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.
Categories: math.MG