Journal of Convex Analysis 21 (2014), No. 3, 727--743
Copyright Heldermann Verlag 2014
Refinements of the Brunn-Minkowski 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
[Abstract-pdf]
\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 $1/p$.
Keywords: Brunn-Minkowski inequality, projections, p-tangential bodies.
MSC: 52A20, 52A40; 52A39
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