Refinements of the Brunn-Minkowski inequality
This is a new contribution of the successful research group \textit{Geometría Diferencial y Convexa} from University of Murcia, in which one of the most important researcher is M. A. Hernández Cifré.NEWLINENEWLINEThe authors present an example of two convex bodies \(K\) and \(L\) in \(\mathbb{R}^n\) with the same projection over a \((n-2)\)-dimensional linear plane such that \(\forall \lambda\in[0,1]\) NEWLINE\[NEWLINE\mathrm{vol}((1-\lambda)K+\lambda L)^{1/2}<(1-\lambda)\mathrm{vol}(K)^{1/2}+\lambda \mathrm{vol}(L)^{1/2}.NEWLINE\]NEWLINE From one of the main results of the paper (Theorem 1.6) two consequences can be obtained: 1) the concavity in \(\lambda\) of the volume function \(\mathrm{vol}((1-\lambda)K+\lambda L)\) when two convex bodies \(L\) and \(K\) have the same projection over a hyperplane, and 2) the famous Brunn-Minkowki inequality.NEWLINENEWLINEIn addition, for \(k\in\{1,\dots,n\}\) and every two convex bodies \(K\) and \(L\), it is proved that the inequality NEWLINE\[NEWLINE\mathrm{vol}((1-\lambda)K+\lambda L)\geq(1-\lambda)^{k}\mathrm{vol}(K)+\lambda^{k} \mathrm{vol}(L)NEWLINE\]NEWLINE is true if one of the following conditions holds: a) some \((n-k)\)-dimensional projection of \(K\) and \(L\) are the same; b) the volumes of some of such as projections are equal; c) some suitable hypothesis about sections is satisfied.NEWLINENEWLINEFinally, for particular families of sets called \(p\)-tangential bodies, it is proved the Brunn-Minkowski inequality with exponent \(1\leq p\leq n-1\) and some general Brunn-Minkowski inequality for quermassintegrals.
scientific article; zbMATH DE number 6353708