Convex transversals

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): "Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?" We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are scaled copies of a convex polygon. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments (or convex polygons) in 2D if the vertices of the transversal are restricted to a given set of points. We also consider stabbing with vertices of a regular polygon - a problem closely related to approximate symmetry detection: Given a set of disks in the plane, is it possible to find a point per disk so that the points are vertices of a regular polygon? We show that the problem can be solved in polynomial time, and give an algorithm for an optimization version of the problem.