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

What OpenAI claims

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 Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0 < p < 1. (Lean)

Paper

1. The logarithmic Brunn–Minkowski conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every 0 < p < 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.