On neighborly families of convex bodies
We say that a family of convex bodies in \(E^d\) is neighborly if the intersection of every two of them is \((d-1)\)-dimensional. The paper gives an answer to the question of \textit{B. Grünbaum} [Isr. J. Math. 1, 5--10 (1963; Zbl 0192.26604)] who asked how many members can a neighborly family of centrally symmetric convex bodies in \(E^d\) have? Namely, it is proved that for every \(d\geq 3\) there exists a sequence of neighborly centrally symmetric convex bodies which are pairwise affinely equivalent and which have prescribed volumes. What is more, those bodies have a large group of affine automorphisms. A similar fact is shown about neighborly families of similar polytopes whose diameters are prescribed.
scientific article; zbMATH DE number 2101782