Covering n-Segment Unit Arcs Is Not Sufficient

In 1974 Gerriets and Poole conjectured for n = 3 that a convex set in the plane which contains a congruent copy of every n-segment polygonal arc of unit length must be a cover for the family of all unit arcs. We disprove this general conjecture by describing for each positive integer n a convex region Rn that contains a ongruent copy of every n-segment unit arc but not a congruent copy of every unit arc.