG

Dario Cordero-Erausquin , Alexandros Eskenazis This is my paper · ORCID
classification math.FAmath.MG MSC 52A4052A2028C2060D0547F10
keywords Brunn–MinkowskiinequalityB-inequalityweightedmarginalslog-concavemeasuresrotationallyinvarianthereditaryconvexityBorell–Brascamp–Liebinequalitiessymmetricconvexsets
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 concavity principle for weighted marginals under rotational symmetry. If $d\mu(x)=e^{-w(|x|)}dx$ with $w$ increasing and $t\mapsto w(e^t)$ convex, then for every concave $\Phi$ with even sections on a convex set $\Omega$ with symmetric sections and every $\beta>0$, the function $\varphi(t)=(\int_{\Omega_t}\Phi(t,x)^\beta\,d\mu(x))^{1/(\beta+n)}$ is concave on its support. This is the functional form of the dimensional Brunn–Minkowski inequality for such measures, and it applies even when $\mu$ itself is not log-concave, for instance for weights like $(1+|x|^\alpha)^{-\beta}$. The same machinery yields a Prékopa-type log-concavity theorem for weighted marginals that contains the B-inequality for rotationally invariant measures, as well as a weighted Poincaré–Brascamp–Lieb inequality.

What carries the argument

The load-bearing object is the hereditary convexity condition (Definition 8): an even measure $\mu=e^{-W}dx$ satisfies inequality (59), namely $\int(\|\nabla^2 u\|_{\mathrm{HS}}^2+\langle\nabla^2 W\,\nabla u,\nabla u\rangle)\,d\nu \ge (\int L_\mu u\,d\nu)^2/\int L_\mu(|x|^2/2)\,d\nu$ for every even probability measure $\nu$ log-concave with respect to $\mu$ and every even smooth $u$ with Neumann data. That condition is exactly the curvature lower bound needed to make the second derivative of the marginal function nonpositive. The second-derivative formula itself (Proposition 7) is derived by a mixed geometric-functional $L^2$ computation: support-function perturbations of the convex sections (Lemmas 5 and 6) are combined with the Bochner–Reilly identity and an elliptic PDE solved by a function $u$, following the line of Brascamp–Lieb and Nguyen but extended to non-product domains. Proposition 9 proves hereditary convexity for rotationally invariant $\mu$ using a weighted Poincaré inequality for even functions imported from [25, Theorem 4].

What would settle it

Search over admissible rotationally invariant weights $w$, even log-concave measures $\nu$ on smooth symmetric supports, and even functions $u$ with compatible Neumann data: a single instance in which the left side of (59) is smaller than the right side for an allowed weight would contradict Proposition 9 and remove the support for Theorem 1. A more direct but harder test is to find any convex $\Omega$ with symmetric sections and concave even $\Phi$ for which the function $\varphi$ in (8) fails to be concave.

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Extended reading notes

Core claim

Restated on the paper's own terms, the central result is: for every increasing $w$ with $t\mapsto w(e^t)$ convex, the measure $d\mu(x)=e^{-w(|x|)}dx$ on $\mathbb{R}^n$, every convex $\Omega\subset\mathbb{R}^{n+1}$ whose sections $\Omega_t$ are symmetric, and every concave $\Phi:\Omega\to\mathbb{R}_+$ that is even in $x$ on each section, the marginal function $\varphi(t)=(\int_{\Omega_t}\Phi(t,x)^\beta\,d\mu(x))^{1/(\beta+n)}$ is concave on its support for every $\beta>0$, whenever the integral converges. The paper also proves that for $\kappa\in[0,1]$, if $V\in C^2$ has even sections and satisfies the Hessian lower bound (14), then $\alpha(t)=\int_{\mathbb{R}^n} e^{-V(t,x)}\,d\mu(x)$ is log-concave; the case $\kappa=1$ contains the functional B-theorem, and hence the B-inequality, for rotationally invariant measures. These statements are established by approximation from smooth cases and by showing that rotationally invariant measures of the stated class satisfy a spectral property the paper calls hereditary convexity.

Load-bearing premise

The proof rests on a single spectral bound: for every symmetric log-concave perturbation of the measure and every symmetric test function, a curvature integral always dominates the square of an average divided by a dimension factor; for rotationally invariant weights this bound is imported from an earlier theorem, and if it failed for an admissible weight, the concavity conclusion would no longer follow.

