A constant bound for geometric permutations of disjoint unit balls
An oriented line is called a line transversal for a set \(S\) of bodies in the \(d\)-dimensional euclidean space if it intersects every member of \(S\). If the objects in \(S\) are pairwise disjoint convex bodies, then a line transversal of \(S\) induces two linear orderings, which are the orders in which the line meets the members of \(S\). The two orders induced by a line transversal are essentially the same (one is the reverse of the other) and so they are together called a single geometric permutation. In the planar case the maximum number of geometric permutations of a family \(S\) with \(n\) members is known to be \(2n-2\). However for dimension \(d>2\) there is a big gap between the upper and the lower bound. This gap motivated to consider a natural specialized family of convex bodies: unit balls. The authors prove that the maximum number of geometric permutations for a family of \(n\) pairwise disjoint unit balls in the \(d\)-dimensional euclidean space is four, for \(n\) a sufficiently large constant that depends on the dimension \(d\).
scientific article; zbMATH DE number 1969081