Brunn-Minkowski and Prékopa-Leindler's inequalities under projection assumptions

Brunn-Minkowski's theorem says that vol((1−λ)K+λL)1/n, for K, L convex bodies, is a concave function in λ, and assuming a common hyperplane projection of K and L, it is known that the volume itself is concave. The 'a priori' natural hypothesis of a common (n−k)-plane projection of the sets turned out in the end not to imply the (1/k)-th concavity of the volume function. In this paper we show which is the, somehow, best projection type assumption that is needed in order to get concavity for vol((1−λ)K+λL)1/k, characterizing also the equality case in the corresponding inequality. Moreover, we consider the same problem for its functional analogue: the Prékopa-Leindler inequality.