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

We show that there exists an ϵ(n) > 0, depending only on the dimension n,s o that for any symmetric convex body K in the ϵ(n)-neighborhood of Bn2 (in the C2 metric), the log-Brunn-Minkowski inequality |λK +0 (1 − λ)|≥ |K|λ|L|1−λ holds. The proof is based on the previous results from [7], as well as an additional “third derivative” argument, which allows us to establish a uniform neighborhood. As a consequence, we conclude that the uniform cone volume measure determines a symmetric convex body uniquely, provided that it is in a fixed neighborhood of any ball.