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

Abstract

We show that for any log-concave measure $μ$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $λ\in [0,1],$ $$μ((1-λ) K+λL)^{c_n}\geq (1-λ) μ(K)^{c_n}+λμ(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ 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.