# Maximal perimeter and maximal width of a convex small polygon

@inproceedings{Bingane2021MaximalPA, title={Maximal perimeter and maximal width of a convex small polygon}, author={Christian Bingane}, year={2021} }

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 of convex small n-gons, n = 2s with s ≥ 4, and show that their perimeters and their widths are within O(1/n8) and O(1/n5) of the maximal perimeter and the maximal width, respectively. From this result, it follows that Mossinghoff’s conjecture on the diameter graph of a convex small 2s-gon with maximal…

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## References

SHOWING 1-10 OF 15 REFERENCES

Tight bounds on the maximal perimeter and the maximal width of convex small polygons

- Mathematics
- 2020

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ vertices are not known when $s \ge 4$. In this paper, we construct a…

Isodiametric Problems for Polygons

- MathematicsDiscret. Comput. Geom.
- 2006

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.

On convex polygons of maximal width

- Mathematics
- 2000

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,…

An Isodiametric Problem for Equilateral Polygons

- Mathematics
- 2008

Abstract. The maximal perimeter of an equilateral convex polygon with unitdiameter and n = 2 m edges is not known when m ≥ 4. Using experimentalmethods, we construct improved polygons for m ≥ 4, and…

Most Reinhardt polygons are sporadic

- Mathematics
- 2014

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…

A Discrete Isoperimetric Problem

- Mathematics
- 1997

We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (π/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence)…

The Small Octagons of Maximal Width

- Computer ScienceDiscret. Comput. Geom.
- 2013

The paper answers an open problem introduced by Bezdek and Fodor and uses a recent version of the deterministic and reliable global optimization code IBBA based on interval and affine arithmetics to guarantee a certified numerical accuracy.

OPTIGON: Extremal small polygons.

- https://github.com/cbingane/optigon,
- 2020