Covering the Edges of a Complete Geometric Graph with Convex Polygons

Given a set $P$ of $n \geq 3$ 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 ${n \choose 2}$ straight line segments determined by the points of $P$. We show that if $n$ is odd, the answer is $\frac{n^2-1}{8}$ regardless of the choice of $P$. The answer in the case where $n$ is even depends on the choice of $P$ and not only on $n$. The talk will focus on presenting the various aspects of the problem and the proof, as well as on their combinatorial context.
This is a joint work with Oren Yerushalmi.