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An algorithm to find maximum area polygons circumscribed about a convex polygon

Markus Ausserhofer, Susanna Dann, Zsolt Lángi, Géza Tóth

TL;DR

An algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time is presented and a conjecture of Farris is disproved, for the special case of regular n-gons.

Abstract

A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time. As an application, we disprove a conjecture of Farris. Moreover, for the special case of regular n-gons we find an explicit solution.

An algorithm to find maximum area polygons circumscribed about a convex polygon

TL;DR

An algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time is presented and a conjecture of Farris is disproved, for the special case of regular n-gons.

Abstract

A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time. As an application, we disprove a conjecture of Farris. Moreover, for the special case of regular n-gons we find an explicit solution.

Paper Structure

This paper contains 6 sections, 6 theorems, 5 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Geometric properties of maximum area circumscribed polygons
  3. An algorithm to find the maximum area circumscribed polygons
  4. Combinatorial properties of maximum area circumscribed polygons
  5. An application to statistics
  6. Remarks and questions

Key Result

Theorem 1

For any $i,j$ with $i \leq k \leq i+n$, let $Q$ be a convex polygon circumscribed about $P$ of maximal area containing $S_{k}, S_{k+1}, \ldots ,S_{i+n}$ on its boundary: that is, these edges of $P$ are used by $Q$. Then for every $j =i+2, i+3, \ldots,k-1$ either (a) $p_j$ is the midpoint of the side

Figures (8)

  • Figure 1: Replacing $L$ by $L'$ increases the area of $Q$
  • Figure 3: An illustration of (ii) for the case $i=4$
  • Figure 4: An illustration of the proof of (i) for the case $i=3$
  • Figure 5: An illustration of the proof of (i) for the case $i=4$
  • Figure 6: An illustration of the proof of (iii) for the case $i=4$
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1
  • proof
  • Remark 1
  • Definition 2
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 2
  • Theorem 4
  • ...and 9 more