A bound on a convexity measure for point sets

TL;DR

This paper establishes a new combinatorial upper bound on the convexity measure of planar point sets based on their polygonizations, and discusses a related conjecture for optimal bounds.

Contribution

It introduces a nontrivial upper bound on the convexity measure for point sets and proposes a conjecture for the best possible bound based on combinatorial analysis.

Findings

Derived a nontrivial upper bound for the convexity measure.

Connected convexity to polygonizations with bounded interior angles.

Posed a conjecture for the optimal upper bound.

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 \pi. 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.