Intersections of convex bodies (Q5941879)

For convex bodies \(K_0,K_1,\dots,K_m \subset\mathbb{R}^n\) and for \(\rho=(r_1, \dots,r_n) \in(\mathbb{R}^n)^m\), let \(\Phi(\rho): =K_0\cap \bigcup^m_{i =1} (K_1+r_i)\) and \(D:=\{\rho: \Phi(\rho) \neq\emptyset\}\). Then \(D\) is convex, and the family \(\{\Phi(\rho)\}_{\rho\in D}\) is concave. If \(\max_\rho \dim\Phi (\rho)=k\geq 1\), then for the mean cross-sectional measures \(W_\nu (\Phi (\rho))\), \(\nu=0,1, \dots,k-1\), the functions \(\rho\mapsto \root{k-\nu} \of {W_\nu (\Phi(\rho))}\), \(\rho\in D\), are concave. \{Reviewer's remark: This is essentially known. The first part is contained in a paper by the reviewer [Monatsh. Math. 74, 258-272 (1970; Zbl 0194.53803)] and the second part follows from A. D. Aleksandrov's generalization of the Brunn-Minkowski theorem\}.