Extremal convex polygons inscribed in a given convex polygon

TL;DR

This paper introduces efficient algorithms for finding extremal convex polygons inscribed in a convex polygon, focusing on minimum area and perimeter solutions, and explores related variants.

Contribution

It provides the first linear and cubic time algorithms for minimum area and perimeter inscribed convex polygons in convex polygons.

Findings

Minimum area inscribed convex polygon can be found in O(n) time.

Minimum perimeter inscribed convex polygon can be found in O(n^3) time.

The paper explores additional variants of the inscribed polygon problem.

Abstract

A convex polygon $Q$ is inscribed in a convex polygon $P$ if every side of $P$ contains at least one vertex of $Q$. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex $n$-gon in $O(n)$ and $O(n^3)$ time, respectively. We also investigate other variants of this problem.