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 $p\in S$ 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 $p\in S$. We show that for every point set S, $c(S)$ is always at least $\frac{1}{\sqrt{2}}{n}^{\frac{3}{2}}-O(n)$. This bound is shown to be almost tight. Consequently, every set of n points admits a star-shaped polygonization with at least $\sqrt{\frac{n}{2}}-O(1)$ convex angles.