The logarithmic Brunn-Minkowski inequality

Support functions and Wulff bodies; the logarithmic Brunn-Minkowski inequality for origin-symmetric convex bodies and its reduction to a variance bound for soft polytopes; the additive inequality for powers between 0 and 1. Moment pairs; adjoint identities; the operator A and its test functions. The tensor estimate, with the difference written as a sum of squares. Density of the even test functions. The variance bound for soft polytopes and the logarithmic Brunn-Minkowski inequality, from two named hypotheses; the Gaussian case without hypotheses. The course is an exposition of OpenAI's proof (2026) of the logarithmic Brunn-Minkowski inequality; it proves the inequality from two statements of the literature that it names as hypotheses.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Wulff bodies and the reduction to a variance bound
2. Moment coordinates and their adjoints
3. The tensor estimate
4. Density of the even tests
5. The variance bound and the inequality

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• The Lebesgue integral in several variables with dominated and monotone convergence; compact convex sets.