A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

Abstract

We show that for any log-concave measure $\mu$ on ${\mathbb{R}}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda \in [0,1],$ $$\mu ((1-\lambda )K+\lambda L{)}^{c_n}\ge (1-\lambda )\mu (K{)}^{c_n}+\lambda \mu (L{)}^{c_n},$$ where $c_n\ge n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures.

arXiv comment19 pages; This was initially part of arXiv:2103.11433, but that paper was split into two papers. A minor correction to the exposition was fixed in May 2026. Namely, minor changes were made to Proposition 4.2 and to the argument at the beginning of the proof of Theorem A