Finding all Maximal Area Parallelograms in a Convex Polygon

TL;DR

This paper presents an efficient method to find all maximum and locally maximal area parallelograms within a convex polygon, using structural properties to achieve an optimal quadratic time complexity.

Contribution

It introduces a novel structural analysis of locally maximal area parallelograms, enabling all such shapes to be computed in quadratic time and proving their interleaving property.

Findings

All LMAPs can be computed in O(n^2) time.

Number of LMAPs is linear in the number of polygon edges.

Applications include maximum symmetric convex bodies and Heilbronn triangle problem.

Abstract

Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant of computing the maximum area parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs) in a given convex polygon. By proving and utilizing several structural properties of the LMAPs, we compute all of them (including all the MAPs) in $O(n^2)$ time, where $n$ denotes the number of edges of the given polygon. In addition, we prove that the LMAPs interleave each other and thus the number of LMAPs is $O(n)$. We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body inside a convex polygon, and the simplest case of the Heilbronn triangle problem.