Tight bounds on the maximal perimeter of convex equilateral small polygons

TL;DR

This paper constructs convex equilateral small polygons with a power-of-two number of sides, providing tight bounds on their maximal perimeter and improving upon previous results, especially for the case of 16 sides.

Contribution

It introduces a new family of polygons that closely approximate the maximum perimeter, surpassing known bounds and resolving the case for 16-sided polygons.

Findings

Perimeters within O(1/n^4) of the maximum

Constructed polygons outperform previous bounds

Proved Mossinghoff's 16-gon is suboptimal

Abstract

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s \ge 4$, and show that their perimeters are within $O(1/n^4)$ of the maximal perimeter and exceed the previously best known values from the literature. In particular, for the first open case $n=16$, our result proves that Mossinghoff's equilateral hexadecagon is suboptimal.