On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body

Abstract:

We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality $W_i^2\geq W_{i-1}W_{i+1}$ which relates the quermassintegrals of a convex body $K$ to those of an arbitrary hyperplane projection of $K$. A consequence is the following fact: for any convex body $K$, for any $(n-1)$-dimensional subspace $E$ of ${\mathbb R}^n$ and any $t>0$, \begin{equation*}\frac {|P_E(K)+tD_E|}{|P_E(K)|}\leq \frac {|K+2tD_n|}{|K|},\end{equation*} where $D$ denotes the Euclidean unit ball and $|\cdot |$ denotes volume in the appropriate dimension.