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Sporadic Reinhardt polygons | | Kevin G. Hare
; Michael J. Mossinghoff
; | | Date: |
19 Mar 2012 | | Abstract: | Let $n$ be a positive integer, not a power of two. A extit{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 extit{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 $rgeq2$. 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. | | Source: | arXiv, 1203.4107 | | Services: | Forum | Review | PDF | Favorites | |
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