A New Methodology in Geometric Transversal Theory

Ricky Pollack

Abstract

We (in joint work with J. E. Goodman) describe a new encoding of a family of mutually disjoint compact convex sets that captures many of its combinatorial properties and use it to give a new proof of the Edelsbrunner-Sharir theorem that a collection of $n$ mutually disjoint compact convex sets in the plane cannot be met by straight lines in more than $2n-2$ combinatorially distinct ways. The encoding generalizes our encoding of planar point configurations by "allowable sequences" of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement $\cal A$ of separators and double tangents the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations induced by pseudoline transversals compatible with $\cal A$.We also describe another aproach via topological affine planes which yields a proof of a similar generalization of the