The different ways of stabbing disjoint convex sets

Abstract

We construct a family ofn disjoint convex set in ℝd having (n/(d−1))d−1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ℝd, and construct a family ofd+1 translates in ℝd admitting (d+1)!/2 geometric permutations.