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

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Pith's one-line read This paper proves that mixed volumes of a convex body with its own reflection are bounded by the simplex value, settling a 1938 conjecture, and gives the sharp L_p analogue with simplex extremizers.

desk verdict The inequality part of Godbersen is real and elegant; the polytope equality proof has a fixable but real gap in Lemma 4.4. read the letter →

arxiv 2607.20387 v1 pith:ZJE6DXQC submitted 2026-07-22 math.MG

Jan Kotrbatý , Mohamed A. Mouamine This is my paper · ORCID
classification math.MG MSC 52A4052A3952B11
keywords Godbersen'sconjecturemixedvolumeRogers-ShephardinequalityL_pBrunn-MinkowskitheoryFireysumconvexpolytopessimplexextremizersdifferencebody
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 settles a conjecture from 1938 about the volume of a convex body mixed with its own reflection about the origin. It proves that for every convex body K in R^n and every k, the mixed volume V(K[k], -K[n-k]) is at most the binomial coefficient (n choose k) times the volume of K, and that among polytopes equality occurs only for simplices. The proof works dimensionally: after embedding K diagonally into R^{2n}, mixed-volume monotonicity gives the binomial bound term by term, which also yields a short proof of the classical Rogers-Shephard inequality. The same estimate is then used to prove the long-open L_p version of Rogers-Shephard for convex bodies containing the origin, with equality characterized precisely by simplices having a vertex at the origin. These results matter because they replace partial and asymptotic bounds by sharp constants in all dimensions and settle the equality cases in the polytopal and L_p settings.

What carries the argument

The central device is a factorization identity for mixed volumes in complementary orthogonal subspaces of R^{2n}: for bodies L_i in E and M_j in E^⊥, the full mixed volume factors as (N choose k)^{-1} times the product of the two lower-dimensional mixed volumes, with constants normalized by Lebesgue measure. Applied to the diagonal embedding Δ(x)=(x,x) and the anti-diagonal embedding Δ̃(x)=(x,-x), mixed-volume monotonicity (ΔK ⊂ K^2) yields the binomial bound in Proposition 3.2. Equality for polytopes is then read from support conditions of the mixed surface-area measure, which force face incidence relations that only a simplex can satisfy. For the L_p result, the Firey L_p sum is represente

What would settle it

Compute V(K[2], -K[2]) for a convex body in R^4 and compare with 6 vol(K): a value above 6 would refute the inequality, and a non-simplex polytope attaining 6 would refute the equality classification.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes two theorems. Theorem 1.1: for any convex body K in R^n and 0 ≤ k ≤ n, V(K[k], -K[n-k]) ≤ (n choose k) vol(K); if K is a polytope with nonempty interior and 0 < k < n, equality holds exactly when K is a simplex. Theorem 1.2: if 0 is in K and p ∈ (1, ∞] with 1/p + 1/q = 1, then vol(K +_p (-K)) ≤ sum_{k=0}^n (n choose k)^2 (n/q choose k/q)^{-1} vol(K), with equality exactly when K is a simplex with a vertex at the origin. These are sharp refinements of the Rogers-Shephard bound, and the equality statements identify the simplex as the unique extremal shape in the relevant settings.

Load-bearing premise

The equality half of the L_p theorem rests, in the final step of Theorem 1.2, on a cited but not re-proved result that V(K, -K, ..., -K) = n vol(K) forces K to be a simplex; if that k=1 characterization had other extremizers, the equality classification in Theorem 1.2 would not follow.

Editorial extensions

If this is right

  • For every 0 < λ < 1, the volume ratio of (1-λ)K + λ(-K) is maximized by simplices; this confirms a separate conjecture that was previously known only in low dimensions.
  • The equality result for polytopes is the first full solution of the sharp form of the conjecture in dimensions n ≥ 4, where earlier results covered only special classes such as bodies of constant width and anti-blocking bodies.
  • The L_p Rogers-Shephard inequality now holds for all convex bodies containing the origin and all p > 1; previously it was known only in the plane or for restricted classes, and equality forces a simplex with a vertex at the origin.
  • The method also proves the inequality part of the higher-order generalization of Godbersen's conjecture and gives a new proof of Schneider's higher-order inequality.
  • Quantitative stability of the Rogers-Shephard inequality reduces to stability of the k=1 mixed-volume inequality and hence to the Minkowski measure of symmetry.

