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

Abstract

For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $μ$ on $\mathbb R^n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R^n$, and all $λ\in[0,1]$, $$ μ(λK+(1-λ)L)^{p_n} \ge λμ(K)^{p_n}+(1-λ)μ(L)^{p_n}, $$ where $$ p_n\ge \frac{c}{n^2\ln n} $$ for some absolute constant $c>0$.