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Lutwak-Petty projection inequalities for Minkowski valuations and their duals

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Pith's one-line read For even, zonal measures, balls maximize the mixed volume product defining generalized projection bodies, extending Lutwak's inequalities and their duals.

desk verdict Solid generalizations of the Lutwak–Petty and Leng–Lu inequalities, but the proof of Theorem 1.2 has a dropped nth power after (5.5) that makes the printed argument too weak; it is a repairable typo, not a fatal flaw. read the letter →

arxiv 1908.01634 v1 pith:AHWX3GHI submitted 2019-08-05 math.MG

Astrid Berg , Franz E. Schuster This is my paper · ORCID
classification math.MG MSC 52A2052A3952A4044A12
keywords MinkowskivaluationsprojectionbodiesintersectionmixedinequalitieszonalmeasuresaffinequermassintegralsBusemann–PettycentroidinequalityLeng–Lu
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 extends two classical families of volume inequalities in convex geometry — Lutwak's mixed projection inequalities and the Leng–Lu intersection inequalities — to polarizations of Minkowski valuations generated by even, zonal measures on the unit sphere. The central assertion is that Euclidean balls maximize the relevant volume products, with equality characterized by homothetic ellipsoids in the discrete projection-body case. If true, this shows that the known affine isoperimetric inequalities are the strongest members of a whole family of Euclidean inequalities, and that the classical projection and intersection body results are sharp within these families. The dual half of the paper introduces radial Minkowski valuations that generalize intersection bodies and proves the analogous maximization, recovering the Busemann and Leng–Lu inequalities as special cases.

What carries the argument

The load-bearing object is the Minkowski valuation Φ_µ defined by h(Φ_µ K,u) = ∫_{∂K} h(Z_µ(u), ν_K(x)) $dH^{{n-1}}$(x), where Z_µ is the zonoid generated by an even, zonal measure µ; its polarization gives the mixed operators Φ_µ(K_1,...,K_{n-1}). On the dual side, the radial Minkowski valuations Ψ_τ are defined by ρ(Ψ_τ L, ·) = ρ(L,·)^{n-1} * Rτ, with R the spherical Radon transform. Lemmas 3.2 and 4.1 express Φ_µ,i and Ψ_τ,i as averages of the classical projection bodies Π_i and intersection bodies I_i over rotations; these integral representations, combined with Jensen and Hölder inequalities, carry the volume product estimates and identify the classical inequalities as the strongest in their families.

What would settle it

For n = 3 and an even, zonal measure µ that is neither discrete nor a scalar multiple of spherical Lebesgue measure, choose two convex bodies K_1, K_2 and numerically compute V_3(Φ_µ,*(K_1,K_2)) V_3(K_1) V_3(K_2); a value greater than the same product for Euclidean balls would disprove Theorem 1.2. Alternatively, for a star body L and a zonal τ, check whether V_n(Ψ_τ,i L)/V_n(L)^i exceeds V_n(I_i L)/V_n(L)^i, which would contradict Corollary 5.7 or Theorem 1.5.

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Extended reading notes

Core claim

The paper's main discovery is Theorem 1.2 and its dual Theorem 1.5. For every even, zonal measure µ on $S^{{n-1}}$, the mixed volume product V_n(Φ_µ,*(K_1,...,K_{n-1})) V_n(K_1)···V_n(K_{n-1}) is maximized by Euclidean balls; when µ is discrete, the maximizers are precisely homothetic ellipsoids, and the new radial valuations Ψ_τ,i satisfy the analogous inequality against intersection bodies and dual affine quermassintegrals. The proofs pass through a mixed-volume inequality (Theorem 5.1) and a generalized Busemann–Petty centroid inequality (Theorem 5.2), and through Jensen-type averaging arguments that single out the classical projection and intersection bodies as the strongest inequalities in the family.

Load-bearing premise

The proofs treat Theorem 1.1 from the authors' earlier article as a black box, so the new mixed inequalities inherit its validity; if that volume-product maximization fails for some even, zonal measure, the conclusions of Theorems 1.2 and 5.2 collapse.

Editorial extensions

If this is right

  • Corollary 5.3: for 1 ≤ i ≤ n−2, the volume product V_n(Φ_µ,i^* K) V_n(K)^i is maximized by Euclidean balls, interpolating between classical isoperimetric inequalities for quermassintegrals and the Lutwak–Petty projection inequalities.
  • Theorem 1.4: with µ(S^{n-1}) = 1/2, the polar volume of Φ_µ,i K is dominated by that of the classical projection body Π_i K, and both are dominated by the reciprocal of the affine quermassintegral A_{n-i}(K), so Lutwak's affine quermassintegral conjecture would imply these inequalities.
  • For every even, zonal τ with τ(S^{n-1}) = κ_{n-1}, the volume of the new radial valuation Ψ_τ,i L is at most V_n(I_i L), so the Busemann and Leng–Lu intersection inequalities dominate the whole family.
  • Theorem 3.7 and Corollary 5.5: an L_p analogue of the Busemann–Petty centroid inequality holds for generalized centroid bodies Γ_µ,p, and the polar volume product V_n(Γ_µ,p^* L) V_n(L) is maximized by balls, extending the L_p Busemann–Petty and Blaschke–Santaló results.

