Small polygons with large area

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...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Bingane, Christian, Mossinghoff, Michael J
Format: Article
Language:English
Subjects:
Asymptotic series
Lower bounds
Polygons
Online Access:Get full text
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Summary: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.
ISSN:2331-8422