大多数Reinhardt多边形是零星的
Most Reinhardt polygons are sporadic
摘要：
\textit{Reinhardt polygon}是凸$n$ -边形，对于不是$2$的幂的$n$，其在三个不同的几何优化问题中是最优的，例如，其具有相对于其直径的最大周长。一些这样的多边形表现出特定的周期性结构;其他的被称为\textit{sporadic}。先前的工作已经完整地描述了周期性情况，并表明对于所有形式为$n=pqr$的$n$，都出现零星的Reinhardt多边形，其中$p$和$q$是不同的奇素数，而$r\geq2$。我们发现，（二面角等价类）零星莱因哈特多边形的数量超过了几乎所有的$n$的周期，并发现这首先发生在$n=105$。当$n=2pq$具有$p$和$q$不同的奇素数时，我们还确定了稀疏Reinhardt多边形个数的一个公式.
英文摘要：
A \textit{Reinhardt polygon} is a convex $n$-gon that, for $n$ not a power of $2$, is optimal in three different geometric optimization problems, for example, it has maximal perimeter relative to its diameter. Some such polygons exhibit a particular periodic structure; others are termed \textit{sporadic}. Prior work has described the periodic case completely, and has shown that sporadic Reinhardt polygons occur for all $n$ of the form $n=pqr$ with $p$ and $q$ distinct odd primes and $r\geq2$. We show that (dihedral equivalence classes of) sporadic Reinhardt polygons outnumber the periodic ones for almost all $n$, and find that this first occurs at $n=105$. We also determine a formula for the number of sporadic Reinhardt polygons when $n=2pq$ with $p$ and $q$ distinct odd primes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
