On the perimeters of simple polygons contained in a disk

TL;DR

This paper provides a simplified proof for the maximum perimeter of simple polygons with an odd number of sides inscribed in a disk, extending the result to hyperbolic and spherical geometries.

Contribution

It offers a shorter, more straightforward proof of the maximum perimeter configuration and generalizes the result to hyperbolic and small spherical disks.

Findings

Maximum perimeter is achieved by an isosceles triangle with a multiple edge

The proof is simplified compared to previous work

Results extend to hyperbolic and small spherical disks

Abstract

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.