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L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies

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Pith's one-line read Asymmetric L2-zonoids obey a sharp Rogers–Shephard bound of factor 2^n, equivalent to a projection comparison for Euclidean balls.

desk verdict Solid, carefully proved advance on Lp-Rogers-Shephard for L2-zonoids and planar centered bodies; remaining general-p cases are cleanly left open. read the letter →

arxiv 2607.03582 v1 pith:SPHZA77Q submitted 2026-07-03 math.MG math.FA

Matthieu Fradelizi , Auttawich Manui , Mark Meyer , Cheikh Saliou Ndiaye This is my paper
classification math.MGmath.FA MSC 52A2052A21
keywords Rogers-ShephardinequalityasymmetricLp-zonoidsFireysummationprojectionsreverseBrascamp-Liebsymmetricconvexbodies
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

Classical Rogers–Shephard inequalities bound the volume of a convex body against the volume of its difference body or its convex hull with its reflection. This paper lifts those bounds to Firey Lp-sums and asks how much they improve when the body is restricted to asymmetric Lp-zonoids or to bodies with a center of symmetry. For asymmetric Lp-zonoids the desired sharp inequality |Z ⊕p −Z| ≤ 2n |Z| is shown to be equivalent to a comparison of the projected volumes of the full ℓq-ball and its positive orthant onto any n-dimensional subspace. The comparison (and therefore the zonoid inequality) is proved completely when p = 2, together with equality cases; partial results cover hyperplane and line projections, local maximizers, and averages. For origin-containing bodies that possess a center of symmetry a smaller constant is conjectured and is established for asymmetric L1-zonoids, which settles the planar case.

What carries the argument

The continuous reverse Brascamp–Lieb inequality applied to an isotropic generating measure of the L2-zonoid (after a linear transformation that makes the measure isotropic). The inequality produces the volume bound; its equality characterization forces the support of the measure onto an orthonormal basis, which translates into the geometric equality cases.

What would settle it

Produce an asymmetric L2-zonoid that is not a linear image of the positive orthant ball yet saturates |Z ⊕2 −Z| = 2n |Z|, or an n-plane whose projected Euclidean ball exceeds 2n times the projected positive orthant.

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Extended reading notes

Core claim

Every asymmetric L2-zonoid Z in Rn satisfies |Z ⊕2 −Z| ≤ 2n |Z|, with equality if and only if Z lies in a hyperplane or is the L2-sum of n segments based at the origin. Equivalently, for every n-dimensional subspace E of Rm the volume of the orthogonal projection of the Euclidean ball Bm2 is at most 2n times the volume of the projection of the positive orthant of that ball, with equality precisely when E is spanned by positive combinations of disjoint groups of coordinate vectors.

Load-bearing premise

The L2 argument relies on the continuous reverse Brascamp–Lieb inequality and its equality characterization once the generating measure has been made isotropic; if that characterization misses non-Gaussian optimizers, the equality cases would be incomplete.

Editorial extensions

If this is right

  • The factor 2n is sharp for every asymmetric L2-zonoid and is attained exactly on linear images of the positive orthant ball.
  • The same factor holds for all unconditional convex bodies under hyperplane and line projections.
  • Centrally symmetric planar bodies satisfy the conjectured Lp-bound, with equality on parallelograms having a vertex at the origin.
  • Averaged (intrinsic-volume) versions of the projection inequality hold for all 1-unconditional zonoids.

Reading between the lines

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

  • The zonoid–projection equivalence may let other projection inequalities be imported directly into the Firey-sum setting.
  • The local-maximizer theorem suggests that the positive orthant ball is a strict local maximizer among asymmetric Lp-zonoids for every p > 1.
  • A reverse Brascamp–Lieb argument that works for q ≠ 2 would resolve the remaining open cases of the main conjecture.
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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

0 major / 7 minor

Summary. The paper studies sharp Rogers–Shephard-type volume comparisons under Firey Lp-summation for two restricted classes: asymmetric Lp-zonoids and origin-containing bodies with a center of symmetry. For asymmetric Lp-zonoids it formulates Conjecture 1 (|Z ⊕p (−Z)| ≤ 2^n |Z|) and proves its equivalence (for zonotopes) to a projection inequality comparing |PE B_q^m| and |PE(B_q^m ∩ R_+^m)| (Conjecture 2 / Theorem 13). The projection inequality and the zonoid inequality are established in full, with equality cases, when p = 2 (Theorems 1–2), via discrete and continuous reverse Brascamp–Lieb inequalities after isotropic positioning. Partial affirmative results for general p are given for hyperplane and line projections, a local maximizer near the model body B_q^n ∩ R_+^n, and an averaged (intrinsic-volume) form for 1-unconditional zonoids. For bodies with a center of symmetry the authors conjecture a sharper constant κ_n,p^(s) and prove it for asymmetric L1-zonoids by shadow-system reduction (Theorem 7), which settles the planar case.

