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 |K+θL|1n≥φn(θ)(|K|1n+|L|1n),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 34n≤θ≤1, characterizing the equality cases.