Most Reinhardt polygons are sporadic

@article{Hare2014MostRP,
  title={Most Reinhardt polygons are sporadic},
  author={Kevin G. Hare and Michael J. Mossinghoff},
  journal={Geometriae Dedicata},
  year={2014},
  volume={198},
  pages={1-18}
}
A 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 sporadic. Prior work has described the periodic case completely, and has shown that sporadic Reinhardt polygons occur for all n of the form $$n=pqr$$n=pqr with p and q distinct odd primes and $$r\ge 2$$r≥2. We show that… Expand
6 Citations

Figures and Tables from this paper

Tight bounds on the maximal perimeter of convex equilateral small polygons
A small polygon is a polygon of unit diameter. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 4$. In this paper, we construct a family ofExpand
The equilateral small octagon of maximal width
A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds theExpand
Maximal perimeter and maximal width of a convex small polygon
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with n = 2s sides are unknown when s ≥ 4. In this paper, we construct a family ofExpand
Characterizing gonality for two-component stable curves
It is a well-known result that a stable curve of compact type over $\mathbb{C}$ having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersectionExpand
A note on the maximal perimeter and maximal width of a convex small polygon
  • Fei Xue, Yanlu Lian, Jun Wang, Yuqin Zhang
  • Mathematics
  • 2021
The polygon P is small if its diameter equals one. When n = 2, it is still an open problem to find the maximum perimeter or the maximum width of a small n-gon. Motivated by Bingane’s series of works,Expand
On very stablity of principal G-bundles
Let X be a smooth irreducible projective curve. In this note, we generalize the main result of Pauly and Peón-Nieto (Geometriae Dedicata 1–6, 2018) to principal G-bundles for any reductive linearExpand

References

SHOWING 1-10 OF 10 REFERENCES
Sporadic Reinhardt Polygons
TLDR
This work completely characterize the integers for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $$n=pqr$$ with $$p$$ and $$q$$ distinct odd primes and $$r\ge 2$$, and it is proved that a positive proportion of the Reinhardt polygamy with sides is sporadic for almost all integers. Expand
Enumerating isodiametric and isoperimetric polygons
TLDR
An exact formula is obtained for E(n) if and only if n=p or n=2p for some odd prime p, and the precise value of E( n) is computed for several integers by enumerating the sporadic polygons that occur in the extremal set. Expand
On convex polygons of maximal width
Abstract. In this paper we consider the problem of finding the n-sided ( $n\geq 3$) polygons of diameter 1 which have the largest possible width wn. We prove that $w_4=w_3= {\sqrt 3 \over 2}$ and,Expand
Isoperimetric Polygons of Maximum Width
TLDR
Two mathematical programs are proposed to determine the maximum width when n=2s with s≥3 and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n=8, 16, and 32. Expand
Isodiametric Problems for Polygons
TLDR
It is shown that the values obtained cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c1. Expand
A $1 Problem
TLDR
The proof of [3] is based on the maximum principle, which is available in its full generality only for (cooperative systems of) second order equations, and the radially decreasing part of the claim allows an o.d.e. approach to get a negative answer to the question whether or not positive solutions are radially symmetric. Expand
Inequalities for convex polygons and Reinhardt polygons
  • Mat . Prosveshchenye
  • 2007
Inequalities for convex polygons and Reinhardt polygons
  • Mat. Prosveshchenye (3) 11
  • 2007
On the factorization of cyclic groups
Extremale Polygone gegebenen Durchmessers.