Sharp bounds on geometric permutations of pairwise disjoint bails in R-d

We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in R-d, is Theta (n(d-1)) . This improves substantially the upper bound of O (n(2d-2)) know n for general convex sets [9]. We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks in the plane is two, impr oving the previous upper bound of three given in [5].