Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies

We consider the problem of finding the best function ${\phi }_n:[0,1]\to \mathbb{R}$ such that for any pair of convex bodies $K,L\in {\mathbb{R}}^n$ the following Brunn-Minkowski type inequality holds $$\mathrm{\mid }K{+}_{\theta }L{\mathrm{\mid }}^{\frac{1}{n}}\ge {\phi }_n(\theta )(\mathrm{\mid }K{\mathrm{\mid }}^{\frac{1}{n}}+\mathrm{\mid }L{\mathrm{\mid }}^{\frac{1}{n}}),$$ 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 ${\left(\frac{3}{4}\right)}^n\le \theta \le 1$, characterizing the equality cases.