Some Discrete Properties of the Space of Line Transversals to Disjoint Balls

Attempts to generalize Helly’s theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in Rd to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euclidean balls in Rd, more precisely the inter-related notions of cone of directions, geometric permutations and Helly-type theorems, and discuss some algorithmic applications.