  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
$|\cdot|$ 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 $K$.
