Maximal perimeter and maximal width of a convex small polygon

TL;DR

This paper introduces a new approach to construct convex small polygons with a large perimeter and width for n=2^s sides, surpassing previous bests and challenging existing conjectures.

Contribution

The paper proposes a novel method to construct convex small polygons with maximal perimeter and width for n=2^s sides, outperforming previous results and providing insights into Mossinghoff's conjecture.

Findings

Constructed polygons with n=64 sides close to theoretical maximums.

Method outperforms previous known polygons in literature.

Challenged Mossinghoff's conjecture for s ≥ 4.

Abstract

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we propose an approach to construct convex small $n$-gons of large perimeter and large width when $n=2^s$ with $s\ge 2$. Assuming the existence of an axis of symmetry, a convex small $n$-gon is described as a composition of $n/2$ and both its perimeter and its width are given as functions of a single variable. By selecting the composition that minimizes the violation of a cycle constraint by a particular solution, the $n$-gons constructed outperform the best $n$-gons found in the literature. For example, for $n=64$, the perimeter and the width obtained are within $10^{-22}$ and $10^{-12}$ of the maximal perimeter and the maximal width, respectively. From our results, it appears that Mossinghoff's conjecture on the diameter graph of a convex small $2^s$-gon with maximal perimeter is not true when $s \ge 4$.