Geometric permutations of higher dimensional spheres (Q1883581)

Given a set \(A\) of pairwise convex objects in \(R^d\), a line \(\ell\) that intersects every object in \(A\) is said to be a {line transversal} of \(A\). Such a transversal defines two linear orders of the objects in \(A\) along \(\ell\) (from both directions); these two orders count as one {geometric permutation}. Studying the maximum number of different geometric permutations a set of objects may have is a problem that has received much attention, and only in \(R^2\) is completely settled. In this paper the authors give tight results for sets of spheres with bounded radius ratio in \(R^d\), \(d\geq 3\). As a particular case, they show that the maximum number of geometric permutations of \(n\) congruent spheres in \(R^d\), \(d\geq 3\), is at most \(4\), matching an independently discovered upper bound for this case by \textit{Y. Zhou} and \textit{S. Suri} [Comput. Geom. 26, No. 1, 3--20 (2003; Zbl 1039.52013)].