The Brunn-Minkowski theorem says that \mathrm{vol}\bigl((1-\lambda)K+\lambda L\bigr)^{1/n} v o l ( ( 1 − λ ) K + λ L ) 1 / n , for K,L K , L convex bodies, is a concave function in \lambda λ , and assuming a common hyperplane projection of K K and L L , it was proved that the volume itself is concave. In this paper we study refinements of Brunn-Minkowski inequality, in the sense of ‘enhancing’ the exponent, either when a common projection onto an ( n-k n − k )-plane is assumed or for particular families of sets. In the first case, we show that the expected result of concavity for the k k -th root of the volume is not true, although other Brunn-Minkowski type inequalities can be obtained under the ( n-k n − k )-projection hypothesis. In the second case, we show that for p p -tangential bodies, the exponent in Brunn-Minkowski inequality can be replaced by 1/p 1 / p .