Reading between the lines

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

  • Beyond the paper: the domination results (1.7) and (1.8) suggest that the classical projection and intersection bodies are not just special cases but the sharp members of whole one-parameter families; this could be tested numerically for intermediate zonal measures, where the inequalities predict a monotone volume product as µ interpolates between spherical Lebesgue measure and a discrete measure.
  • Beyond the paper: the same rotation-averaging technique used in Theorem 1.4 might extend to L_p and Orlicz settings beyond the zonal class, since Lemma 3.6 provides the needed averaging identity for Γ_µ,p and the paper already proves the L_p analogue for centroid bodies.
  • Beyond the paper: if Lutwak's affine quermassintegral conjecture (Conjecture 2.1) is ever proved, Theorem 1.4 will immediately upgrade every generalized Lutwak–Petty inequality to an affine invariant one, whereas the present paper already shows the reverse direction fails at the level of these families.
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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 / 6 minor

Summary. The paper generalizes Lutwak's mixed projection volume inequalities and the Leng-Lu intersection-body inequalities to polarizations of Minkowski valuations generated by even, zonal measures on the sphere. The main results are Theorem 1.2, asserting that among K1,...,K_{n-1} the product Vn(Φ^{µ,*}(K1,...,K_{n-1}))Vn(K1)···Vn(K_{n-1}) is maximized by Euclidean balls, with equality characterized by homothetic ellipsoids in the discrete case; Theorem 1.3, a Busemann-Petty type centroid inequality for the generalized centroid bodies Γ^{µ}; Theorem 1.4, domination of polar projection bodies by affine quermassintegrals; and Theorem 1.5, the dual inequalities for new radial Minkowski valuations Ψ^{τ}_i, with a comparison to the classical intersection bodies I_i. The proofs use Hölder, Jensen, the Aleksandrov-Fenchel inequality, spherical convolution, and Radon transforms, and they rely on Theorem 1.1 of the authors' companion paper [23] and on polarization results from [59].

Significance. If the stated results hold, this is a substantial contribution to affine isoperimetric theory: it unifies the Lutwak-Petty inequalities, the Busemann intersection inequality, the Leng-Lu inequalities, and their recent Minkowski-valuation generalizations, and it identifies the classical projection-body inequalities as the strongest members of a large family. The paper also contains a new integral representation and explicit constants, and it connects the results to Lutwak's conjecture on affine quermassintegrals and to Grinberg's inequalities. The proofs are analytic and do not involve parameter fitting or definitional circularity; the main dependency on Theorem 1.1 of [23] is an external input rather than a reduction of the new inequalities to themselves. The paper is significant for convex geometry and geometric tomography, provided the two technical issues raised below are corrected.

major comments (2)
  1. [Section 5, proof of Theorem 1.2, display after (5.5)] The displayed inequality after (5.5) is missing the nth power. From (5.5) one has V(K1,...,K_{n-1}, Γ^{µ}Φ^{µ,*}(K1,...,K_{n-1})) = 1/(n+1), and applying (2.7) gives V(...)^n ≥ Vn(K1)···Vn(K_{n-1})Vn(Γ^{µ}Φ^{µ,*}(...)). The correct consequence is therefore (1/(n+1))^n ≥ Vn(K1)···Vn(K_{n-1})Vn(Γ^{µ}Φ^{µ,*}(...)). Combining this with Theorem 5.2 yields (1/(n+1))^n ≥ [Vn(Γ^{µ}Bn)/Vn(Bn)] Vn(K1)···Vn(K_{n-1})Vn(Φ^{µ,*}(...)). The printed line uses 1/(n+1) instead of its nth power; since n ≥ 3 this is strictly weaker and, together with (5.4), gives only Vn(Φ^{µ,*}(...))∏Vn(K_i) ≤ (n+1)^{n-1}Vn(Φ^{µ,*}Bn)Vn(Bn)^{n-1}, not the asserted sharp inequality. The proof is repairable by restoring the power, but as written it does not establish Theorem 1.2.
  2. [Section 4, Theorem 1.5, equation (1.8)] The constant in the right-hand inequality of (1.8) is inconsistent with the proof and appears to be inverted. The derivation at the end of Section 5 gives Vn(I_iL) ≤ (κ_{n-1}^n κ_n / κ_i^n) ∫_{Gr_{n,i}} Vi(L∩E)^n dν_i(E) = (κ_{n-1}^n / κ_n^{n-1}) \tilde{A}_{n-i}(L)^n. With the constant κ_n^{n-1}/κ_{n-1}^n as printed, the inequality is false already for L = Bn and i = n-1, where Vn(IBn) = κ_n κ_{n-1}^n but the printed right side equals κ_n^{2n-1}/κ_{n-1}^n. The statement of Theorem 1.5 should be corrected to match the factor derived in the proof.
minor comments (6)
  1. [Section 5, proof of Theorem 1.2] The reference 'Theorem (5.1)' should be 'Theorem 5.1'.
  2. [Section 5, proof of Theorem 5.6] The sentence 'replacing K by ϑK' should read 'replacing L by ϑL', since the equivariance is applied to the star body L.
  3. [Section 5, proof of Theorem 1.5] The phrase 'using (5.11, followed by integration' is missing a closing parenthesis; it should read 'using (5.11), followed by integration'.
  4. [Page 24] The sentence 'it is an open problem wether ...' contains a typo; 'wether' should be 'whether'.
  5. [Section 2, definition of dual affine quermassintegrals] In the displayed definition of \tilde{A}_{n-i}(L), the integration domain is written as Gr_{n,k}; it should be Gr_{n,i}.
  6. [Section 5, equation (5.3)] The constant in (5.3) should be typeset unambiguously as (n+1)^n Vn(Γ^{µ}L) Vn(Φ^{µ,*}Bn) Vn(Bn)^{n-1}; the current rendering 'κn−1 n' is hard to read and could be mistaken for κ_{n-1}^n rather than κ_n^{n-1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the proof chain reduces to independent prior theorems and standard inequalities, not to its own conclusions.

