A combinatorial property on angular orders of plane point sets
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 is an element of 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 is an element of S. We show that for every point set S. c(S) is always at least 1 3 1/root 2n(3/2) - 0(n). This bound is shown to be almost tight. Consequently, every set of n points admits a star-shaped polygonization with at least root n/2 - 0(1) convex angles. (C) 2011 Elsevier B.V. All rights reserved.
point set star-shaped polygon convex angle combinatorial problems