Lines Pinning Lines

A line l is a transversal to a family F of convex polytopes in ℝ3 if it intersects every member of F. If, in addition, l is an isolated point of the space of line transversals to F, we say that F is a pinning of l. We show that any minimal pinning of a line by polytopes in ℝ3 such that no face of a polytope is coplanar with the line has size at most eight. If in addition the polytopes are pairwise disjoint, then it has size at most six.