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On unbalanced difference bodies and Godbersen's conjecture

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Pith's one-line read This paper proves that Godbersen's conjecture holds on average for every convex body, with equality only for simplices.

desk verdict Solid short note: new binomial-weighted and uniform-average mixed-volume inequalities that would follow from Godbersen's conjecture, with a small but real equality-case bug at λ=0 and λ=1. read the letter →

arxiv 2412.05308 v2 pith:NX4GBJQM submitted 2024-11-27 math.MG

Shiri Artstein-Avidan , Eli Putterman This is my paper · ORCID
classification math.MG MSC 52A3952A40
keywords convexgeometrymixedvolumesGodbersen'sconjectureRogers-ShephardinequalitysimplexextremalityunbalanceddifferencebodycovariogramBrunn-Minkowski
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 establishes unconditional bounds on weighted averages of Godbersen's mixed volumes $V(K[j],-K[n-j])$, quantities that measure how a convex body and its reflection interact. The main theorem, Theorem 2, shows that for every convex body $K\subset\mathbb{R}^n$ and every $\lambda\in[0,1]$, the sum $\sum_{j=0}^n \lambda^j(1-\lambda)^{n-j}V(K[j],-K[n-j])$ is at most $\operatorname{Vol}(K)$, with equality only for simplices. Averaging this over $\lambda$ gives Corollary 3: $\frac{1}{n+1}\sum_{j=0}^n \binom{n}{j}^{-1}V_j\le \operatorname{Vol}(K)$, a uniform version of Godbersen's conjecture "on average" that does not require any single $V_j$ to obey the conjecture. A second theorem generalizes the Rogers-Shephard inequality by bounding another weighted sum of mixed volumes, recovering the difference-body bound at one end and the known $j=1,n-1$ cases at the other. If correct, the results do not settle Godbersen's conjecture, but they show its extremal behavior—simplex maximality—survives in integrated form for every convex body.

What carries the argument

Throughout, the paper works with Rogers-Shephard-type constructions: the $(n+1)$-dimensional body $C=\operatorname{conv}(\{0\}\times(1-\lambda)K\cup\{1\}\times(-\lambda K))$ in Lemma 8, whose volume is computed by slicing and expanding in mixed volumes, and a $(2n+1)$-dimensional body $T$ built from two scaled copies of $K$. The engine is Lemma 9, Rogers and Shephard's section-projection inequality, which bounds the product of the volume of a section and the volume of an orthogonal projection of a convex body. Fubini slicing of $C$ converts the bound on $\operatorname{Vol}(C)$ into exactly the binomial-weighted sum of Theorem 2. Theorem 5 uses an "unbalanced covariogram" $f(x)=\operatorname{Vol}((\lambda K+x)\cap K)/\operatorname{Vol}(\lambda K)$ and a local Steiner formula to produce a second weighted sum. Equality analysis is carried by the homothety statement in Lemma 9: equality forces all relevant sections to be homothetic, which pins down the simplex.

What would settle it

Search for a convex body $K$ that is not a simplex and a value $\lambda\in(0,1)$ for which every section $\lambda(1-\lambda)K\cap(\lambda(1-\lambda)K+y)$ is homothetic to $\lambda(1-\lambda)K$ as $y$ varies; if such a body exists, the equality part of Lemma 8 and hence Theorem 2 would fail. Equivalently, numerically evaluate the sum in Theorem 2 for a non-simplex such as a cube or an ellipsoid and check whether equality holds for any $\lambda$; any equality would falsify the theorem.

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Extended reading notes

Core claim

The central claim is that several families of linear combinations of the mixed volumes $V_j=V(K[j],-K[n-j])$ are maximized, for fixed volume, by the $n$-simplex, even though the individual inequalities $V_j\le \binom{n}{j}\operatorname{Vol}(K)$ (Godbersen's conjecture) remain open. Theorem 2 proves the binomial-weighted sum $\sum_{j=0}^n \lambda^j(1-\lambda)^{n-j}V_j\le\operatorname{Vol}(K)$ for all $\lambda\in[0,1]$, with equality if and only if $K$ is a simplex; Theorem 5 proves an analogous bound for a second weighted sum involving $V(K[n-j],-K[j])$, again sharp exactly for simplices. Theorems 2 and 5 would both be immediate consequences of Godbersen's conjecture, and the paper shows conversely that both follow from the unbalanced difference body conjecture stated as Conjecture 6, which it proves in dimensions up to five. The method identifies the simplex as the unique maximizer in these averaged senses, so the extremal content of Godbersen's conjecture is correct even though the pointwise form remains unresolved.

Load-bearing premise

The load-bearing premise is that Rogers and Shephard's equality characterization in the section-projection inequality is complete: if equality holds in Lemma 9, then all relevant sections are homothetic; the inequalities themselves do not depend on this, but the assertions that equality occurs only for simplices do.

