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

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |\lambda K+_0 (1-\lambda)L|\geq |K|^{\lambda}|L|^{1-\lambda} \end{equation*} (where $\mathrm{\mid }\cdot \mathrm{\mid }$ 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 $K$.