  We consider the problem of finding the best function
$\varphi_n:[0,1]\to\mathbb{R}$ such that for any pair of convex bodies
$K,L\in\mathbb{R}^n$ the following Brunn-Minkowski type inequality holds $$
|K+_\theta
L|^\frac{1}{n}\geq\varphi_n(\theta)(|K|^\frac{1}{n}+|L|^\frac{1}{n}), $$ where
$K+_\theta L$ is the $\theta$-convolution body of $K$ and $L$. We prove a sharp
inclusion of the family of Ball's bodies of an $\alpha$-concave function in its
super-level sets in order to provide the best possible function in the range
$\left(\frac{3}{4}\right)^n\leq\theta\leq1$, characterizing the equality cases.
