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

We consider the problem of finding the best function φ n : [ 0 , 1 ] → R such that for any pair of convex bodies K , L ∈ R n the following Brunn-Minkowski type inequality holds | K + θ L | 1 n ≥ φ n ( θ ) ( | K | 1 n + | L | 1 n ) , where K + θ L 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 3 4 n ≤ θ ≤ 1 , characterizing the equality cases.