Significance. The work cleanly extends classical Rogers–Shephard inequalities into the Firey Lp setting for two natural subclasses where sharper constants are expected. The equivalence between the zonoid inequality and a concrete projection comparison for orthants of ℓ_q-balls is a useful structural contribution. The complete resolution for p = 2, including equality characterization for both zonotopes and general zonoids (via Barthe’s continuous reverse Brascamp–Lieb and the Böröczky–Kalantzopoulos–Xi equality cases), is a solid, self-contained advance. The shadow-system proof for asymmetric L1-zonoids recovers the planar symmetric case and fits the existing Campi–Gronchi / Bianchini–Colesanti toolkit. Partial results (hyperplanes, lines, local maximizers, averaged form) give concrete evidence toward the general conjectures and are of independent interest in the geometry of projections of unconditional bodies.

minor comments (7)
  1. Title and running heads contain missing spaces (e.g., “FORL p-ZONOIDS”, “Lp-ROGERS–SHEPHARD”). Clean typography throughout.
  2. Remark 1 correctly notes that the approximation argument from Conjecture 2 yields the inequality for general asymmetric Lp-zonoids but does not automatically transfer equality cases; Theorem 2 supplies the missing equality analysis only for p = 2. A short forward pointer from Remark 1 to Theorem 2 would help the reader.
  3. In the statement of Conjecture 2 the equality partition condition is lengthy; a brief parenthetical that it means E is spanned by n positive-cone combinations of disjoint blocks of the standard basis would improve readability.
  4. Lemma 18 (Lewis-type isotropic positioning for L2-zonoids) is standard but the push-forward construction of μ_TZ could be written more explicitly for non-specialists.
  5. Conjecture 5 (uniqueness of equality for planar symmetric bodies) is stated only at the very end and is not numbered in the introduction; either promote it to the main list of conjectures or mark it clearly as an open uniqueness question.
  6. A few bibliographic items are listed as “In preparation” ([21], [36]); if they remain unpublished at acceptance, replace with arXiv identifiers or remove non-essential citations.
  7. Notation for the Firey sum is written both ⊕_p and ⊕p; standardize to one form.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: derivations rest on external reverse Brascamp-Lieb inequalities and standard convex-geometry tools, not on self-definitional or fitted quantities.

full rationale

The paper's central claims (Theorems 1-2 for p=2, Proposition 3 for hyperplanes, Theorem 7 for L1-zonoids) are ordinary mathematical inequalities derived from previously published external results: Barthe's reverse Brascamp-Lieb (discrete and continuous forms, Theorems 16 and 19), Lewis isotropic positioning, Campi-Gronchi shadow systems, and classical Rogers-Shephard. The equivalence of Conjectures 1 and 2 (Theorem 13) is a direct linear-algebraic reduction via singular-value decomposition and support-function identities (Lemmas 8-12), not a circular redefinition. Equality cases are obtained by applying the external equality characterizations of reverse BL and then reducing via the elementary n-segment computation of Proposition 15. Self-citations (e.g., [20], [21], [37]) supply background or related volume formulae but are never load-bearing for the new inequalities; the target bounds are not recovered from parameters fitted to the same data. No quantity is defined in terms of the quantity later claimed as a prediction. The derivation chain is therefore self-contained against external mathematical benchmarks and exhibits no circular reduction.

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

The paper works entirely inside classical convex geometry. All background results (Firey Lp-sums, reverse Brascamp-Lieb, mixed volumes, shadow systems, isotropic measures) are taken from the published literature. No free parameters are fitted and no new physical or geometric entities are postulated beyond the standard definitions of asymmetric Lp-zonoids (already introduced by Weberndorfer).

assumptions (4)
  • standard math Firey Lp-sum of origin-containing convex bodies is well-defined via support functions (or the equivalent non-convex formula of Lutwak-Yang-Zhang) and coincides with Minkowski sum for p=1 and convex hull for p=∞.
    Used throughout; recalled in Section 1.1 and Lemma 8.
  • standard math Continuous reverse Brascamp-Lieb inequality of Barthe (Theorem 19) and its equality characterization hold for isotropic measures and functions satisfying hypothesis (H).
    Invoked as a black box in the proof of Theorem 2.
  • standard math Volume of a shadow system of convex bodies is a convex function of the parameter (Rogers-Shephard).
    Used in the proof of Theorem 7 to reduce to parallelotopes.
  • domain assumption Asymmetric Lp-zonoids are precisely the Hausdorff limits of finite Lp-sums of origin-containing segments, equivalently bodies whose support function is the Lp-norm of a positive measure on the sphere.
    Definition taken from Weberndorfer; used to pass from zonotopes to zonoids by approximation.

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Pith. "Pith review of L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies." pith.science (2026). https://pith.science/paper/SPHZA77Q

@misc{pith2026260703582,
  author       = {Pith},
  title        = {Pith review of: L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPHZA77Q}},
  note         = {Machine review of arXiv:2607.03582}
}
abstract

We study generalizations of the classical Rogers--Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers--Shephard inequality for asymmetric $L_p$-zonoids in $\mathbb{R}^n$ is equivalent to proving a sharp inequality between the volumes of projections of $B_q^m\cap \mathbb{R}^m_+$ and $B_q^m$ onto an $n$-dimensional subspace $E$, where $q$ is the H\"older conjugate of $p$. We conjecture that the inequality is sharp when the subspace $E$ is a coordinate subspace. We fully establish this inequality along with equality conditions in the case $p =2$. For general $p$, we prove it in the case $n=m-1$, $n=1$, and discuss several particular cases, including an averaged version and a local version of the inequality. We then turn to the setting of convex bodies having a center of symmetry. Rogers and Shephard also proved a sharp version of their inequality for bodies in this class. We conjecture a similar bound for the $L_p$-summation, and we establish our conjecture for the particular case of asymmetric $L_1$-zonoids, which, in particular, proves our conjecture in the planar case.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality

    math.MG 2026-07 accept novelty 8.0 of 10

    Godbersen's 1938 conjecture is proved: V(K[k],−K[n−k]) ≤ C(n,k) vol(K) for all convex bodies, with equality characterizations, and it yields the sharp L_p Rogers–Shephard inequality.

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