Result 091, Convex and metric geometry

Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture

Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive L_p Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0\lt p\lt 1$ .

Lean formalization Proof

The bigger picture

Why it matters

The manuscript reports a sharp volume inequality for symmetric convex shapes, which contain every line segment between their points. It bounds how small an intermediate shape can be when distances to supporting planes are combined through geometric averages.

What changes?

The claimed logarithmic Brunn-Minkowski inequality covers origin-symmetric convex bodies in every dimension: compact convex shapes unchanged by reflection through the origin. For ordinary Lebesgue volume, it also gives the additive Lp Brunn-Minkowski inequality for full-dimensional such bodies for every p strictly between zero and one. This family uses power-based combinations of supporting distances. The manuscript further reports the logarithmic inequality and scalar-dilation B-conjecture for every even log-concave Radon measure: a reflection-symmetric, locally finite measure satisfying a logarithmic concavity condition.

What does that help mathematicians do?

Combined with Saroglou's transfer theorem and a reduction to the subspace supporting a measure, the claimed geometric inequality yields control over mass under uniform enlargement. Specifically, for a fixed origin-symmetric convex body, the logarithm of its measure is concave as a function of the logarithm of the dilation factor. Researchers can therefore bound the mass at an intermediate scale from masses at two other scales. This concerns scalar dilation, not arbitrary direction-dependent stretching.

Are there practical applications?

The immediate value is foundational: the reported result connects volume comparisons for symmetric convex bodies with scale-dependent mass comparisons for a broad class of measures. For probability measures in that class, these masses are probabilities of lying inside dilated symmetric regions. This provides a theoretical probability bound, rather than a demonstrated computational method or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The logarithmic Brunn–Minkowski conjecture

September 23, 2026 20 pages Main result formalized in Lean

We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric L_p Brunn–Minkowski inequality for every $0\lt p\lt 1$ . Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation $(B)$ -conjecture.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{The logarithmic Brunn--Minkowski conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf}{OAI:The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/091.md.

Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The logarithmic Brunn–Minkowski conjecture

Scope

The logarithmic Brunn–Minkowski conjecture asserts that logarithmic interpolation of origin-symmetric convex bodies preserves the geometric-mean lower bound for volume. The formalized result proves, for every dimension $n\ge1$ , such bodies $K,L\subset\mathbb R^n$ , and $0\le t\le1$ , that their logarithmic Wulff combination has volume at least $|K|^{1-t}|L|^t$ . It assumes neither boundary smoothness nor coordinatewise unconditionality.

Comparator links

Result	Comparator statement
Logarithmic Brunn–Minkowski inequality	LogBrunnMinkowski.lean

At a glance

Subject: Convex and metric geometry
Manuscripts: 1
Pages: 20
Dated: Sep 23, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Logarithmic Brunn–Minkowski conjectureProved
Scalar-dilation B-conjectureProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

Most similar results

087, Convex and metric geometryThe Mahler conjectures, functional inequalities and polar-product symplectic width
093, Convex and metric geometryDimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
088, Convex and metric geometrySharp projection-body inequalities and a counterexample to simplex maximization
101, Convex and metric geometryThe sharp simplex conjecture for isotropic constants
072, Real and complex analysisBrennan's conjecture and the integral-means spectrum
092, Convex and metric geometryThe optimal order of convex-body covering density

By similarity of summaries and abstracts.

On the map

This result, outlined, and its most similar results among all 372. Open the full map.

Why it matters

What changes?

What does that help mathematicians do?

Are there practical applications?

Manuscript

The logarithmic Brunn–Minkowski conjecture

Lean formalization

Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture

Scope

Comparator links

Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture