Small polygons with large area

TL;DR

This paper improves the lower bounds on the maximum area of small polygons with an even number of sides greater than or equal to 14, refining asymptotic estimates and confirming optimality for certain small cases.

Contribution

It provides an improved lower bound for the maximal area of small polygons with even sides n ≥ 14, refining the asymptotic expansion's 1/n^3 term.

Findings

New lower bounds for n ≥ 14

Optimal polygons identified for n=6,8,10,12

Refinement of asymptotic expansion terms

Abstract

A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $O(1/n^3)$. For $n=6$, $8$, $10$, and $12$, the polygon we construct has maximal area.