Covering the Edges of a Complete Geometric Graph with Convex Polygons

Given a set P of m >= 3 points in general position in the plane, we want to find the smallest possible number of convex polygons with vertices in P such that the edges of all these polygons contain all the (m 2) straight line segments determined by the points of P. We show that if m is odd, the answer is (m(2) - 1)/8 regardless of the choice of P. In this case there is even a partitioning of the edges of the complete geometric graph on m vertices into (m(2) - 1)/8 convex polygons. The answer in the case where m is even depends on the choice of P and not only on m. Nearly tight lower and upper bounds follow in the case where m is even.