Editorial extensions

If this is right

  • Corollary 3 gives, for any convex body of volume one, $\frac{1}{n+1}\sum_{j=0}^n \binom{n}{j}^{-1}V_j\le 1$, so Godbersen's conjecture holds on average over $j$ for every body.
  • The median of the normalized quantities $\binom{n}{j}^{-1}V_j$ is less than two, and Corollary 4 gives a Markov-type statement: for at least $k$ of the $n-1$ interior indices, $V_j\le \frac{n-1}{n-k}\binom{n}{j}$ when $\operatorname{Vol}(K)=1$.
  • Theorems 2 and 5 recover the known boundary cases $j=1$ and $j=n-1$ of Godbersen's conjecture, namely $V(K,-K[n-1])\le n\operatorname{Vol}(K)$.
  • Taking $\lambda=1$ in Theorem 5 recovers the classical Rogers-Shephard inequality for the difference body, while Theorem 2 supplies a new proof of the earlier pointwise bound from reference [2].
  • Conjecture 6, which asks whether the unbalanced difference body $D_\lambda K=(1-\lambda)K-\lambda K$ is volume-maximized by simplices, is proved in dimensions $n\le 5$, so simplex maximality of $D_\lambda K$ is verified in low dimensions.

Reading between the lines

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

  • A direct consequence the authors leave implicit is that differentiating Theorem 2 with respect to $\lambda$ yields polynomial inequalities in $\lambda$ whose coefficients constrain the differences $V_j-\binom{n}{j}$; these may be easier to test than the individual conjecture.
  • The reduction in Section 4 suggests a program for Godbersen's conjecture: if the unbalanced difference-body ratio $\operatorname{Vol}(D_\lambda K)/\operatorname{Vol}(K)$ can be shown to be maximized by the simplex in all dimensions, then both Theorem 2 and Theorem 5 follow by integration; the low-dimensional verification for $n\le 5$ supports but does not prove this.
  • One could test Conjecture 6 numerically in dimension six or seven for random polytopes; since Theorem 2 and Theorem 5 already hold unconditionally, such tests would probe only the stronger unbalanced-difference-body statement, not the paper's main results.
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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 manuscript studies Godbersen's conjecture via weighted sums of the mixed volumes V_j = V(K[j], -K[n-j]). Its main results are Theorem 2, a binomial-type bound sum_{j=0}^n lambda^j (1-lambda)^{n-j} V_j <= Vol(K), and Theorem 5, a more intricate double-sum bound, both with equality claimed exactly for simplices. Corollary 3 integrates Theorem 2 to obtain a uniform average form of Godbersen's conjecture. Section 4 shows how Theorems 2 and 5 would follow from the unbalanced difference body conjecture (Conjecture 6) and proves Conjecture 6 for n=4,5 by reducing to the known j=1 bound and the Rogers-Shephard inequality. Section 5 gives a new proof of an inequality from [2]. The arguments are non-circular and use Rogers-Shephard section-projection inequalities, a covariogram/Brunn-Minkowski argument, and the local Steiner formula.

Significance. Assuming the equality statements are corrected, the paper gives genuine progress on a longstanding conjecture: it proves a Godbersen-type bound on a uniform average for every body, generalizes Rogers-Shephard in a different direction, and settles the unbalanced difference body conjecture in dimensions 4 and 5. The proofs are self-contained and do not assume the conjectures they address; Theorem 7 is reduced to classical inequalities. The main weakness is the endpoint degeneracy in the equality statements, which is local and fixable rather than fatal to the inequalities themselves.

major comments (2)
  1. [Section 2, Theorem 2 and Lemma 8] The equality characterization in Theorem 2 is false at lambda=0 and lambda=1. Setting lambda=0 in (5) leaves only the j=0 term, which equals Vol(-K)=Vol(K), so every convex body attains equality; the lambda=1 endpoint is analogous. Lemma 8 inherits the same problem: at lambda=0 the body C is a pyramid over K with volume Vol(K)/(n+1) for every K. In the proof this degeneracy is visible because the argument divides by Vol(lambda(1-lambda)K), which vanishes at the endpoints, so the homothety-based equality analysis is not valid there. The inequality (5) is unaffected, but the equality statement should be restricted to lambda in (0,1), with the endpoints noted as trivial equality cases or excluded.
  2. [Section 3, Theorem 5] The equality claim in Theorem 5 is false at lambda=0. At lambda=0 the double sum in (6) reduces to the m=n term, whose value is exactly Vol(K), so equality holds for every K. The proof also degenerates at both endpoints: B=lambda(K-K) is {0} at lambda=0 and K'=(1-lambda)K is {0} at lambda=1, so the local Steiner formula (9), which requires 0 in int(B), is not directly applicable. The correct statement is that equality characterizes simplices for lambda in (0,1); the case lambda=1 then follows from the Rogers-Shephard inequality, and lambda=0 is trivial.
minor comments (3)
  1. [Section 2, equality case of Lemma 8] In the displayed formula for T_{theta,y}, the last set should be theta*lambda*K intersected with ((1-theta)(1-lambda)K - y), not with (1-theta)(1-lambda)K; the subsequent sentence appears to use the corrected version.
  2. [Section 4, Theorem 7] In the n=5 case the sentence that equality follows from the equality case of Rogers-Shephard alone is slightly compressed; when alpha-beta > 0 it also uses equality in V_1 <= 5, whose equality case gives the same simplex conclusion. Expanding this sentence would avoid an apparent gap.
  3. [Abstract and Introduction] The phrase "with equality if and only if K is a simplex" in the abstract and in the statements of Theorems 2 and 5 is overbroad; it should be qualified to lambda in (0,1) after the endpoint issue is resolved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: The theorems are derived from Rogers–Shephard, Brunn–Minkowski, and the local Steiner formula, not from the conjectures they address.

