On empty convex polygons in a planar point set
Rom Pinchasi, Radoš Radoičić, Micha Sharir · 2004
Let P be a set of n points in general position in the plane. Let Xk(P) denote the number of empty convex κ-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities and inequalities are new, except for a couple that have been proved earlier using a considerablymore complex machinery from matroid and polytope theory,algebraic topology and commutative algebra. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.