, Volume 49, Issue 3, pp 540-557

Sporadic Reinhardt Polygons

Purchase on Springer.com

$39.95 / €34.95 / £29.95*

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.