Geometric Permutations Induced by Line Transversals through a Fixed Point

A line transversal of a family S of n pairwise disjoint convex objects is a straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in Rd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K (n − 1, d − 1) and K (n, d − 1), where K (n, d) =d i=0 ( n−1 i ) = (nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d − 1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K (n, d − 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations. ∗ Work on this paper has been partially supported by a grant from the U.S.–Israeli Binational Science Foundation. The first author’s work on this paper has also been supported by NSF Grant CCR-9972568 and NSF ITR Grant CCR-0081964. Part of the work was performed while he was visiting the Mathematical Sciences Research Institute, Berkeley, California. Part of the work on this paper was performed while the second author was a Ph.D student at Tel-Aviv University, under the supervision of Micha Sharir, and later a postdoctoral fellow at the Mathematical Sciences Research Institute, Berkeley, California. 286 B. Aronov and S. Smorodinsky