Upper bounds on geometric permutations for convex sets

Abstract

Let A be a family of n pairwise disjoint compact convex sets in Rd. Let {Mathematical expression}. We show that the directed lines in Rd, d ≥ 3, can be partitioned into {Mathematical expression} sets such that any two directed lines in the same set which intersect any A′⊆A generate the same ordering on A′. The directed lines in R2 can be partitioned into 12 n such sets. This bounds the number of geometric permutations on A by 1/2φd for d≥3 and by 6 n for d=2. © 1990 Springer-Verlag New York Inc.