Sporadic Reinhardt polygons (Q1953052)

A Reinhardt polygon is a convex \(n\)-gon that is optimal in the following optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter (where \(n\) is a positive integer, not a power of 2). For almost all \(n\), there are many Reinhardt \(n\)-gons, and many of them exhibit a special periodic structure. The authors study sporadic Reinhardt polygons (that do not possess such periodic structure). The most interesting result is the following: Sporadic Reinhardt \(n\)-gons occur precisely when \(n=pqr\), where \(p\) and \(q\) are distinct odd primes and \(r \geq 2\).