Helly-type theorems for polygonal curves

We prove the following intersection and covering Helly-type theorems for boundaries of convex polygons in the plane. • Let S be a set of points in the plane. Let n4. If any 2n+2 points of S can be covered by the boundary of a convex n-gon, then S can be covered by the boundary of a convex n-gon. The value of 2n+2 is best possible in general. If n=3, 2n+2 can be reduced to 7. • Let be a finite collection of boundaries of convex n-gons, n5. If any 3n−3 members of have non-empty intersection, then has non-empty intersection. The value 3n−3 is best possible in general. For n=3 and 4, the value 3n−3 can be reduced to 8 and 10, respectively.