Brunn-Minkowski Inequality for θ-Convolution Bodies via Ball's Bodies

We consider the problem of finding the best function φn:[0, 1] → R such that for any pair of convex bodies K, L ∈ Rn the following Brunn-Minkowski type inequality holds [Formula: see text] where K+θL is the θ-convolution body of K and L. We prove a sharp inclusion of the family of Ball's bodies of an α-concave function in its super-level sets in order to provide the best possible function in the range (¾)n ≤ θ ≤ 1, characterizing the equality cases.