A bound on a convexity measure for point sets

Abstract

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most π. We can thus talk about the convexity of a set of points in terms of the minimum, taken over all polygonizations, of the maximum interior angle. The main result presented here is a nontrivial combinatorial upper bound of this min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

An upper bound for min-max angle of polygons

“…Here, we propose an upper bound for min-max value of the angles in polygonization and demonstrate that this bound is tight. Based on our knowledge to date, much less attention has been paid to this aspect so far [25]. The rest of the paper is as follows: In the section 2, notations, definitions and some basic lemma are presented.…”