Maximal perimeter and maximal width of a convex small polygon
Abstract
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we propose an approach to construct convex small $n$-gons of large perimeter and large width when $n=2^s$ with $s\ge 2$. Assuming the existence of an axis of symmetry, a convex small $n$-gon is described as a composition of $n/2$ and both its perimeter and its width are given as functions of a single variable. By selecting the composition that minimizes the violation of a cycle constraint by a particular solution, the $n$-gons constructed outperform the best $n$-gons found in the literature. For example, for $n=64$, the perimeter and the width obtained are within $10^{-22}$ and $10^{-12}$ of the maximal perimeter and the maximal width, respectively. From our results, it appears that Mossinghoff's conjecture on the diameter graph of a convex small $2^s$-gon with maximal perimeter is not true when $s \ge 4$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.11831
- arXiv:
- arXiv:2106.11831
- Bibcode:
- 2021arXiv210611831B
- Keywords:
-
- Mathematics - Metric Geometry;
- Mathematics - Combinatorics;
- Mathematics - Optimization and Control;
- 52A40;
- 52A10;
- 52B55;
- 90C26