Reading between the lines

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

  • Because the equality classification in Theorem 1.1 is proved only for polytopes, the question of whether non-simplex convex bodies can attain equality remains open; continuity of mixed volumes does not by itself transfer the polytope result, since an approximating sequence need not attain equality.
  • The p = ∞ case of Theorem 1.2, namely vol(conv(K ∪ -K)) ≤ 2^n vol(K), is a concrete specialization that could be tested by computation for high-dimensional polytopes; its equality case is a simplex with a vertex at the origin.
  • The factorization argument is not obviously limited to the pair K, -K; a natural test is whether the same monotonicity yields a sharp bound for V(K[k], L[n-k]) when L is contained in a scaled copy of K, which would extend the conjecture to nonsymmetric pairs.
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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 / 4 minor

Summary. The paper proves Godbersen's conjecture: for every convex body K⊂R^n and every 0≤k≤n, the mixed volume V(K[k],−K[n−k]) is at most binom(n,k) vol_n(K). For polytopes with nonempty interior and 0<k<n, it further characterizes equality by K being a simplex. The proof of the inequality is a short argument in R^{2n} using the diagonal embedding, the mixed-volume splitting lemma, and monotonicity of mixed volumes. The equality case for polytopes is obtained by converting the equality chain into vanishing of certain integrals η_{r,j}, then using face incidence conditions to force simpliciality and finally a simplex. In the last section, the paper derives an L_p Rogers–Shephard inequality for bodies containing the origin, with sharp constant Σ (n choose k)^2 (n/q choose k/q)^{-1} vol_n(K), and equality for simplices with a vertex at the origin.

Significance. If the proofs are completed, this settles a conjecture from 1938 and provides a clean, unified derivation of both the classical Rogers–Shephard inequality and its L_p analogue. The inequality part (Proposition 3.2) is elegant and appears correct; the derivation of the L_p inequality by integrating the discrete inequality is conceptually attractive. The equality classifications are substantial and are the main vulnerability of the paper: they depend on delicate face-incidence arguments and on an external characterization of equality for k=1. The manuscript also explicitly claims that the method extends to higher-order settings, which increases its potential impact. However, the equality proof as written contains a false cone identity and an induction gap, so the paper is not yet in acceptable form.

major comments (2)
  1. [§4, Lemma 4.4 (Eq. (15))] The proof asserts C=(ι1C1+ι2C2)∩(∆C0). This is false: the definition of C only requires v1+v2∈C0, whereas ∆C0 imposes v1=v2. Consequently, u∈relint C need not lie in relint(∆C0), and the span formula (15) is not justified as written. Since (15) is used to conclude w∈spanC, Lemma 4.4 is unproved. This lemma is load-bearing for the polytope equality case of Theorem 1.1. The likely repair is to replace ∆C0 by δ^{-1}(C0)={(v1,v2): v1+v2∈C0} and to justify (15) via relative interiors of the two cones; such a correction must be supplied.
  2. [§4, proof of Theorem 1.1 (induction step)] The induction step is stated for all 1≤k≤n−1, but the induction hypothesis applies in dimension n−1 only to nontrivial indices 1≤m≤n−2. For k=n−1, Lemma 4.6 propagates the vanishing of η only to j'=n−1, which is the trivial index for a facet E, so it does not imply that E is a simplex. Thus the conclusion that K is simplicial is not established for k=n−1. The gap is fixable by using the symmetry V(K[k],−K[n−k])=V((−K)[n−k],K[k]) to reduce equality for k=n−1 to the case k=1, but this reduction is not stated in the manuscript.
minor comments (4)
  1. [§4, Lemma 4.3] The claim 'F1∩F2=∅ implies f0≤n−1' is false: if F0=K, then u1+u2=0, not necessarily u=0; for a cube with u=(e1,−e1), opposite facets satisfy the displayed dimension identities and f0=n. This assertion is not actually needed, since r≥0 already follows from the choice s≥n, but the passage should be corrected.
  2. [§5, Lemma 5.1] In the definition of a1,a2,b1,b2, the factors (1−t)^{-1/q} and t^{-1/q} are undefined at t=0 or t=1. This can be handled by assuming 0<t<1 and using continuity (or by treating the endpoints separately), but it should be stated.
  3. [§4, Lemma 4.7] In the proof, the set A_i is used but not defined; it should presumably be V(E_i). Also, the sentence 'Then for i≠j, A_i∩A_j is an (n−2)-dimensional face' needs clarification.
  4. [General] Some notation is introduced informally (e.g., K^2 for K×K, and the binomial notation with real arguments). A brief note fixing these conventions would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main inequalities are derived from standard mixed-volume monotonicity with exact constants; self-citations are contextual only.