full rationale

The derivation chain is self-contained. Theorem 2 is obtained by slicing the Rogers–Shephard body C and expanding the integrand by the mixed-volume binomial expansion; the only geometric input is Lemma 9, whose equality case is a published external theorem, not a restatement of the paper's conclusion. Theorem 5 is derived from the covariogram estimate f≥(1−d_B)^n, which uses only Brunn–Minkowski concavity, and from the local Steiner formula (9) of Schneider–Weil; the equality check for the simplex uses the known mixed volumes of the simplex and Chu–Vandermonde. Godbersen's conjecture and Conjecture 6 are never assumed: Section 4 explicitly runs in the converse direction, showing that Conjecture 6 would imply Theorems 2 and 5, so it cannot make the proofs circular. Self-citations to [2] and [4] are to published external results with stated hypotheses and are not used as an unverified uniqueness chain. I note one non-circular mathematical caveat: the equality characterizations in Theorems 2 and 5 degenerate at λ=0 and λ=1, where the left side is Vol(K) for every K and the proof divides by Vol(λ(1−λ)K); this is a correctness and statement issue, not circularity, and it does not affect the inequalities themselves.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data. The auxiliary bodies C and T and the unbalanced difference body D_lambda K are elementary convex-geometric constructions, not postulated entities needing independent evidence. All substantive inputs are standard theorems from the cited literature.

assumptions (7)
  • standard math Rogers-Shephard inequality for sections and projections: Vol(P_{E⊥}T) Vol(T∩E) ≤ C(m,j) Vol(T), with equality forcing homothetic sections (Lemma 9).
    Used in Lemma 8 and Section 5. The equality case is quoted from [2] and drives the equality iff simplex statements.
  • standard math Local Steiner formula (9): ∫ g(d_B(x,K)) dx = Vol(K)g(0) + Σ_{m=0}^{n-1} V(K[m],B[n-m]) C(n,m) ∫ (n-m)t^{n-1-m} g(t) dt for monotone g, from [13].
    Used to evaluate the integral in the proof of Theorem 5. The authors state it holds by approximation for monotone g.
  • standard math Brunn-Minkowski inequality and its equality condition.
    Used to establish the 1/n-concavity of the covariogram function f in Theorem 5 and the homothetic equality case.
  • standard math Known j=1 and j=n-1 case of Godbersen's conjecture: V(K,-K[n-1]) ≤ n Vol(K), from the inclusion K ⊂ n(-K) after centering, cited to Schneider [11].
    Used in Theorem 7 for n=4,5 and to show that Theorem 2 contains the known boundary cases.
  • standard math Rogers-Shephard inequality for the difference body: Vol(K-K) ≤ C(2n,n) Vol(K), with equality iff simplex.
    Used in the Theorem 7 reduction and as the baseline for Theorem 5 at lambda=1.
  • standard math K is the closure of the convex hull of its exposed points [12, Theorem 1.4.7].
    Step 1 of Proposition 10, showing that equality in Theorem 5 forces K to be a polytope.
  • standard math [9, Lemma 4]: if L ∩ (L+y) are all homothetic as y varies, then L is a simplex.
    Final step of the equality case of Lemma 8, converting homothetic sections into simplexity.

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Pith. "Pith review of On unbalanced difference bodies and Godbersen's conjecture." pith.science (2026). https://pith.science/paper/NX4GBJQM

@misc{pith2026241205308,
  author       = {Pith},
  title        = {Pith review of: On unbalanced difference bodies and Godbersen's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NX4GBJQM}},
  note         = {Machine review of arXiv:2412.05308}
}
abstract

The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\binom{n}{j}$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n + 1} \sum_{j = 0}^n \binom{n}{j}^{-1} V_j \le 1$, showing that Godbersen's conjecture holds ''on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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