full rationale

The paper's central new inequalities (Theorems 1.2, 1.3, 1.5, 3.7, 5.2, and their corollaries) are derived from explicit inputs: the prior sharp inequality Theorem 1.1 from [23] (Haberl-Schuster, overlapping with the second author), the polarization/existence theorem (1.2) from [59] (Schuster), the Aleksandrov-Fenchel inequality (2.7), Hölder's inequality, Jensen's inequality, Radon transform identities, and Grinberg's/Busemann-Straus inequalities. None of these inputs is defined in terms of the target volume products, and no parameter is fitted to any subset of the claimed inequalities. Theorem 1.1 is the special case of Theorem 1.2 obtained when all K_i coincide, but the paper proves Theorem 1.2 from Theorem 1.1 via Theorem 5.1 and Theorem 5.2, not the reverse; Theorem 1.1 is an independently established result, so its use is a dependency rather than a circular reduction. The operators Φ_μ, Γ_μ, and Ψ_τ are introduced by explicit integral formulas (1.1), (3.10), and (4.2), and the polarization formula (1.2) is a quoted theorem, not an ansatz that smuggles in the desired inequality. I found no displayed equality or inequality that is, by construction, the target statement in a different notation. A separate mathematical concern is that the line after (5.5) appears to drop the nth power in the bound obtained from (2.7); this is a proof gap or typo, not a circularity. Therefore the circularity score is 0.

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

The central claims rest on prior theorems by the same group (notably Theorem 1.1 of [23] and polarization results from [59]) plus standard inequalities from convex geometry. No free parameters are fitted, and no ad hoc entities are introduced.

assumptions (6)
  • domain assumption Theorem 1.1 from [23]: for every even, zonal measure µ, Vn(Φµ,*K) Vn(K)^{n-1} is maximized by Euclidean balls, with explicit equality conditions.
    Used as a black box in the proof of Theorem 5.2 (after equation (5.2)) and in Theorem 1.2.
  • domain assumption Existence of a symmetric polarization Φµ(K1,...,K_{n-1}) for each Φµ, as in formula (1.2), proved in [59].
    Defines the mixed Minkowski valuations used throughout the paper.
  • standard math Aleksandrov-Fenchel inequality (2.7): V(K1,...,Kn)^n ≥ Vn(K1)...Vn(Kn).
    Used in the proof of Theorem 1.2 to relate mixed volumes to products of volumes.
  • standard math Grinberg and Busemann-Straus inequalities for dual affine quermassintegrals (2.11).
    Used to derive Corollary 5.7 from Theorem 1.5.
  • standard math Blaschke-Santalo inequality (2.9): Vn(K) Vn(K*) ≤ κ_n^2 for origin-symmetric K.
    Used to prove Corollary 5.5.
  • standard math Busemann-Petty centroid inequality for the case of discrete µ.
    Serves as the base case in the proof of Theorem 5.2.

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Pith. "Pith review of Lutwak-Petty projection inequalities for Minkowski valuations and their duals." pith.science (2026). https://pith.science/paper/AHWX3GHI

@misc{pith2026190801634,
  author       = {Pith},
  title        = {Pith review of: Lutwak-Petty projection inequalities for Minkowski valuations and their duals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHWX3GHI}},
  note         = {Machine review of arXiv:1908.01634}
}
read the original abstract

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.