full rationale

The main inequality (5) is not taken as input: Proposition 3.2 obtains it by comparing V(K^2[n], ι1K[k], ι2K[n−k]) with V(∆K[n], ι1K[k], ι2K[n−k]) through monotonicity of mixed volumes, Lemma 2.1, and the exact identities (9)–(11). The target quantity appears only after the comparison, not in the hypothesis, and the constant C(n,k) is computed exactly, not fitted. Proposition 3.1 is the same independent argument for Rogers–Shephard. Theorem 1.2 follows by Fubini integration of the already proved inequality (5); the coefficient Σ_k C(n,k)^2 (n/q choose k/q)^{-1} is obtained from an explicit beta integral. No fitted parameter is later renamed as a prediction. The equality analysis for polytopes is internal (Lemmas 4.2–4.7) apart from the classical k=1 simplex characterization cited to Grünbaum [19, §6.1] and the simplex verification in [35,15] for the L_p case; those are independent external facts, not author-supplied inputs chosen to force the conclusion. Self-citations [24] and [17] are used only for context and motivation; the proof does not rely on them. A reviewer-flagged correctness concern in Lemma 4.4 (the asserted identity C=(ι1C1+ι2C2)∩(∆C0) and the span formula (15)) would be a proof gap if it stands, but it is not a circular reduction of the theorem to its assumptions. Thus the circularity score is low.

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

No free parameters: all constants emerge from mixed-volume identities and beta-function integrals. The axioms are standard theorems of mixed volume theory (splitting formula, positivity, monotonicity) plus two external classical results: the k=1 equality characterization of Minkowski's measure of symmetry (Grünbaum), and the fact that simplices with a vertex at the origin attain equality in the L_p inequality (cited to [7],[35],[15]). No invented entities are introduced.

assumptions (6)
  • standard math Mixed-volume splitting formula: (N choose k)V(L_1,...,L_k,M_1,...,M_{N-k}) = V^E(L_1,...,L_k)V^{E⊥}(M_1|E⊥,...,M_{N-k}|E⊥) for L_i⊂E, M_j⊂E⊥ (Lemma 2.1, from [44, Thm 5.3.1]).
    Used in Prop 3.1 and 3.2 to reduce mixed volumes in R^{2n} to products in the diagonal and anti-diagonal subspaces.
  • standard math Positivity and support characterization of mixed volumes: V(K_1,...,K_n)>0 iff dim(Σ_{i∈I}K_i) ≥ |I| for all I; support formula (7) for mixed surface area measure.
    Used in Prop 3.2 (vanishing terms in the expansion) and throughout Section 4 to identify when a vector lies in the support of a mixed surface area measure (13).
  • standard math Monotonicity of mixed volume: K⊂L implies V(K,K_2,...,K_n) ≤ V(L,K_2,...,K_n).
    The entire proof of Prop 3.1/3.2 and the equality-case chain in Section 4 rest on this.
  • domain assumption Characterization of equality in the k=1 case: V(K,−K[n−1]) = n vol_n(K) iff K is a simplex (Minkowski measure of symmetry; Grünbaum [19, §6.1]).
    Used in the proof of Theorem 1.2 to conclude from equality in (5) for all k (in particular k=1) that K is a simplex.
  • standard math The L_p-Minkowski sum representation K+_p L = { (1-t)^{1/q} x + t^{1/q} y : x∈K,y∈L,t∈[0,1] } for p>1, q=p/(p-1) (Lutwak–Yang–Zhang [33, Lemma 2]).
    Used in Section 5 to express D_pK as the union of C_t and to prove the concavity of |I_z|.
  • domain assumption Simplices with a vertex at the origin attain equality in the L_p-Rogers–Shephard inequality (6) (Bianchini–Colesanti [7]; Manui–Ndiaye–Zvavitch [35, Lemma 23]; Fradelizi–Manui–Meyer–Ndiaye [15, Thm 1]).
    Used in the equality part of Theorem 1.2 to assert sufficiency; the proof says 'well known and not difficult to verify' and cites [35,15].

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Pith. "Pith review of Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality." pith.science (2026). https://pith.science/paper/ZJE6DXQC

@misc{pith2026260720387,
  author       = {Pith},
  title        = {Pith review of: Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJE6DXQC}},
  note         = {Machine review of arXiv:2607.20387}
}
abstract

We prove that the mixed volume of a convex body with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the Rogers-Shephard inequality. We also prove that, among convex polytopes, simplices are the only extremizers. Finally, we use this inequality to prove the $L_p$-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.

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