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Let A be a family of n pairwise disjoint compact convex sets in R a. Let =2Y,=o",_l(m-1)i . We show that the directed lines in R d, d>-3, can be ~d (m) o((")t partitioned into ,1 2 sets such that any two directed lines in the same set which intersect any A'c_ A generate the same ordering on A'. The directed lines in R 2 can be partitioned into 12n such sets. This bounds the number of geometric permutations on A by ½qbd ((~))ford>-3andby6nford=2.doi:10.1007/bf02187777 fatcat:s7a3s6yrqvfc5etyoin57qvkxa