Editorial extensions

If this is right

  • Corollary 2: for each $\beta>0$, the measure $\nu_\beta=\Phi^\beta d\mu$ on a symmetric convex set $C$ satisfies $\nu_\beta(\lambda K+(1-\lambda)L)^{1/(\beta+n)}\ge \lambda\nu_\beta(K)^{1/(\beta+n)}+(1-\lambda)\nu_\beta(L)^{1/(\beta+n)}$ for all symmetric convex $K,L$.
  • Theorem 4 with $\kappa=1$ yields the functional B-theorem, the log-concavity of $t\mapsto\int e^{-V(e^t x)-W(x)}\,dx$, and hence the B-inequality for rotationally invariant measures; Remark 13 notes that this provides a new proof of the B-theorem for such measures.
  • Theorem 3 gives a weighted Poincaré–Brascamp–Lieb inequality (12) for even functions under the same class of rotationally invariant weights.
  • The approximation argument repairs a gap in earlier local proofs of dimensional Prékopa theorems in the range $\beta>0$, where only product-set domains were handled, and extends the conclusion to general convex domains with symmetric sections.

Reading between the lines

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

  • If the hereditary convexity question posed for all even log-concave measures has an affirmative answer, the same proof structure would deliver both the B-conjecture and the dimensional Brunn–Minkowski conjecture for general even measures; rotational invariance is the sufficient case established here.
  • The second-derivative formula (45) is a general calculus for weighted marginals that is likely to transfer to other functionals; the paper's question about weighted torsional rigidity is the natural next target.
  • For non-rotationally invariant even log-concave measures, inequality (59) can be probed directly on quadratic test functions $u$; the first even counterexample, if it exists, would show exactly where symmetry assumptions are needed.
  • The proof uses that $\Phi^\beta d\mu$ is more log-concave than $\mu$, so the negative-exponent regime of Question 19 will require new spectral inequalities rather than a simple adaptation of this argument.
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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 / 5 minor

Summary. The paper develops a local L2/Bochner method for weighted marginal functions of the form φ(t)=(∫_{Ω_t} Φ(t,x)^β dμ(x))^{1/(β+n)}. It introduces a spectral condition called 'hereditary convexity' (Definition 8) and shows, in Theorem 10, that this condition suffices for a functional dimensional Brunn–Minkowski inequality; Proposition 9 then asserts that rotationally invariant measures with density e^{-w(|x|)}, where t↦w(e^t) is increasing and convex, satisfy the condition. This yields Theorem 1, the main concavity principle, and Corollary 2 for weighted Brunn–Minkowski inequalities. A second application gives Theorem 4, a log-concavity preservation principle for weighted marginals with a Hessian lower bound, which contains a functional B-inequality for rotationally invariant measures. Theorem 3 derives a weighted Poincaré–Brascamp–Lieb inequality from Theorem 1. The paper also discusses negative exponents, questions on torsional rigidity, and reprints Brascamp–Lieb's proof of the Gaussian B-theorem in an appendix.

Significance. If all claims hold, the paper is a substantial contribution: it upgrades the geometric dimensional Brunn–Minkowski inequality for rotationally invariant measures of [27,25] to a functional version, and it provides a unified hereditary-convexity framework that also yields the B-inequality. The second-derivative formula in Proposition 7 and the approximation chain in the proof of Theorem 10 are technically valuable, and the authors are transparent about the points where the proof relies on the earlier spectral result [25, Theorem 4]. The main concern is that this cited result is not stated in sufficient detail for the reader to verify that it covers the bounded-support, arbitrary-Neumann-data case required in the proof of Proposition 9; this issue is load-bearing for Theorem 1. The paper also contains a display error in Theorem 3 and an invalid illustrative counterexample in the introduction.

major comments (2)
  1. [Section 4.1 (Proposition 9, inequality (63)) and Section 4.2 (Step 1, inequality (74))] The proof of Proposition 9 applies [25, Theorem 4] to v = u + (λ/2)|x|² and concludes (63), but the cited theorem is not stated, and no argument is given for the case allowed in Definition 8(ii) where the measure ν has bounded smooth support U and the function u has arbitrary Neumann data on ∂U. This is exactly the case needed in Step 1 of the proof of Theorem 10 via inequality (74), since the measure ν there is supported on the bounded set Ω0 and u satisfies the nonzero Neumann condition (44); the boundary terms in the weighted Reilly formula (21) are then generally present. Please state Theorem 4 of [25] and either verify that its hypotheses cover the bounded-support Neumann-data situation or supply a proof of (63) in that situation. Without this, the passage from (63) to the hereditary-convexity inequality (59), and hence the central estimate (74), is not justified as written.
  2. [Section 6, Theorem 3, Eq. (12) and proof, Eq. (103)] The displayed inequality (12) contains ⟨(-∇²Φ)∇g,∇g⟩ in the first integral, but the proof establishes ⟨(-∇²Φ)^{-1}∇g,∇g⟩. The inverse is what follows from the Cauchy–Schwarz step in the construction of Φ_ε in (99)–(100) and is what appears in (103). As printed, Theorem 3 states a different and unproved inequality; please correct (12) by adding the inverse in the first integrand.
minor comments (5)
  1. [Section 1.2] The proposed disproof of [60, Conjecture 6.2] using translated indicator functions with M→∞ is not valid as stated for a finite measure μ: by dominated convergence, both ∫ f_M dμ and ∫ g_M dμ tend to zero, so the inequality in question would hold in the limit. The example should be replaced or the argument amended.
  2. [Section 4.1, display (62)] The displayed implication 'μ verifies the dimensional Brunn–Minkowski conjecture ⇒ μ is even hereditarily convex' appears to have the direction opposite to the surrounding discussion and to Theorem 10, which derives the functional Brunn–Minkowski statement from hereditary convexity. Please clarify the intended logical direction.
  3. [Section 4.2, Step 2] In the approximation argument, the boundary value of Ψ_{1/k} on ∂Ω_{k,m} is η+1/m, whereas Step 1 is formulated for a boundary value η. This is harmless after rescaling the constant, but it should be said explicitly.
  4. [Throughout] There are several typos that should be corrected in revision: 'Luster nik' (§1.1), 'heriditarily' (Definition 8 and §4.1), 'indespensable' (§2.1), and 'assets' (Remark 14).
  5. [Section 5, proof of Theorem 4] Formula (93) is obtained by a limiting argument from Proposition 7 with Φ=(1-β^{-1}V)_+ and β→∞; a short direct derivation, or a precise statement of the approximation used for the unbounded domain, would improve readability and rigor.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the concavity theorem is derived in-paper from a second-derivative formula plus an imported spectral estimate, and the cited [25] inequality is independent content rather than a restatement of the conclusion.

