Refinements of the Brunn-Minkowski Inequality

\def\vol{\mathrm{vol}} The Brunn-Minkowski theorem says that $\vol\bigl((1-\lambda)K+\lambda L\bigr)^{1/n}$, for $K,L$ convex bodies, is a concave function in $\lambda$, and assuming a common hyperplane projection of $K$ and $L$, 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 ($n-k$)-plane is assumed or for particular families of sets. In the first case, we show that the expected result of concavity for the $k$-th root of the volume is not true, although other Brunn-Minkowski type inequalities can be obtained under the ($n-k$)-projection hypothesis. In the second case, we show that for $p$-tangential bodies, the exponent in Brunn-Minkowski inequality can be replaced by $1/p$.