Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons

TL;DR

This paper constructs small polygons with maximal area for even numbers of vertices, improving previous bounds and explicitly solving the case for six vertices, advancing understanding of geometric optimization.

Contribution

It introduces a new construction method for small polygons that achieves higher maximal areas, refining Mossinghoff's bounds for all even n ≥ 6.

Findings

Constructed small polygons with maximal area for all even n ≥ 6

Improved the area bounds by O(1/n^5) over previous Mossinghoff polygons

Established the maximal area small 6-gon

Abstract

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value of a one-variable function. We show that, for all even $n\ge 6$, the area obtained improves by $O(1/n^5)$ that of the best prior small $n$-gon constructed by Mossinghoff. In particular, for $n=6$, the small $6$-gon constructed has maximal area.