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The (B)-conjecture for even log-concave measures: log-concavity of t -> mu(e^t K)

Let μ\mu be an even log-concave measure on Rn\mathbb R^n and KK an origin-symmetric convex body. The (B)-conjecture asks whether t↦μ(etK)t\mapsto\mu(e^tK) is log-concave on R\mathbb R. Cordero-Erausquin, Fradelizi and Maurey (2004) proved it for the Gaussian measure, and the question for general even log-concave measures remained open; Saroglou (2016) showed that the logarithmic Brunn-Minkowski inequality in a given dimension implies it for every even log-concave density in that dimension. Is t↦μ(etK)t\mapsto\mu(e^tK) log-concave for every even log-concave measure μ\mu and every origin-symmetric convex body KK?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Convex geometry; log-concave measures
Posed by
Not named in the manuscript; it cites Cordero-Erausquin, Fradelizi and Maurey (2004) for the conjecture's Gaussian case
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Corollary 8.1: for every n≥1n\ge1, every even log-concave Radon measure μ\mu on Rn\mathbb R^n (finite or infinite total mass, possibly supported on a subspace) and origin-symmetric convex bodies K,LK,L, the log-Brunn-Minkowski inequality holds for μ\mu; taking LL a dilate of KK gives log-concavity of t↦μ(etK)t\mapsto\mu(e^tK). This is the scalar-dilation form; the stronger diagonal (B)-property for general linear dilations is not claimed except where Saroglou's equivalence for cubes applies.

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. This entry is Corollary 8.1 of the log-Brunn-Minkowski manuscript (September 23, 2026), obtained by combining its main theorem with Saroglou's published transfer theorem and a short support-subspace reduction written by the model.

Verification

No independent mathematician has checked this yet. Checked here: Corollary 8.1 was read; it gives μ(W[hK1−λhLλ])≥μ(K)1−λμ(L)λ\mu(W[h_K^{1-\lambda}h_L^\lambda])\ge\mu(K)^{1-\lambda}\mu(L)^\lambda for every even log-concave Radon measure, hence the scalar-dilation (B)-property. The deduction cites Saroglou's transfer theorem (Mathematika 2016, Theorem 3.1) and adds a reduction for measures supported on proper subspaces. The Lean formalization (comparator LogBrunnMinkowski, OAI.LogBrunnMinkowski.main, listed in lean/formalization.yaml) states only the Lebesgue volume inequality, read here; the transfer to measures and the (B)-conjecture are not formalized. Not rebuilt here. The proof was not refereed. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the Lebesgue inequality, and the step to the (B)-conjecture uses Saroglou's published transfer.

Sources

Changelog1 change

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