full rationale

I walked the derivation chain and found no step in which a claimed output is equivalent, by construction or by self-citation, to its own input. Theorem 1 is proved via Theorem 10, which combines the local second-derivative representation (45) of Proposition 7 with the hereditary-convexity spectral estimate (59) of Definition 8. Proposition 7 is proved in-paper from the Bochner-Reilly identity (21), Lemmas 5-6, and integration by parts, without assuming the concavity of the marginal. Proposition 9 verifies (59) for rotationally invariant log-concave measures using inequality (63), quoted from the authors' prior work [25, Theorem 4]. This is a load-bearing self-citation, but [25] is a published, parameter-free theorem whose assumptions (rotationally invariant log-concave measures, even ν, even u) do not include the target result; inequality (63) is a spectral estimate on Hessian norms, not a restatement of the concavity of φ(t). The later estimates (72)-(77) use joint concavity of Φ, Cauchy-Schwarz, and (59); none of these is equivalent to the conclusion. Theorem 3 is deduced from Theorem 1, not used to prove it, and Theorem 4 uses (59) under a different Hessian condition (14), so there is no feedback of the output into the input. The one caveat, namely the exact boundary generality of [25, Theorem 4] when ν has bounded support and u has nonzero Neumann data, would be a correctness or coverage gap if real, but it is not a circularity: a missing external hypothesis does not make the argument a tautology. Under the hard rules, without a demonstrated reduction-by-construction, the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results involve no data fitting. The key structural input is the hereditary convexity inequality (59), which for the theorem's range of measures is established by invoking [25, Theorem 4] from the authors' prior work. No new entities are postulated.

assumptions (4)
  • standard math Classical elliptic regularity: for a C^2 domain and C^1 data, the Neumann problem L_ν u = f has a C^2 solution under the compatibility condition (20).
    Invoked in Section 2.1 to define u in Proposition 7 and in the proof of Theorem 10.
  • standard math Weighted Reilly/Bochner identity (21) for manifolds with boundary, cited from [41, Theorem 2.1].
    Used to derive the Brascamp-Lieb inequality and the second derivative formula in Proposition 7.
  • domain assumption Poincaré inequality for even functions under rotationally invariant log-concave measures, [25, Theorem 4].
    Load-bearing in Proposition 9; if false, hereditary convexity for rotationally invariant measures fails. This is prior work by one of the present authors.
  • standard math Stability of concavity under uniform limits and pointwise limits of marginals, including dominated convergence in approximation steps.
    Used in Steps 2 and 3 of the proof of Theorem 10.

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Pith. "Pith review of Concavity principles for weighted marginals." pith.science (2026). https://pith.science/paper/6OLMCFDB

@misc{pith2026250616941,
  author       = {Pith},
  title        = {Pith review of: Concavity principles for weighted marginals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OLMCFDB}},
  note         = {Machine review of arXiv:2506.16941}
}
abstract

We develop a general framework to study concavity properties of weighted marginals of $\beta$-concave functions on $\mathbb{R}^n$ via local methods. As a concrete implementation of our approach, we obtain a functional version of the dimensional Brunn-Minkowski inequality for rotationally invariant log-concave measures. Moreover, we derive a Pr\'ekopa-type concavity principle with rotationally invariant weights for even log-concave functions which encompasses the B-inequality.

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