On characterizing collections arising from N-gons in the plane (Q916091)

Consider a convex n-gon P in the plane with vertices labelled 1,...,n and a point x in general position in the interior of P. With P and x is associated the collection \({\mathcal F}(x)\) of all 3-element sets \(\{\) i,j,k\(\}\) of \(\{\) 1,...,n\(\}\) for which x lies in the triangle with vertices labelled i, j, k. The authors completely characterize those collections of 3-element subsets of \(\{\) 1,...,n\(\}\) which can arise this way. They also give a characterization for a related problem.