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The logarithmic Brunn-Minkowski conjecture for origin-symmetric convex bodies

For origin-symmetric convex bodies K,L⊂RnK,L\subset\mathbb R^n with support functions hK,hLh_K,h_L and 0≤λ≤10\le\lambda\le1, the logarithmic combination is the Wulff body (1−λ)K+0λL={x:⟨x,u⟩≤hK(u)1−λhL(u)λ ∀u∈Sn−1}(1-\lambda)K+_0\lambda L=\{x:\langle x,u\rangle\le h_K(u)^{1-\lambda}h_L(u)^\lambda\ \forall u\in S^{n-1}\}. Boroczky, Lutwak, Yang and Zhang (2012, Problem 1.1) conjectured the log-Brunn-Minkowski inequality ∣(1−λ)K+0λL∣≥∣K∣1−λ∣L∣λ|(1-\lambda)K+_0\lambda L|\ge|K|^{1-\lambda}|L|^\lambda, which strengthens the multiplicative Brunn-Minkowski inequality and implies the symmetric LpL_p Brunn-Minkowski inequality for 0<p<10<p<1. They proved it in the plane; it was known for unconditional bodies (Saroglou), bodies with enough common reflection symmetries, complex unit balls, locally near the ball and for zonoids (local form). Does the inequality hold for all origin-symmetric convex bodies in every dimension?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Convex geometry; Lp Brunn-Minkowski theory
Posed by
Karoly J. Boroczky, Erwin Lutwak, Deane Yang and Gaoyong Zhang
Year posed
2012
Years open
14y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every n≥1n\ge1, origin-symmetric convex bodies K,LK,L and 0≤λ≤10\le\lambda\le1, ∣W[hK1−λhLλ]∣≥∣K∣1−λ∣L∣λ|W[h_K^{1-\lambda}h_L^\lambda]|\ge|K|^{1-\lambda}|L|^\lambda. Corollary 1.2 gives the symmetric LpL_p Brunn-Minkowski inequality for full-dimensional origin-symmetric bodies and every 0<p<10<p<1, via the monotone implication recorded by Boroczky-Lutwak-Yang-Zhang. The proof reduces to a variance bound for sums of even powers of linear forms, proved in moment-measure coordinates. It does not treat equality cases, non-symmetric bodies (where the inequality is false in general), or the uniqueness question for the even logarithmic Minkowski problem.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 23, 2026) is the whole family; it also derives the (B)-conjecture, listed as a separate entry.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Problem 1.1 of Boroczky-Lutwak-Yang-Zhang; it is the conjecture verbatim, for arbitrary origin-symmetric convex bodies in every dimension, with no smoothness or unconditionality assumption. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator LogBrunnMinkowski, declaration OAI.LogBrunnMinkowski.main). The comparator statement ComparatorChallenges/LogBrunnMinkowski.lean was read here: for n≥1n\ge1, compact convex origin-symmetric K,L⊂RnK,L\subset\mathbb R^n with nonempty interior and 0≤t≤10\le t\le1, it asserts vol(K)1−tvol(L)t≤vol(W)\mathrm{vol}(K)^{1-t}\mathrm{vol}(L)^t\le\mathrm{vol}(W) where WW is the Wulff body of hK1−thLth_K^{1-t}h_L^t. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript itself notes that Stancu announced the all-dimensional inequality in an Oberwolfach report (2018) with details deferred.

Sources

Changelog1 change

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