A combinatorial property on angular orders of plane point sets

Abstract

We study the following combinatorial property of point sets in the plane: For a set S of n points in general position and a point p ∈ S consider the points of S − p in their angular order around p. This gives a star-shaped polygon (or a polygonal path) with p in its kernel. Define c ( p ) as the number of convex angles in this star-shaped polygon around p, and c ( S ) as the sum of all c ( p ) , for p ∈ S . We show that for every point set S, c ( S ) is always at least 1 2 n 3 2 − O ( n ) . This bound is shown to be almost tight. Consequently, every set of n points admits a star-shaped polygonization with at least n 2 − O ( 1 ) convex angles.