Sporadic Reinhardt Polygons

Abstract

Let $$n$$ be a positive integer, not a power of two. A 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, which do not possess this structured form. We call these polygons 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 $$r\ge 2$$. We also prove that a positive proportion of the Reinhardt polygons with $$n$$ sides is 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.