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

Corollary 8.1: for every $n\ge1$ , every even log-concave Radon measure $\mu$ on $\mathbb R^n$ (finite or infinite total mass, possibly supported on a subspace) and origin-symmetric convex bodies $K,L$ , the log-Brunn-Minkowski inequality holds for $\mu$ ; taking $L$ a dilate of $K$ gives log-concavity of $t\mapsto\mu(e^tK)$ . 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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: The (B)-conjecture for even log-concave measures: log-concavity of t -> mu(e^t K)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Corollary 8.1 was read; it gives $\mu(W[h_K^{1-\lambda}h_L^\lambda])\ge\mu(K)^{1-\lambda}\mu(L)^\lambda$ 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.

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof (OAI.LogBrunnMinkowski.main)

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Lean proof (OAI.LogBrunnMinkowski.main)

lean artifact · pending

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This event attributed to Unreleased internal OpenAI model

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

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

The logarithmic Brunn-Minkowski conjecture for origin-symmetric convex bodies evidence for this event

Lean proof (OAI.LogBrunnMinkowski.main) evidence for this event

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