A note on the quantitative local version of the log-Brunn-Minkowski inequality

Abstract

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |λK+_0 (1-λ)L|\geq |K|^λ|L|^{1-λ} \end{equation*} (where $|\cdot |$ is the Lebesgue measure and ${+}_0$ is the so-called log-addition) holds when $K\subset {\mathbb{R}}^n$ is a ball and $L$ is a symmetric convex body in a suitable $C^2$ neighborhood of $K$.

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