
Journal of Convex Analysis 21 (2014), No. 3, 727743 Copyright Heldermann Verlag 2014 Refinements of the BrunnMinkowski Inequality María A. Hernández Cifre Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain mhcifre@um.es Jesús Yepes Nicolás Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain jesus.yepes@um.es [Abstractpdf] \def\vol{\mathrm{vol}} The BrunnMinkowski 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 BrunnMinkowski inequality, in the sense of `enhancing' the exponent, either when a common projection onto an ($nk$)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 BrunnMinkowski type inequalities can be obtained under the ($nk$)projection hypothesis. In the second case, we show that for $p$tangential bodies, the exponent in BrunnMinkowski inequality can be replaced by $1/p$. Keywords: BrunnMinkowski inequality, projections, ptangential bodies. MSC: 52A20, 52A40; 52A39 [ Fulltextpdf (162 KB)] for subscribers only. 