Sporadic Reinhardt polygons

TL;DR

This paper characterizes the existence of sporadic Reinhardt polygons, which are non-periodic optimal polygons, showing they occur for certain composite numbers and constitute a positive proportion of all Reinhardt polygons.

Contribution

It provides a complete characterization of integers with sporadic Reinhardt polygons and introduces a construction method to count them for various n.

Findings

Sporadic Reinhardt polygons exist if and only if n=pqr with p,q odd primes and r≥2

A positive proportion of Reinhardt polygons are sporadic for almost all n

A new construction method estimates the number of sporadic polygons for specific n

Abstract

Let $n$ be a positive integer, not a power of two. A \textit{Reinhardt polygon} is a convex $n$-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all $n$, there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$, additional Reinhardt polygons exist that do not possess this structured form. We call these polygons \textit{sporadic}. We completely characterize the integers $n$ for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $n=pqr$ with $p$ and $q$ distinct odd primes and $r\geq2$. We also prove that a positive proportion of the Reinhardt polygons with $n$ sides are sporadic for almost all integers $n$, and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of $n$ by a construction that we introduce.